MRA Based Wavelet Frames on Local Fields of ...

MRA Based Wavelet Frames on Local Fields of Positive Characteristic

A thesis submitted to the Central University of Jammu in partial fulfilment of the requirements for the award of the degree of

Doctor of Philosophy in Mathematics by

Mohd Younus Bhat under the supervision of

Dr. Pavinder Singh

Dr. Firdous A. Shah

(Supervisor)

(Co-Supervisor)

Department of Mathematics, Central University of Jammu, Jammu-180011, Jammu and Kashmir, India. May, 2016

i

Dedicated to my Parents

ii

Contents

1 Introduction

1

1.1

Multiresolution Analysis on R . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Preliminaries on LFPC . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

Wavelets on LFPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5

Organisation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Periodic Wavelet Frames on LFPC

27

2.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2

PWF via Unitary Extension Principle . . . . . . . . . . . . . . . . . . 29

2.3

Dual PWF via Mixed Extension Principle . . . . . . . . . . . . . . . 37

3 Minimum Energy Wavelet Frames on LFPC

47

3.1

Construction of MEWF on LFPC . . . . . . . . . . . . . . . . . . . . 48

3.2

Polyphase Characterization of Wavelets . . . . . . . . . . . . . . . . . 54

3.3

Decomposition and Reconstruction Algorithms of MEWF . . . . . . . 60 iii

Contents

4 Semi-orthogonal Wavelet Frames on LFPC

63

4.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2

Construction of SWF on LFPC . . . . . . . . . . . . . . . . . . . . . 65

5 Tight Framelet Packets on LFPC

73

5.1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2

TFP via Wavelet Spaces Wj,` . . . . . . . . . . . . . . . . . . . . . . 75

5.3

TFP for L2 (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4

TFP on LFPC via MRA Space VJ . . . . . . . . . . . . . . . . . . . . 83

6 Nonuniform Wavelets/Vector-valued Wavelets

87

6.1

Nonuniform Multiresolution Analysis on LFPC . . . . . . . . . . . . . 89

6.2

Nonuniform Wavelet Packets on LFPC . . . . . . . . . . . . . . . . . 92 6.2.1

Orthogonal Properties of Nonuniform Wavelet Packets . . . . 99

6.3

Vector-valued NUMRA on LFPC . . . . . . . . . . . . . . . . . . . . 104

6.4

Sufficient Condition for the Scaling Function

. . . . . . . . . . . . . 116

Publications

121

Bibliography

122

iv

Chapter 1 Introduction “A teacher can never truly teach unless he is still learning himself. A lamp can never light another lamp unless it continues to burn its own flame. The teacher who has come to the end of this subject, who has no living traffic with his knowledge but merely repeats his lessons to his students, can only load their minds; he can not quicken them.” Rabindranath Tagore

This chapter is structured in the following manner. In Section 1.1, we discuss in brief wavelets and multiresolution Analysis on the Euclidean field R. In Section 1.2, we provide preliminaries of local fields of positive characteristic(LFPC) and their Fourier analysis. In Section 1.3, we discuss the wavelet structure of local fields. In Section 1.4, we review the notable work that has been done so far in this field. In Section 1.5, we, discuss the strategy and organisation of rest of the thesis.

1.1

Multiresolution Analysis on R

Historically, the concept of “ondelettes” or “wavelets” started to appear more frequently only in the early 1980’s. This new concept can be viewed as a synthesis of 1

1.1 Multiresolution Analysis on R

various ideas originating from different disciplines including mathematics (CalderónZygmund operators and Littlewood-Paley theory), physics (the coherent states formalism in quantum mechanics and the renormalization group), and engineering (quadratic mirror filters, sideband coding in signal processing, and pyramidal algorithms in image processing). In 1982, Jean Morlet, a French geophysical engineer, discovered the idea of the wavelet transform, providing a new mathematical tool for seismic wave analysis. In Morlet’s analysis, signals consist of different features in time and frequency, but their high-frequency components would have a shorter time duration than their low-frequency components. In order to achieve good time resolution for the high-frequency transients and good frequency resolution for the low-frequency components, Morlet [148, 149] first introduced the idea of wavelets as a family of functions constructed from translations and dilations of a single function called the “mother wavelet” ψ(t). They are defined by 1 t−b ψa,b (t) = p ψ , a, b ∈ R, a |a|

a 6= 0,

(1.1.1)

where a is called a scaling parameter which measures the degree of compression or scale, and b a translation parameter which determines the time location of the wavelet. If |a| < 1, the wavelet (1.1.1) is the compressed version (smaller support in time-domain) of the mother wavelet and corresponds mainly to higher frequencies. On the other hand, when |a| > 1, ψa,b (t) has a larger time-width than ψ(t) and corresponds to lower frequencies. Thus, wavelets have time-widths adapted to their frequencies. This is the main reason for the success of the Morlet wavelets in signal processing and time-frequency signal analysis. It may be noted that the resolution of wavelets at different scales varies in the time and frequency domains as governed by the Heisenberg uncertainty principle. At large scale, the solution is coarse in the time domain and fine in the frequency domain. As the scale a decreases, the resolution in the time domain becomes finer while that in the frequency domain becomes coarser. Definition 1.1.1 (Wavelet). A wavelet is a function ψ ∈ L2 (R) which satisfies the

2


condition ∞

ˆ 2 ψ(ξ)

−∞

|ξ|

Z Cψ ≡

dξ < ∞,

(1.1.2)

ˆ where ψ(ξ) is the Fourier transform of ψ(t). If ψ ∈ L2 (R), then ψa,b (t) ∈ L2 (R) for all a, b. For Z Z ∞

2 −1 ∞ t − b 2 ψ(x) 2 dx = ψ 2 .

ψa,b (t) = a dt = a −∞ −∞ The Fourier transform of ψa,b (t) is given by Z − 1 ∞ −iξt 1 t−b 2 ˆ ˆ ψa,b (ξ) = a e ψ dt = a 2 e−ibξ ψ(aξ). a −∞

(1.1.3)

(1.1.4)

Definition 1.1.2 (Continuous Wavelet Transform). If ψ ∈ L2 (R), and ψa,b (t) is given by (1.1.1), then the integral transformation Wψ defined on L2 (R) by Z ∞

Wψ f (a, b) = f, ψa,b = (1.1.5) f (t) ψa,b (t) dt −∞

is called a continuous wavelet transform of f (t). First, the kernel ψa,b (t) in (1.1.3) plays the same role as the kernel exp(−iξt) in the Fourier transform. However, unlike the Fourier transformation, the continuous wavelet transform is not a single transform but any transform obtained in this way. Like the Fourier transformation, the continuous wavelet transformation is linear. Second, as a function of b for a fixed scaling parameter a, Wψ f (a, b) represents the detailed information contained in the signal f (t) at the scale a. In fact, this interpretation motivated Morlet et al. [148, 149] to introduce the translated and scaled versions of a single function for the analysis of seismic waves. Using the Parseval relation of the Fourier transform, it also follows from (1.1.5) that

1 Dˆ ˆ E Wψ f (a, b) = f, ψa,b = f , ψa,b Z ∞ p2π 1 ˆ ˆ = |a| f (ξ) ψ(aξ) eibξ dξ, 2π −∞ 3

by (1.1.4).


This means that n o Z F Wψ f (a, b) =

∞

p ˆ e−ibξ Wψ f (a, b) db = |a| fˆ(ξ) ψ(aξ).

(1.1.6)

−∞

The Haar wavelet [94] is one of the classic examples. It is defined by  1   1, 0 ≤ t <   2 1 ψ(t) = −1, ≤t 0 and ζ0 , ζ1 , ζ2 , . . . , ζr−1 be as above. We define a character χ on K as follows: ( exp(2πi/p), µ = 0 and j = 1, χ(ζµ p−j ) = 1, µ = 1, . . . , r − 1 or j 6= 1.

(1.2.4)

We also denote the test function space on K by Ω, that is, each function f in Ω is a finite linear combination of functions of the form 1k (x − h), h ∈ K, k ∈ Z, where 1k is the characteristic function of Bk . Then, it is clear that Ω is dense in Lp (K), 1 ≤ p < ∞, and each function in Ω is of compact support and so is its Fourier transform.

1.3

Wavelets on LFPC

In order to able to define the concepts of multiresolution analysis(MRA) and wavelets on local fields, we need analogous notions of translation and dilation. Since S −1 p D = K, we can regard p−1 as the dilation (note that |p−1 | = q) and since j∈Z

{u(n) : n ∈ N0 } is a complete list of distinct coset representatives of D in K, the set {u(n) : n ∈ N0 } can be treated as the translation set. Note that unlike the standard wavelet theory on the real line, the translation set is not a group. Definition 1.3.1. A function f on K will be called integral-periodic if f x + u(k) = f (x) for all k ∈ N0 . Definition 1.3.2. For j ∈ Z and y ∈ K, we define the dilation operator Dj and the translation operator Ty as follows: Dj f (x) = q j/2 f (p−j x) and Ty f (x) = f (x − y), f ∈ L2 (K). Definition 1.3.3. A finite set {ψ` : ` = 1, 2, . . . , L} ⊂ L2 (K) is called a set of basic wavelets of L2 (K) if the system j/2 q ψ` p−j x − u(k) : 1 ≤ ` ≤ L, k ∈ N0 , j ∈ Z forms an orthonormal basis for L2 (K). 13

1.3 Wavelets on LFPC

The most elegant method to construct wavelets is based on MRA which is a family of closed subspaces of a Hilbert space satisfying certain properties. A generalization of classical theory of MRA on local fields of positive characteristic was considered by Jiang et al.[108]. Analogous to the Euclidean case, following is the definition of uniform multiresolution analysis on the local field K of positive characteristic. Definition 1.3.4. Let K be a local field of positive characteristic p > 0 and p be a prime element of K. An MRA of L2 (K) is a sequence of closed subspaces {Vj : j ∈ Z} of L2 (K) satisfying the following properties: (a) Vj ⊂ Vj+1 for all j ∈ Z; (b)

S

Vj is dense in L2 (K);

j∈Z

(c)

T

Vj = {0};

j∈Z

(d) f (x) ∈ Vj if and only if f (p−1 x) ∈ Vj+1 for all j ∈ Z; (e) There is a function φ ∈ V0 , called the scaling function, such that φ x − u(k) : k ∈ N0 forms an orthonormal basis for V0 . Given an MRA {Vj : j ∈ Z}, we define another sequence {Wj : j ∈ Z} of closed subspaces of L2 (K) by Wj = Vj+1 Vj . These subspaces also satisfy f (x) ∈ Wj

if and only if f (p−1 x) ∈ Wj+1

for all j ∈ Z.

Moreover, they are mutually orthogonal, and we have the following orthogonal decompositions: ! L2 (K) =

M

Wj = V0

M M j≥0

j∈Z

14

Wj

.

(1.3.1)


Since φ ∈ V0 ⊂ V1 and φ p−1 x − u(k) : k ∈ N0 is a orthonormal basis of V1 , there exists {hk } ∈ l2 (N0 ) such that √ X φ(x) = q hk φ p−1 x − u(k) .

(1.3.2)

k∈N0

On taking the Fourier transform of (1.3.2), we have ˆ ˆ φ(x) = m0 (pξ) φ(pξ),

(1.3.3)

1 X m0 (ξ) = √ hk χk (ξ) q k∈N

(1.3.4)

where 0

is an integral periodic function in L2 (D) and is often called the refinement symbol of ˆ = 1. Therefore by letting ξ = 0 in (1.3.3) and (1.3.4), φ. Observe that χk (0) = φ(0) P we obtain hk = 1. Further, it is proved in [13, 107] that a function φ ∈ L2 (K) k∈N0

generates an MRA in L2 (K) if and only if ˆ = 1, ˆ = lim φ(ξ) φ(0)

ξ ∈ K,

(1.3.5)

X 2 φˆ ξ + u(k) ∈ L∞ (D).

(1.3.6)

ξ→0

and k∈N0

As in case of Rn there exists q − 1 functions {ψ1 , ψ2 , . . . , ψq−1 } in L2 (K) such that their translates and dilations form an orthonormal bases of Wj , i.e., for all j ∈ Z, Wj = span q j/2 ψ` p−j x − u(k) : k ∈ N0 , 1 ≤ ` ≤ q − 1 .

(1.3.7)

Since ψ` ∈ W0 ⊂ V1 , 1 ≤ ` ≤ q − 1, there exist a sequence {h`k } ∈ l2 (N0 ) such that ψ` (x) =

√ X ` q hk φ p−1 x − u(k) ,

1 ≤ ` ≤ q − 1.

(1.3.8)

k∈N0

Equation (1.3.8) can be written in the frequency domain as 1 X ` ˆ ψˆ` (ξ) = √ h χk (pξ)φ(pξ) q k∈N k 0

ˆ = m` (pξ) φ(pξ), 15

(1.3.9)


where 1 X ` m` (ξ) = √ h χk (ξ), 1 ≤ ` ≤ q − 1 q k∈N k

(1.3.10)

0

are the integral periodic function in L2 (D) and are called the framelet symbols or wavelet masks. With framelet symbols m0 (ξ), m1 (ξ), . . . , mL (ξ), we formulate the q × q matrix M(ξ) as:     M(ξ) =   

m0 ξ + pu(0) m0 ξ + pu(1) m1 ξ + pu(0) m1 ξ + pu(1) .. .. . . mq−1 ξ + pu(0) mq−1 ξ + pu(1)

 . . . m0 ξ + pu(q − 1)  . . . m1 ξ + pu(q − 1)   . .. ..  . .  . . . mq−1 ξ + pu(q − 1)

The matrix M(ξ) is known as the modulation matrix. Let Z = {u(n) : n ∈ N0 }, where {u(n) : n ∈ N0 } is a complete list of (distinct) coset representation of D in K + . Then n o X z u(n) 2 < ∞ l2 (Z) = z : Z → C : n∈N0

is a Hilbert space with the inner product X hz, wi = z u(n) w u(n) . n∈N0

Moreover, the Fourier transform on l2 (Z) is a map ∧ : l2 (Z) → L2 (D) defined by X zˆ(ξ) = z u(n) χu(n) (ξ), z u(n) ∈ Z n∈N0

and its inverse is Z

z u(n) = hf, χu(n) i =

f (x)χu(n) (x) dx,

f ∈ L2 (D).

D

For all z, w ∈ Z, we have Parseval’s relation: Z X hz, wi = z u(n) w u(n) = zˆ(ξ)w(ξ) ˆ dξ = hˆ z , wi, ˆ n∈N0

D

16

1.4 Background

and Plancherel’s relation: Z X

2

z = z u(n) 2 = zˆ(ξ) 2 dξ = zˆ 2 . n∈N0

D

For given Ψ := ψ1 , . . . , ψL ⊂ L2 (K), define the wavelet system ` FΨ = ψj,k : 1 ≤ ` ≤ L, j ∈ Z, k ∈ N0 , (1.3.11) ` (x) = q j/2 ψ` p−j x − u(k) . The wavelet system FΨ is called a wavelet where ψj,k frame if there exist positive numbers 0 < A ≤ B < ∞ such that L X X

2 X

2 `

f, ψj,k ≤ B f 2 , A f 2≤ 2

for all f ∈ L2 (K).

(1.3.12)

`=1 j∈Z k∈N0

The largest constant A and the smallest constant B satisfying (1.3.12) are called the lower and upper frame bound, respectively. A wavelet frame is a tight wavelet frame if A and B are chosen so that A = B and then the set Ψ := {ψ1 , ψ2 , . . . , ψL } is called frame wavelet for the corresponding tight wavelet frame. Also in this case, the generators ψ1 , ψ2 , . . . , ψL are referred as tight framelets. If a frame ceases to be a frame when any element is removed, then the frame is said to be exact. If only the right-hand inequality in (1.3.12) holds, then FΨ is called a Bessel sequence. Furthermore, the wavelet frame is called a Parseval or normalized tight wavelet frame if A = B = 1, i.e., L X X X

2 2 ` f, ψj,k = f , 2


(1.3.13)

`=1 j∈Z k∈N0

The collection FΨ defined by (1.3.11) is said to be a semi-orthogonal wavelet frame if

1.4

` ψj,k , ψj`0 ,k0 = 0,

whenever j 6= j 0 ∈ Z, k, k 0 ∈ N0 , 1 ≤ ` ≤ L.

Background

The theory of wavelets is a comparatively new development in the field of mathematics. Wavelet is interdisciplinary in origin. During last 30 years voluminous 17

1.4 Background

literature has accumulated regarding application of wavelet to data analysis, image compression and enhancement, computer vision, sub band coding, filtering of signals and many other fields. Wavelets are not based on any “bright new idea” but on a concept that already existed under various forms in many different fields. Formalization of “Wavelet theory” is the result of collective effort of mathematicians, physicists, engineers etc. Wavelet methods are refinement of Fourier analysis, where mathematical foundation has been provided by Grossman and Morlet. In 1910, A. Haar gave the first example of an orthonormal wavelet on R but because of the poor frequency localization of the resulting orthonormal basis, they are not of much use in practice. In 1981, while trying to further understand the Hardy spaces, Stromberg [185] obtained a wavelet of L2 (R) by modifying a basis constructed earlier by Franklin in 1927. We refer to [191] for a detailed discussion of the Stromberg wavelet. In the early eighties, Morlet introduced the continuous wavelet transform. Grossman obtained an inversion formula for this transform and along with Morlet explored several applications. Meyer [144] constructed an example of an infinitely differentiable wavelet such that its Fourier transform also had this property. This construction was generalized to higher dimensions by Lemarie and Meyer [127]. The concept of multiresolution analysis (MRA) was developed by Meyer and Mallat [140, 147]. Daubechies used this concept to construct compactly supported wavelets with arbitrarily high, but fixed, regularity. The wavelets have poor frequency localization. To overcome this disadvantage, Coifman, Meyer and Wickerhauser [50] constructed wavelet packets from a wavelet associated with an MRA. Cohen, Daubechies and Feauveau in [47] introduced the concept of biorthogonal wavelets. Wavelets and multiresolution analyses were also studied extensively in the higher dimensional cases Rn , see [35, 56, 101, 138, 191] and references therein. The concept of wavelet has been extended to many different setups by several authors. Dahlke [53] introduced it on locally compact abelian groups (see also [72, 102]). It was generalized to abstract Hilbert spaces by Han, Larson, Papadakis and Stavropoulos [97, 183]. Lemarie [125] extended this concept to stratified Lie 18

1.4 Background

groups. Recently, R. L. Benedetto and J. J. Benedetto [19] developed a wavelet theory for local fields and related groups. In [20], R. L. Benedetto proved that Haar and Shannon wavelets exist and, in fact, both are the same for such a group. The concept of frames in a general Hilbert space was orgionally introduced by Duffin and Schaeffer [67] in the context of non-harmonic Fourier series. An important example about frame is wavelet frame, which is obtained by translating and dilating a finite family of functions. A wavelet frame is a generalization of an orthonormal wavelet basis by introducing redundancy into a wavelet system. By sacrificing orthonormality and allowing redundancy, the tight wavelet frames become much easier to construct than the orthonormal wavelets. Tight wavelet frames provide representations of signals and images in applications, where redundancy of the representation is preferred and the perfect reconstruction property of the associated filter bank algorithm, as in the case of orthonormal wavelets, is kept. The main tools for construction and characterization of wavelet frames are the several extension principles, the unitary extension principle (UEP) and oblique extension principle (OEP) as well as their generalized versions, the mixed unitary extension principle (MUEP) and the mixed oblique extension principle (MOEP). They give sufficient conditions for constructing tight and dual wavelet frames for any given refinable function which generates an MRA. These essential methods were firstly introduced by Ron and Shen in [158] and in the fundamental work of Daubechies et al.[59] for scalar refinable functions. The resulting tight wavelet frames are based on an MRA, and the generators are often called framelets. To mention only a few references on tight wavelet frames, the reader is referred to [32, 65, 66, 96, 98, 119, 120, 166]. These methods of construction of wavelet frames are generalized from onedimension to higher-dimension, tight frames to dual frames, from single scaling function to a scaling function vector. More importantly, the setup of tight wavelet frames provides great flexibility in approximating and representing periodic functions. Using periodization techniques, Zhang [197] constructed a dual pair of periodic wavelet frames for L2 [0, 1] under the assumption that the support of the wavelet

19

1.4 Background

function ψ in the frequency domain is contained in [−π, −ε] ∪ [ε, π], ε > 0. In 2009, Zhang and Saito [198] constructed general periodic wavelet frames using extension principles. More precisely, they proved that under some decay condition, the periodization of any wavelet frame constructed by the unitary extension principle is a periodic wavelet frame, and the periodization of any pair of dual wavelet frames constructed by the mixed extension principle is a pair of dual periodic wavelet frames. More results in this direction can be found in [77, 86, 87, 128] and in the references therein. In order to decrease the computational complexity and numerical instability of the wavelet frames during the course of decomposition and reconstruction of functions, Chui and He [40] proposed the concept of minimum-energy wavelets frames in L2 (R). Later on, Huang and Cheng [103] studied the construction and characterization of the minimum-energy wavelet frames with arbitrary integer dilation factor. These studies were continued by Huang and his colleagues in [104, 200], where they have studied the problem of constructing minimum-energy frames for multiwavelet and bivariate wavelet systems with refinable functions associated with arbitrary dilation matrix. On the other hand, Gao and Cao [85] have constructed dyadic minimum-energy wavelet frames on the interval [0, 1] and have established a necessary condition for a finite number of functions {ψ1 , ψ2 , . . . , ψL } to be a minimumenergy wavelet frame for L2 [0, 1]. On the other hand, the most efficient way to construct orthonormal wavelet is to construct it from an orthonormal MRA. Since the use of an MRA has proven to be a very efficient tool in wavelet theory mainly because of its simplicity, it is of interest to try to generalize this notion as much as possible while preserving its connection with wavelet analysis. In this connection, Benedetto and Li [20] considered the dyadic semi-orthogonal frame multiresolution analysis of L2 (R) with a single scaling function and successfully applied the theory in the analysis of narrow band signals. The characterization of the dyadic semi-orthogonal frame multiresolution analysis with a single scaling function admitting a single frame wavelet whose dyadic dilations of the integer translates form a frame for L2 (R) was obtained independently by

20

1.4 Background

Benedetto and Treiber [21] by a direct method, and by Kim et al. [117] using the theory of shift-invariant spaces. Later on, Xiaojiang [193] extended the results of Benedetto and Li’s theory of frame multiresolution analysis to higher dimensions with arbitrary expansive matrix dilations, and has established the necessary and sufficient conditions to characterize semi-orthogonal multiresolution analysis frames for L2 (Rn ). Ehler and Han [68] constructed biframes from multivariate refinable functions. Later on Ehler [69] provided multiresolution structure of pair of dual wavelet frames for a pair of Sobolve spaces. The traditional wavelet frames [32] provide poor frequency localization in many applications as they are not suitable for signals whose domain frequency channels are focused only on the middle frequency region. Therefore, in order to make more kinds of signals suited for analyzing by wavelet frames, it is necessary to extend the concept of wavelet frames to a library of wavelet frames, called framelet packets or wavelet frame packets. The original idea of framelet packets was introduced by Coifman et al.[51] to provide more efficient decomposition of signals containing both transient and stationary components. Well known Daubechies orthogonal wavelets are a special of wavelet packets. Chui and Li [38] generalized the concept of orthogonal wavelet packets to the case of non-orthogonal wavelet packets so that they can be employed to the spline wavelets and so on. Shen [182] generalized the notion of univariate orthogonal wavelet packets to the case of multivariate wavelet packets. Other notable generalizations are the wavelet packets and framelet packets on a positive half-line R+ [159, 160, 168, 169], orthogonal wavelet packets [46, 48, 50, 55], the vector-valued wavelet packets [33, 178], biorthogonal wavelet packets [47, 175], nonuniform wavelet packets [11, 174] and the tight framelet packets on Rd [135]. The past decade has witnessed a tremendous interest in the problem of constructing wavelet bases on various spaces other than R [2, 7, 18, 23, 24, 25, 27, 28, 36, 56, 57, 58, 61, 62, 64, 65, 101, 150, 189, 191, 196] namely, positive halfline R+ [78, 79, 75, 88, 161, 163, 167] Abstract Hilbert spaces [177, 183], Cantor dyadic groups [121, 122], locally compact Abelian groups [53, 72], p-adic fields [3, 4, 5, 6, 42, 71, 111, 112, 113, 114, 115], zero-dimensional Abelian groups [102, 136],

21

1.4 Background

Heisenberg group [195] and Vilenkin groups [73, 74, 76, 77, 137]. Recently, R. L. Benedetto and J. J. Benedetto [19] developed a wavelet theory for local fields and related groups. They did not develop the MRA approach, their method is based on the theory of wavelet sets and only allows the construction of wavelet functions whose Fourier transforms are characteristic functions of some sets. The concept of MRA on a local field K of positive characteristic was introduced by Jiang et al. [108]. They pointed out a method for constructing orthogonal wavelets on local field K with a constant generating sequence. Subsequently, the tight wavelet frames on local fields were constructed by Li and Jiang [129]. They have established necessary condition and sufficient conditions for tight wavelet frame on local fields of positive characteristics in frequency domain. As far as the characterization of wavelets on local fields is concerned, Behera and Jahan [13] have given the characterization of all wavelets associated with MRA on local field K based on results on affine and quasi-affine frames. Recently, Shah [162] introduced the notion of frame multiresolution analysis (FMRA) on local fields of positive characteristic and established a complete characterization of wavelet frames based on the theory of shift invariant spaces. More results in this direction can also be found in [1, 12, 13, 14, 15, 16, 17, 41, 164, 165, 166, 187] and the references therein. The construction of tight wavelet frames on local fields of positive characteristic using extension principles were first reported by Shah and Debnath in [170]. They provide a sufficient condition for finite number of functions {ψ1 , ψ2 , . . . , ψL } to form a tight wavelet frame for L2 (K). Moreover, they also established a complete characterization of wavelet frames on local fields of positive characteristic by virtue q−1 of the modulation matrix M(ξ) = m` pξ + pu(k) `,k=0 formed by the framelet symbols m` (ξ), ` = 0, 1, . . . , L. In recent years, local fields have attracted the attention of several mathematicians, and have found innumerable applications not only in number theory but also in representation theory, division algebras, quadratic forms, and algebraic geometry. As a result, local fields are now consolidated as part of the standard repertoire of contemporary mathematics. It is well known that under some decay conditions,

22

1.4 Background

one uses the method of periodization to construct periodic wavelet bases with the help of wavelet bases. Under some decay condition, the periodization of any wavelet frame constructed by the unitary extension principle (UEP) is a periodic wavelet frame, and the periodization of any pair of dual wavelet frames constructed by the mixed extension principle is a pair of dual periodic wavelet frames. We extended this concept to the local fields of positive characteristic. The minimum-energy wavelet frames reduce the computational, and maintain the numerical stability, and do not need to search dual frames in the decomposition and reconstruction of functions (or signals). Therefore, many people paid more attention to the study of minimumenergy wavelet frames. We, therefore constructed such frames on local fields of positive characteristic. The class of all semi-orthogonal Parseval frame wavelets contains all orthonormal wavelets. Both orthonormal and Parseval frame wavelets are studied extensively over the last two decades even for more general dilations. We therefore, focused our attention to the construction of semi-orthogonal wavelet frame them on local fields of positive characteristic. People are now-a-days studying the concept of a multiresolution analysis with the associated translation set a discrete set which may not be a group. Such constructions are called nonuniform MRA (NUMRA). Characterization of nonuniform wavelets associated with a NUMRA have been obtained. It is well-known that the classical orthonormal wavelet bases have poor frequency localization. Univariate orthogonal wavelet packets have been constructed as an alternative. The fundamental idea of wavelet packet analysis is to construct a library of orthonormal bases, which can be searched in real time for the best expansion with respect to a given application. The standard construction is to start from a multiresolution analysis (MRA) and generate the library using the associated quadrature mirror filters (QMFs). The internal structure of the MRA and the speed of the decomposition schemes make this an efficient adaptive method for simultaneous time and frequency analysis of signals. We therefore constructed nonuniform wavelet packets on local fields of positive characteristic and obtained their properties by virtue of the classical Fourier transform. On the other hand, vector-valued wavelets are a class of generalized multiwavelets and multiwavelets can be generated from the component function in vector-valued wavelets. Vector-

23

1.5 Organisation of the Thesis

valued wavelets and multiwavelets are different in the following sense. Vector-valued wavelets can be used to decorrelate a vector-valued signal not only in the time domain but also between components for a fixed time where as multiwavelets focuses only on the decorrelation of signals in time domain. Moreover, prefiltering is usually required for discrete multiwavelet transform but not necessary for discrete vectorvalued wavelet transforms. We, therefore developed the concept of vector-valued nonuniform multiresolution analysis on local fields of positive characteristic and obtained the characterization of associated wavelets.

1.5

Organisation of the Thesis

The aim of this thesis is to develop a wavelet theory on local fields of positive characteristic. The thesis is mainly divided into two sections. In the first section, which is the main part of the thesis, we construct wavelet frames via MRA approach on local field. In the second section, we develop nonuniform and vector-valued multiresolution analyses besides obtaining necessary and sufficient conditions for such wavelets on local fields of positive characteristic. The thesis is mainly based on the problems that have been discussed in the articles [171, 172, 174, 173, 179, 180, 181]. In Chapter 2, we introduce the concept of periodic wavelet frame on local fields of positive characteristic and show that under weaker conditions, the periodization of any wavelet frame constructed by the unitary extension principle with dilation factor p−1 is a periodic wavelet frame on local fields of positive characteristic. Moreover, based on the mixed extension principle and Fourier-based techniques of the wavelet frames, we present an explicit construction method for a pair of dual periodic wavelet frames on local fields of positive characteristic. In Chapter 3, we introduce the notion of minimum-energy wavelet frame on local fields of positive characteristic and present its equivalent characterizations in terms of their framelet symbols. Furthermore, based on polyphase representation

24


of the framelet symbols, we obtain the necessary and sufficient condition for the existence of minimum-energy wavelet frames on local fields of positive characteristic. We continue the study based on the extension principles and give a polyphase matrix characterization of tight wavelet frames on local fields of positive characteristic. Moreover, we derive the minimum-energy wavelet frame decomposition and reconstruction formulae which are quite similar to those of orthonormal wavelets on local fields of positive characteristic. In Chapter 4, we investigate semi-orthogonal wavelet frames on local fields of positive characteristic and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. Here, we have used the theory of frame multiresolution analysis on local fields to establish equivalent conditions for a finite number of functions {ψ1 , ψ2 , . . . , ψL } in L2 (K) to generate a semi-orthogonal wavelet frame for local fields of positive characteristic. In Chapter 5, we continue the study based on the extension principles and give an explicit construction of a class of tight framelet packets on local fields of positive characteristic. In Chapter 6, we extended our study of research to the nonstandard setting and introduce the concepts of nonuniform and vector-valued nonuniform multiresolution analyses. Inspired by the concept of nonuniform multiresolution analysis on local fields of positive characteristic, we construct the associated orthogonal wavelet packets for such an MRA. More precisely, we show that the collection of all dilations and translations of the wavelet packets is an overcomplete system in L2 (K). Finally, we investigate certain properties of the nonuniform wavelet packets on local fields by virtue of the Fourier transform. In the vector-valued nonuniform setting, we establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of vector-valued nonuniform multiresolution analysis on local fields of positive characteristic starting from a vector refinement mask with appropriate conditions. Here, we generalize the concept of vector-valued multiresolution analysis

25


on Euclidean spaces Rn to vector-valued nonuniform multiresolution analysis on local fields of positive characteristic, in which the translation set acting on the scaling vector associated with the multiresolution analysis to generate the subspace V0 is no longer a group, but is the union of Z and a translate of Z, where Z = {u(n) : n ∈ N0 } is a complete list of (distinct) coset representation of D in K + . We call this a vector-valued nonuniform multiresolution analysis on local fields of positive characteristic. As a consequence of this generalization, we obtain necessary and sufficient condition for such wavelets. In the end, a comprehensive bibliography is included.

26

Chapter 2 Periodic Wavelet Frames on LFPC The advantages of wavelet frames and their promising features in various applications have attracted a lot of interest in recent years. In this chapter, our aim is to extend the notion of wavelet frames to the periodic wavelet frames(PWF) on local fields via extension principles. More precisely, we introduce the concept of periodic wavelet frame on local field K of positive characteristic and prove that under weaker conditions, the periodization of any wavelet frame constructed by the unitary extension principle with dilation p−1 is a periodic wavelet frame on local fields of positive characteristic. Furthermore, based on the mixed extension principle and Fourierbased techniques of the wavelet frames, an explicit constructed method for a pair of dual periodic wavelet frames on local fields of positive characteristic is also given. This chapter is organized as follows. In Section 2.1, we list some basic notations and results required in the subsequent sections. In Section 2.2 and Section 2.3, we construct periodic wavelet frames and dual periodic wavelet frames using unitary extension principle and mixed extension principle, respectively on local fields of positive characteristic.

27

2.1 Preliminaries

2.1

Preliminaries

For j ∈ N0 , let Nj denote a full collection of coset representatives of N0 /q j N0 , i.e., Nj = 0, 1, 2, . . . , q j − 1 ,

j ≥ 0.

Then, N0 =

[

n + q j N0 ,

n∈Nj

and for any distinct n1 , n2 ∈ Nj , we have n1 + q j N0 ∩ n2 + q j N0 = ∅. Thus, every non-negative integer k can uniquely be written as k = rq j + s, where r ∈ N0 , s ∈ Nj . Moreover, for a function f defined on K, we say that a bounded function W : K + → K + is a radial decreasing L1 -majorant of f if |f (x)| ≤ W (x), W ∈ L1 (K + ), and W (0) < ∞. Recently, Shah and Debnath [170] gave an explicit construction scheme for tight wavelet frames on local fields of positive characteristic using unitary extension principles. The following is the fundamental tool they gave to construct tight wavelet frames on local fields. Theorem 2.1.1 (Unitary Extension Principle). Suppose that the refinable function φ and the framelet symbols m0 , m1 , . . . , mL satisfy equations (1.3.4)-(1.3.10). If the modulation matrix M(ξ) satisfies M(ξ)M∗ (ξ) = Iq , a.e. ξ ∈ D. Then, the wavelet ` L system ψj,k given by equation (1.3.11) is a normalized tight wavelet `=1;j∈Z,k∈N 0

frame for L2 (K). The result corresponding to the mixed extension principle is as follows. Theorem 2.1.2 (Mixed Extension Principle). Suppose φ1 and φ2 ∈ L2 (K) are refinable functions that satisfy the conditions of the UEP. Let the refinement masks 28

2.2 PWF via Unitary Extension Principle

be denoted by m0 and m ˜ 0 . Let the corresponding wavelet masks be denoted by m` , m ˜ ` , ` = 1, 2, . . . , L. Define ψ` and ψ˜` , respectively, as ˆ˜ ψˆ˜` (ξ) = m ˜ ` (pξ)φ(pξ).

ˆ ψˆ` (ξ) = m` (pξ)φ(pξ),

(2.1.1)

` L ` L If both ψj,k and ψ˜j,k `=1;j∈Z,k∈N0 are Bessel sequence, and the matrices `=1;j∈Z,k∈N 0

L;q−1 M(ξ) = m` pξ + pu(k) `=0;k=0

and

L;q−1 ˜ M(ξ) = m ˜ ` pξ + pu(k) `=0;k=0

` L ` ˜ satisfy M∗ (ξ)M(ξ) = Iq , a.e. ξ ∈ D. Then, ψj,k , ψ˜j,k is a pair of dual `=1;j∈Z,k∈N0 wavelet frames for L2 (K).

2.2

PWF via Unitary Extension Principle

For any f ∈ L1 (K), we define the periodic version of f as X f per = f (x + u(k)). k∈N0

Then, it is easy to verify that f per is well defined and it is N0 -periodic local integrable function. With the same dilation and translation operators as defined in Definition 1.3.2, we define the periodic wavelet system as per FΨper := φper , ψ`,j,k : 1 ≤ ` ≤ L, j ∈ N0 , k ∈ Nj .

(2.2.1)

First, we present an approach for constructing periodic wavelet frames on local fields of positive characteristic with the help of the unitary extension principle. Theorem 2.2.1. Let m0 (ξ) be the refinement mask of a refinable function φ of an MRA and let m` (ξ), ` = 1, 2, . . . , L be the wavelet masks associated with the basic wavelets given by the equation (1.3.9). Furthermore, let the wavelet system FΨ given by equation (1.3.11) be a normalized tight frame generated by the UEP associated with φ. If {φ, ψ1 , ψ2 , . . . , ψL } ⊂ L2 (K) and φ, ψ1 , ψ2 , . . . , ψL have common radial decreasing L1 -majorant, then the periodic wavelet system FΨper given by equation (2.2.1) generates a normalized tight frame for L2 (D). 29


We split the proof of Theorem 2.2.1 into several lemmas. Lemma 2.2.2. Suppose that the periodic wavelet system FΨper is as in Theorem 2.2.1. Then, for any function f ∈ Ω and given ε > 0, there exists a positive integer J ∈ N such that X

2

f, φper 2 ≤ (1 + ε) f 2 , (1 − ε) f 2 ≤ j,k 2

for all j ≥ J.

(2.2.2)

k∈Nj

Proof. Let E denotes the support of the Fourier coefficients fˆ u(r) : r ∈ N0 of f . Note that E ⊂ N0 is a finite set since f lies in Ω. Therefore, we have f (x) =

X

fˆ u(r) χu(r) (x).

r∈E

Suppose φper j,k (x) =

X

φˆper j,k u(r) χu(r) (x),

(2.2.3)

r∈N0

where the Fourier coefficients of the above series are given by j/2 ˆ j j u(r) . φˆper φ p u(r) u(r) = q χ p k j,k

(2.2.4)

Applying Parseval’s formula to the above Fourier series, we obtain X

X X 2 ˆ u(r) φˆper u(r) f, φper 2 = f j,k j,k k∈Nj

k∈Nj

r∈E

X X j/2 2 j j ˆ ˆ = f u(r) q φ p u(r) χk p u(r) k∈Nj

r∈E

X X 2 j ˆ ˆ = dr f , φ χk p u(r) , k∈Nj

r∈E

where dr fˆ, φˆ = q j/2 fˆ u(r) φˆ pj u(r) . As E is a finite set, there exists a positive number N such that E ⊆ D(N ) = k ∈ N0 : |k| ≤ N . Hence, there exists J1 ≥ 0 such that for all j ≥ J1 , the elements of D(N ) lie in different cosets of N0 /q j N0 (see [164]). Thus, the cardinality of E ∩ (k + q j N0 ) is at most 1 for each j ≥ J1 , k ∈ Nj . Consequently, we have

30


X

XX X j ˆ ˆ ˆ ˆ f, φper 2 = d f , φ χ p u(r) d f , φ χk pj u(s) r k s j,k k∈Nj r∈E

k∈Nj

=

XX

s∈E

dr

X ˆ ˆ ˆ ˆ f , φ dr f , φ χk pj u(r − s)

r∈E s∈E

k∈Nj

2 X = qj ds fˆ, φˆ s∈E

X 2 j/2 ˆ j ˆ = q f u(s) φ p u(s) . s∈E

ˆ = 1, therefore there exist a non-negative integer J2 such that ˆ = lim φ(ξ) Since φ(0) ξ→0

2 (1 − ε) ≤ φˆ pj u(s) ≤ (1 + ε),

for all j ≥ J2 .

Assume J = max{J1 , J2 }, then with this choice of j ≥ J, we get X X

X 2 2 ˆ u(s) . f, φper 2 < (1 + ε) f (1 − ε) fˆ u(s) < j,k s∈E

s∈E

k∈Nj

Using equation (1.2.2), we obtain X

2

f, φper 2 ≤ (1 − ε) f 2 . (1 − ε) f 2 ≤ j,k 2 k∈Nj

This completes the proof of Lemma 2.2.2. Lemma 2.2.3. Let m0 (ξ) be the refinement mask of a refinable function φ of an MRA and let m` (ξ), ` = 1, 2, . . . , L be the wavelet masks. Furthermore, let the wavelet system FΨ given by equation (1.3.11) be a normalized tight frame generated by the UEP associated with φ and m1 , m2 , . . . , mL . Then for any f ∈ L2 (K), we have

L X X

X

X

2 ` f, φj+1,k 2 = f, φj,k 2 + f, ψj,k . k∈N0

k∈N0

(2.2.5)

`=1 k∈N0

Proof. For any f ∈ L2 (K) and j ∈ N0 , define the linear operators Pj and Qj as X

Pj f (x) = f, φj,k φj,k (x),

Qj f (x) =

k∈N0

L X X

`=1 k∈N0

31

` ` f, ψj,k ψj,k (x).


Since Ω is dense in L2 (K) and closed under Fourier transform, the set n o Ω0 = f ∈ Ω : fˆ is continuous and compactly supported is also dense in L2 (K). Therefore, it is necessary to prove that

Pj f, f + Qj f, f = Pj+1 f, f ,

(2.2.6)

holds for all functions f ∈ Ω0 . Therefor for all f ∈ Ω0 and j ∈ Z, k ∈ N0 , we obtain the following equality by using Z

j Pj f, f = q

=

the Parseval’s formula 2 X −j ˆ ˆ f p ξ + u(r) φ ξ + u(r) dξ D r∈N0 2 Z X −j j ξ + u(r) dξ. ˆ ˆ φ p f ξ + p u(r)

p−j D

(2.2.7)

r∈N0

Using equation (1.3.3) and the fact that m0 (ξ) is an integral-periodic function to equality (2.2.7), we get 2 Z X

−j j+1 ξ + pu(r) m pj+1 ξ + pu(r) dξ ˆ ˆ Pj f, f = f ξ + p u(r) φ p 0 r∈N0 −j p D Z X X fˆ ξ + p−j p−1 u(r) + u(s) = p−j D

r∈N0 s∈N1

2 j+1 −1 j+1 −1 ˆ ×φ p ξ + p p u(r) + u(s) m0 p ξ + p p u(r) + u(s) dξ Z X X = fˆ ξ + p−j p−1 u(r) + u(s) p−j D

=

r∈N0 s∈N1

2 j+1 −1 j+1 ˆ ×φ p ξ + p p u(r) + u(s) m0 p ξ + pu(s) dξ 2 Z X j Rf,φ u(s), ξ m0 pj+1 ξ + pu(s) dξ,

p−j D

s∈N1

where X j Rf,φ u(s), ξ = fˆ ξ + p−j p−1 u(r) + u(s) φˆ pj+1 ξ + p p−1 u(r) + u(s) . r∈N0

32


Proceeding on the similar lines as above, we can have 2 L Z X

X j Qj f, f = Rf,φ u(s), ξ m` pj+1 ξ + pu(s) dξ. `=1

s∈N1

p−j D

Thus, we have

Pj f, f + Qj f, f =

Z (X

) u(s), ξ m0 pj+1 ξ + pu(s)

j Rf,φ

s∈N1

p−j D

( X

×

j Rf,φ

) u(s0 ), ξ m0 pj+1 ξ + pu(s0 ) dξ

s0 ∈N1

+

L X `=1

Z (X

) j Rf,φ

s∈N1

p−j D

( X

×

u(s), ξ m` pj+1 ξ + pu(s)

j Rf,φ

) u(s0 ), ξ m` pj+1 ξ + pu(s0 ) dξ

s0 ∈N1

=

Z (X X

j Rf,φ

) j u(s), ξ Rf,φ u(s0 ), ξ

s∈N1 s0 ∈N1

p−j D

×

( L X

) m` pj+1 ξ + pu(s0 ) m` pj+1 ξ + pu(s) dξ.

`=0

Since the unitary extension principle condition is equivalent to L X

m` pj+1 ξ + pu(s0 ) m` pj+1 ξ + pu(s) = δs,s0 .

`=0

Therefore, we have

Pj f, f + Qj f, f =

Z

2 u(s), ξ dξ

X j R

f,φ

p−j D

Z = p−j D

s∈N1

X X fˆ ξ + p−j p−1 u(r) + u(s)

s∈N1 r∈N0

2 j+1 −1 ˆ × φ p ξ + p p u(r) + u(s) dξ 33


=

X

Z

s∈N1 −j p D+p−j u(s)

X fˆ ξ + p−j−1 u(r) r∈N0

2 × φˆ pj+1 ξ + u(r) dξ 2 Z X fˆ ξ + p−j−1 u(r) φˆ pj+1 ξ + u(r) dξ = r∈N0 p−j−1 D

= Pj+1 f, f , and hence, we get the desired result. Lemma 2.2.4. Let φ ∈ L2 (K) be a refinable function with refinement mask m0 (ξ), and let m` (ξ), ` = 1, . . . , L be the wavelet masks. Furthermore, let the wavelet system FΨ given by equation (1.3.11) be a normalized tight frame generated by the UEP associated with φ. If {φ, ψ1 , ψ2 , . . . , ψL } ⊂ L2 (K) and φ, ψ1 , ψ2 , . . . , ψL have common radial decreasing L1 -majorant, then we have L X X X

X

f, ψ per 2 . f, φper 2 + f, φper 2 = `,j,k j,k j+1,k

(2.2.8)

`=1 k∈Nj

k∈Nj

k∈Nj

Proof. For any f ∈ Ω and j ∈ N0 , we have 2 2 X X D X D X X

E E 2 per f, φ = f, φj,k x + u(r) . φj,k x + u(r) = f, j,k k∈Nj

k∈Nj

k∈Nj r∈N0

r∈N0

The change of the summation and the integration in the above is reasonable. In fact, we have Z X Z f (x)φj,k x + u(r) dx ≤ kf kL∞ (D) φj,k (x) dx r∈N0 D

K

= f L∞ (D) q j/2

Z

φ(x) dx

K

< ∞. We can also deduce that the series X X XD E ED f, φj,k x + u(r) f, φj,k x + u(s) k∈Nj r∈N0 s∈N0

34


is absolutely convergent. Therefore, the series can be rearranges as follows: X X XD X

ED E f, φper 2 = f, φ x + u(r) f, φ x + u(s) j,k j,k j,k k∈Nj r∈N0 s∈N0

k∈Nj

X X XD

=

f, φj,k

E ED f, φj,k x + u(r) + u(s) . x + u(r)

k∈Nj r∈N0 s∈N0

For s ∈ N0 , we define Fs (x) = f (x)1D+u(s) (x), where 1D+u(s) (x) is the characteristic function of the set D + u(s). Using the fact that φj,k x + u(s) = φj,k−p−j u(s) (x), we have X

f, φper 2 j,k k∈Nj

=

 X X X Z

f (x)φj,k x + u(r) dx

  Z

f (x)φj,k x + u(r) + u(s) dx

 D    Z

 D  X X X Z

k∈Nj r∈N0 s∈N0



   f (x)φj,k x + u(r) dx f (x)φj,k x + u(r) dx =      k∈Nj r∈N0 s∈N0 D D+u(s)    Z X X X Z   = F0 (x)φj,k x + u(r) dx Fs (x)φj,k x + u(r) dx    k∈Nj r∈N0 s∈N0 K X X X K

= F0 , φj,k−p−j u(r) Fs , φj,k−p−j u(r) k∈Nj r∈N0 s∈N0

=

XX

hF0 , φj,k i hFs , φj,k i.

k∈N0 s∈N0

Similarly, for each ` = 1, 2, . . . , L, we have X

X X

` ` f, ψ per 2 = F0 , ψj,k Fs , ψj,k . `,j,k k∈N0 s∈N0

k∈Nj

Application of Lemma 2.2.3 yields L X X

X

f, φper 2 + f, ψ per 2 j,k `,j,k k∈Nj

`=1 k∈Nj

=

XX

hF0 , φj,k i hFs , φj,k i +

L X X X

`=1 k∈N0 s∈N0

k∈N0 s∈N0

35

 

` F0 , ψj,k

` Fs , ψj,k

2.2 PWF via Unitary Extension Principle XX

=

hF0 , φj+1,k i hFs , φj+1,k i

k∈N0 s∈N0

X

f, φper 2 . j+1,k

=

k∈Nj

This completes the proof. Proof of Theorem 2.2.1. For an arbitrary function f ∈ Ω and ε > 0, we can choose J > 0 by Lemma 2.2.2 such that for all j > J, we have X

2 f, φper 2 ≤ (1 + ε) f 2 . (1 − ε) f 2 ≤ j,k 2 k∈Nj

For any j ∈ Z, Lemma 2.2.4 implies that L X

X

X X

2 f, φper 2 = f, φper 2 + f, ψ per . j,k j−1,k `,j−1,k k∈Nj

k∈Nj−1

By repeating this argument on

`=1 k∈Nj−1 2 |hf, φper j−1,k i| , we obtain

P k∈Nj−1

j−1 L X X

X X

f, φper 2 = f, φper 2 + f, ψ per 2 . j,k `,r,k `=1 r=0 k∈Nr

k∈Nj

Therefore, we have j−1 L X X X

2

per 2

f, ψ per 2 ≤ (1 + ε) f 2 . (1 − ε) f 2 ≤ f, φ + `,r,k 2 `=1 r=0 k∈Nr

Letting j → ∞, we obtain L X X

2

2 X f, ψ per 2 ≤ (1 + ε) f 2 . (1 − ε) f 2 ≤ f, φper + `,r,k 2 `=1 r∈N0 k∈Nr

Since > 0 was arbitrary. Hence, it follows that L X X X

f, φper 2 + f, ψ per 2 = kf 2 . `,r,k 2 `=1 r∈N0 k∈Nr

This completes the proof of the Theorem. 36

2.3

Dual PWF via Mixed Extension Principle

2.3


In this section, we construct pairs of dual periodic wavelet frames on local fields of positive characteristic using the mixed extension principle. The following theorem is the main result of this section. Theorem 2.3.1. Suppose φ and φ˜ are two compactly supported refinable functions, and m0 (ξ), m ˜ 0 (ξ) be the corresponding refinement masks. For each ` = 1, 2, . . . , L, let m` (ξ) and m ˜ ` (ξ) be periodic bounded functions. Assume that the ` L ` L wavelet systems ψj,k `=1;j∈Z,k∈N and ψ˜j,k constitutes a pair of dual `=1;j∈Z,k∈N0 0

wavelet frames for L2 (K) generated by the mixed extension principle associated with ˜ m0 , m1 , . . . , mL and m φ, φ, ˜ 0, m ˜ 1, . . . , m ˜ L . Then, FΨper and FΨ˜per are a pair of dual periodic frames for L2 (D), where per FΨper := φper , ψ`,j,k : 1 ≤ ` ≤ L, j ∈ N0 , k ∈ Nj ,

(2.3.1)

and n o per FΨ˜per := φ˜per , ψ˜`,j,k : 1 ≤ ` ≤ L, j ∈ N0 , k ∈ Nj .

(2.3.2)

We need the following lemmas, which are important to the proof of the main result. Lemma 2.3.2. The sequences FΨper and FΨ˜per are both Bessel sequences for L2 (D). Proof. To simplify the expressions in the proof, we let L X X X

f, ψ per 2 , PΨ (f ) = `,j,k `=1 j∈N0 k∈Nj L X X D E 2 X per PΨ˜ (f ) = f, ψ˜`,j,k . `=1 j∈N0 k∈Nj

In order to prove that the sequences FΨper and FΨ˜per are both Bessel sequences, we need to find out two positive numbers C, C˜ such that for any function f ∈ Ω, we have

2 2 PΨ (f ) + f, φper ≤ C f 2

2 2 and PΨ˜ (f ) + f, φ˜per ≤ C˜ f 2 . 37

2.3


For any f ∈ L2 (D), by the Parseval’s formula of the Fourier series, we observe that 2 X

per ˆ ˆ f, φper 2 = f u(s) φ u(s) s∈N0 2 X fˆ u(s) φˆ u(s) χu(s) = s∈N0 X 2 2 X ˆ ˆ φ u(s) (2.3.3) ≤ f u(s) . s∈N0

s∈N0

By the assumptions (1.3.5) and (1.3.6), there exists C1 > 0 such that X 2 φˆ u(s) ≤ C1 s∈N0

Using identity (1.2.2) and the above estimate, we obtain

f, φper 2 ≤ C1 f 2 . 2

(2.3.4)

per Next, we compute PΨ (f ). Using the periodic property of ψ`,j,k i.e.,

` ` ψj,k x + u(s) = ψj,k−p −j u(s) (x), s ∈ N0 , we have PΨ (f ) =

L X X X `=1 j∈N0 k∈Nj

=

≤

≤

=

X D E ` x + u(s) f, ψj,k

!2

s∈N0

2 X Z `   f (x) ψj,k x + u(s) dx s∈N0 D `=1 j∈N0 k∈Nj 2 Z L XX X X f (x) ψ ` x + u(s) dx C2 j,k `=1 j∈N0 k∈Nj s∈N0 D 2 L X X Z X ` C2 f (x) ψ (x) dx j,k `=1 j∈N0 k∈Nj D 2 L X X Z X ` f · 1D (x) ψ (x) dx , C2 j,k `=1 j∈N0 k∈Nj L X X X



K

38

2.3


where the bound C2 is obtained by considering the fact that the union of supports of ψ` , 1 ≤ ` ≤ L is contained in a disk with centre 0 and positive radius. Moreover, by the assumptions, we know that the wavelet system FΨ given by equation (1.3.11) is a Bessel sequence for L2 (K), we can deduce that there exist a positive number C3 such that

2

2 PΨ (f ) ≤ C3 f .1D 2 = C3 f 2 .

(2.3.5)

Combining equations (2.3.4) and (2.3.5), we have

2

2

2 PΨ (f ) + f, φper ≤ (C1 + C3 ) f 2 = C4 f 2 . Similarly, we have

2 2 PΨ˜ (f ) + f, φ˜per ≤ C˜4 f 2 . This completes the proof of Lemma 2.3.2. Lemma 2.3.3. If f, g ∈ Ω, i.e., X fˆ u(r) χu(r) (x), f (x) =

and

g(x) =

r∈N0

X

gˆ u(r) χu(r) (x),

r∈N0

where the sequences fˆ u(r) : r ∈ N0 and gˆ u(r) : r ∈ N0 have only finitely many non-zero terms. Then the following formula holds: hf, gi =

f, φper j,k

L X X

per X per per ˜ φj,k , g + f, ψ`,j,k ψ˜`,j,k , g .

(2.3.6)

`=1 j∈N0 k∈Nj

Proof. We split the proof into three steps. Step 1. We rearrange and rewrite the following series: X

X

per per per f, φper φ˜j,k , g , f, ψ`,j,k ψ˜`,j,k , g . j,k k∈Nj

k∈Nj

First, we infer that

per φ˜j,k , g = f, φper j,k

XD

f, φj,k

! E x + u(r)

r∈N0

=

X XD

XD s∈N

f, φj,k

φ˜j,k

E x + u(s) , g

0 E D E x + u(r) φ˜j,k x + u(s) , g

r∈N0 s∈N0

39

!

2.3


=

X XD

E f, φj,k x + u(r)

r∈N0 s∈N0

D

E ˜ × φj,k x + u(r) + u(s) , g .

(2.3.7)

Then, for any s ∈ N0 , we define Fs (x) = f (x)1D+u(s) (x),

Gs (x) = g(x)1D+u(s) (x),

(2.3.8)

where 1D+u(s) (x) is the characteristic function of the set D + u(s). Since f and g are integral-periodic functions, by equations (2.3.7) and (2.3.8), we have   Z  X X 

per ˜ ,g = x + u(r) dx f, φper φ f (x)φ j,k j,k j,k   r∈N0 s∈N0 D   Z  ˜ × φj,k x + u(r) + u(s) g(x)dx   D

 X X Z

  = f (x)φj,k x + u(r) dx   r∈N0 s∈N0 D     Z   φ˜j,k x + u(r) g(x)dx ×     D+u(s)   X X Z  = F0 (x)φj,k x + u(r) dx   r∈N0 s∈N0 K   Z  ˜ × φj,k x + u(r) Gs (x)dx  

(2.3.9)

K

By summing identity (2.3.9) over the set Nj and noting that φj,k x + u(r) = φj,k−p−j u(r) (x), we have X

f, φper j,k

E X X X

per D φ˜j,k , g = F0 , φj,k−p−j u(r) φ˜j,k−p−j u(r) , Gs k∈Nj r∈N0 s∈N0

k∈Nj

=

X X

F0 , φj,k φ˜j,k , Gs . s∈N0 k∈N0

40

(2.3.10)

2.3


Similarly, for each ` = 1, 2, . . . , L, we have X

per X X

` per ` ψ˜`,j,k , g = F0 , ψj,k ψ˜j,k , Gs . f, ψ`,j,k

(2.3.11)

s∈N0 k∈N0

k∈Nj

Case 2. For any J ≥ 0, we claim that L X X E X

per X per D per

per per ˜ ˜ ψ˜`,j,k , g = φ , g . f, φj,k φj,k , g + f, ψ`,j,k f, φper J+1,k J+1,k `=1 j∈N0 k∈Nj

k∈NJ+1

Taking sum on the R.H.S of (2.3.11) over ` = 0, 1, . . . , L, we have Qj =

L X X X

` F0 , ψj,k

` ψ˜j,k , Gs .

(2.3.12)

`=0 k∈N0 s∈N0

By the Parseval identity and equation (1.3.9), we deduce that

` ` F0 , ψj,k = Fˆ0 , ψˆj,k Z j/2 = q Fˆ0 (ξ) ψˆ` pj ξ χk pj ξ dξ K

= q

j/2

Z

Fˆ0 (ξ) m` pj+1 ξ φˆ pj+1 ξ χk pj ξ dξ

K

= q

j/2

Z

p−j D

X

Fˆ0 ξ + p−j u(r) m` pj+1 ξ + pu(r)

r∈N0

×φˆ pj+1 ξ + pu(r) χk pj ξ dξ.

(2.3.13)

Since m` (ξ), 1 ≤ ` ≤ L are wavelet masks associated with the given wavelets ψ` and each m` (ξ) is bounded periodic function on the disk D, we have ) Z (X ˆ F0 ξ + p−j u(r) m` pj+1 ξ + pu(r) φˆ pj+1 ξ + pu(r) χk pj ξ dξ p−j D

r∈N0

≤ m` ∞

Z ˆ F0 (ξ)φˆ pj+1 ξ dξ < ∞. K

Therefore, the exchange of the integral and the summation is reasonable in the above formula. Again by the periodicity property of the wavelet masks m` , we infer that 41

2.3


` F0 , ψj,k

= q

j/2

Z

p−j D

X X

Fˆ0 ξ + p−j p−1 u(r0 ) + u(t) χk pj ξ

r0 ∈N0 t∈N1

×m` pj+1 ξ + p p−1 u(r0 ) + u(t) φˆ pj+1 ξ + p p−1 u(r0 ) + u(t) dξ Z X X j/2 Fˆ0 ξ + p−j p−1 u(r0 ) + u(t) χk pj ξ = q p−j D

r0 ∈N0 t∈N1

×φˆ pj+1 ξ + p p−1 u(r0 ) + u(t) m` pj+1 ξ + pu(t) dξ. Similarly, we have Z E D ` j/2 ˜ ψj,k , Gs = q p−j D

X X

(2.3.14)

ˆ s ξ + p−j p−1 u(s0 ) + u(t0 ) G

s0 ∈N0 t0 ∈N1

×χk pj ξ φˆ˜ pj+1 ξ + p p−1 u(s0 ) + u(t0 ) ×m ˜ ` pj+1 ξ + pu(t0 ) dξ.

(2.3.15)

Since q j/2 χk pj ξ : k ∈ N0 is an orthonormal basis for L2 (p−j D). Therefore, by equalities (2.3.14) and (2.3.15) together with the Parseval’s formula of Fourier series, we have X

` ` F0 , ψj,k ψ˜j,k , Gs k∈N0 Z (X X −j −1 0 ˆ = F0 ξ + p p u(r ) + u(t) m` pj+1 ξ + pu(t) p−j D

r0 ∈N0 t∈N1

(X X ˆ s ξ + p−j p−1 u(s0 ) + u(t0 ) ×φˆ pj+1 ξ + p p−1 u(r0 ) + u(t) G s0 ∈N0 t0 ∈N1 o ×m ˜ ` pj+1 ξ + pu(t0 ) φˆ˜ pj+1 ξ + p p−1 u(s0 ) + u(t0 ) dξ. Again, by equation (2.3.12), we deduce that X Z X X X X Fˆ0 ξ + p−j p−1 u(r0 ) + u(t) Qj = s∈N0 −j r0 ∈N0 s0 ∈N0 t∈N1 t0 ∈N1 p D

ˆ s ξ + p−j p−1 u(s0 ) + u(t0 ) φˆ pj+1 ξ + p p−1 u(r0 ) + u(t) ×G

42

2.3

Dual PWF via Mixed Extension Principle ( L X × φˆ˜ pj+1 ξ + p p−1 u(s0 ) + u(t0 ) m` pj+1 ξ + pu(t) `=0

o ×m ˜ ` pj+1 ξ + pu(t0 ) dξ. By mixed extension principle condition, we have L X

m` pj+1 ξ + pu(t) m ˜ ` pj+1 ξ + pu(t0 ) = δt,t0 .

`=0

Therefore, we obtain X Z X X X Qj = Fˆ0 ξ + p−j p−1 u(r0 ) + u(t) s∈N0 −j r0 ∈N0 s0 ∈N0 t∈N1 p D

−j −1 0 j+1 −1 0 ˆ ˆ ×Gs ξ + p p u(s ) + u(t) φ p ξ + p p u(r ) + u(t) × φˆ˜ pj+1 ξ + p p−1 u(s0 ) + u(t) dξ. Setting αj (ξ) =

X

Fˆ0 ξ + p−j−1 u(r0 ) φˆ pj+1 ξ + u(r0 ) ,

r0 ∈N0

βj,s (ξ) =

ˆ s ξ + p−j−1 u(r0 ) φˆ˜ pj+1 ξ + u(s0 ) , G

X s0 ∈N0

we conclude that Qj =

XX Z

αj ξ + p−j u(t) βj,s ξ + p−j u(t) dξ

s∈N0 t∈N1 −j p D

=

XX

Z αj (ξ) βj,s (ξ) dξ

s∈N0 t∈N1 −j p D+p−j u(t)

=

X

Z αj (ξ) βj,s (ξ) dξ.

s∈N0 −j−1 p D

Using the Parseval identity of the Fourier series, we obtain Z αj (ξ) βj,s (ξ)dξ p−j−1 D j+1

=p

X

Z

Z αj (ξ) χpj+1 k (ξ) dξ

k∈N0 −j−1 p D

p−j−1 D

43

βj,s (ξ) χpj+1 k (ξ) dξ

2.3


j+1

=p

Z

X

Fˆ0 (ξ) φˆ pj+1 ξ χpj+1 k (ξ) dξ

k∈N0 −j−1 p D

Z

ˆ s (ξ) φˆ˜ pj+1 ξ χpj+1 k (ξ) dξ. G

p−j−1 D

Again by the Parseval identity of the Fourier transform, we obtain Qj =

X X

F0 , φj+1,k

φ˜j+1,k , Gs .

(2.3.16)

k∈N0 s∈N0

Therefore, it follows from equations (2.3.12) and (2.3.16) that X X

F0 , φj+1,k φ˜j+1,k , Gs k∈N0 s∈N0 L X X X

` ` = F0 , ψj,k ψ˜j,k , Gs `=0 k∈N0 s∈N0

=

L X X X X

X

` ` F0 , φj,k φ˜j,k , Gs + ψ˜j,k , Gs . F0 , ψj,k k∈N0 s∈N0

`=1 k∈N0 s∈N0

Using equations (2.3.10) and (2.3.11) in the above identity, we obtain L X X

per X

per X

per per per per ˜ ˜ f, φj+1,k φj+1,k , g = f, φj,k φj,k , g + f, ψ`,j,k ψ˜`,j,k , g . k∈Nj+1

k∈Nj

`=1 k∈Nj

By applying the fact that Nj = {0} , when j = 0, we have L X X

X

per per

per ˜ , g = f, φper φ˜per , g + ψ˜`,0,k , g . f, φper φ f, ψ`,0,k 0,0 0,0 1,k 1,k k∈N1

`=1 k∈N1

In general, for any J ≥ 0, we have L X X X

per

per X

per per per ˜ ˜ ,g + f, φper φ , g = f, φ φ f, ψ`,j,k ψ˜`,j,k , g . J+1,k J+1,k j,k j,k `=1 j∈N0 k∈Nj

k∈NJ+1

(2.3.17) Step 3. For f, g ∈ Ω, we have X

Sj =

per φ˜j,k , g → hf, gi , f, φper j,k

j → ∞.

(2.3.18)

k∈Nj

Since f, g both lies in Ω, therefore, there exists non-negative integer J such that f (x) =

X

fˆ u(s) χu(s) (x),

g(x) =

X s∈N0

s∈N0

44

gˆ u(s) χu(s) (x),

2.3


where fˆ u(s) = gˆ u(s) = 0, s ∈ / NJ . Again, let φper j,k (x) =

X

φˆper j,k u(s) χu(s) (x),

φ˜per j,k (x) =

s∈N0

X ˆper φ˜j,k u(s) χu(s) (x), s∈N0

where φˆper j,k u(s)

= q j/2 φˆ pj u(s) χk pj u(s) ,

= q j/2 φˆ˜ pj u(s) χk pj u(s) . φˆ˜per j,k u(s) Then, for j ≥ J, the Parseval identity of the Fourier series implies that

X X fˆ u(s) q j/2 φˆ pj u(s) χk pj u(s) . f, φper fˆ u(s) φˆper j,k u(s) = m,k = s∈Nj

s∈Nj

Similarly, we have

X φ˜per gˆ u(s) q j/2 φˆ˜ pj u(s) χk pj u(s) , , g = m,k

j ≥ J.

s∈Nj

Hence, we conclude that for j ≥ J, we have    X X j/2 j j ˆ ˆ f u(s) q φ p u(s) χk p u(s) Sj =   s∈Nj k∈Nj   X  × gˆ u(t) q j/2 φˆ˜ pj u(t) χk pj u(t)   t∈Nj

= qj

XX

X fˆ u(s) gˆ u(t) φˆ pj u(s) φˆ˜ pj u(t) χk pj u(s − t)

s∈Nj t∈Nj

=

X

k∈Nj

fˆ u(s) gˆ u(s) φˆ pj u(s) φˆ˜ pj u(s)

s∈Nj

=

X

fˆ u(s) gˆ u(s) φˆ pj u(s) φˆ˜ pj u(s) .

s∈NJ

ˆ˜ = 1, in the above relation, we obtain ˆ = lim φ(ξ) Using the fact that lim φ(ξ) ξ→0

lim Sj =

j→∞

X

ξ→0

X fˆ u(s) gˆ u(s) = fˆ u(s) gˆ u(s) = hf, gi.

s∈NJ

s∈N0

45

2.3


From equations (2.3.17) and (2.3.18), we deduce that

f, φper j,k

L X X X

per per per ˜ φj,k , g + f, ψ`,j,k ψ˜`,j,k , g = hf, gi. `=1 j∈N0 k∈Nj

This completes the proof of the Lemma 2.3.3. Proof of Theorem 2.3.1. By Lemma 2.3.2, it follows that the sequences FΨper and FΨ˜per are both Bessel sequences for L2 (D). By Lemma 2.3.3, we know that for any f, g ∈ oΩ, equality (2.3.6) holds. Since Ω is dense in L2 (D), it follows that n FΨper , FΨ˜per is a pair of dual wavelet frames for L2 (D). This completes the proof.

46

Chapter 3 Minimum Energy Wavelet Frames on LFPC Minimum-energy wavelets frames are constructed as an alternative to overcome the short comings of the computational complexity and numerical instability of the wavelet frames that occur during the course of decomposition and reconstruction of functions. Motivated and inspired by the concept of wavelet frames on local fields of positive characteristic, our aim in this Chapter is to extend the notion of wavelet frames to minimum-energy wavelet frames(MEWF) on local fields via extension principles. More precisely, we introduce the concept of minimum-energy wavelet frame associated with a given refinable function on local field K of positive characteristic and present its equivalent characterizations in terms of their framelet symbols. This Chapter is organised as follows. In Section 3.1, we give a complete characterization of minimum-energy wavelet frames associated with some given refinable functions in terms of their framelet symbols. More precisely, we present a necessary and sufficient condition for the existence of minimum-energy wavelet frame on local field K of positive characteristic. In Section 3.2, we provide the polyphase representation of the wavelet masks. Based on the polyphase representation as ob-

47

3.1 Construction of MEWF on LFPC

tained, we give the necessary and sufficient conditions for minimum-energy wavelet frames on local fields. Finally, in Section 3.3, we derive the minimum-energy wavelet frame decomposition and reconstruction formulae which are quite similar to those of orthonormal wavelets on local fields of positive characteristic.

3.1

Construction of MEWF on LFPC

Definition 3.1.1. Let φ ∈ L2 (K) satisfies φˆ ∈ L∞ and φˆ is continuous at 0, and ˆ = 1. Suppose that φ generates the nested closed subspaces {Vj : j ∈ Z} in the φ(0) sense of Definition 1.3.4 Then, a finite family Ψ = {ψ1 , ψ2 , . . . , ψL } ⊂ V1 is called a minimum-energy wavelet frame associated with φ, if for all f ∈ L2 (K), L X X X X

2 ` hf, φ1,k i 2 = hf, φ0,k i 2 + f, ψ0,k . k∈N0

k∈N0

(3.1.1)

`=1 k∈N0

By the Parseval identity, minimum-energy wavelet frame Ψ must be a tight frame for L2 (K) with frame bound being equal to 1. At the same time identity (3.1.1) is equivalent to X

hf, φ1,k i φ1,k =

k∈N0

X

hf, φ0,k i φ0,k +

k∈N0

L X X

` ` f, ψ0,k ψ0,k , ∀ f ∈ L2 (K). (3.1.2)

`=1 k∈N0

The following theorem presents the equivalent characterizations of the minimum energy wavelet frames associated with refinable function φ on local fields of positive characteristic. Theorem 3.1.2. Suppose that the refinable function φ and the framelet symbols m` (ξ), ` = 0, 1, . . . , L satisfy identities (1.3.3)-(1.3.10). If φˆ is continuous at 0 and φ(x) generates a sequence of nested closed subspaces {Vj : j ∈ Z}. Then the following statements are equivalent: (i) (ii)

Ψ is a minimum-energy wavelet frame associated with φ, M(ξ)M∗ (ξ) = Iq ,

for all ξ ∈ D, 48

(3.1.3)

3.1 Construction of MEWF on LFPC ( (iii) αr,s =

X

hu(r)−qk hu(s)−qk +

L X

k∈N0

) hù(r)−qk hù(s)−qk

− qδr,s = 0, ∀ r, s ∈ N0 .

`=1

(3.1.4) Proof. By using the functional equations (1.3.2) and (1.3.8) and notation αr,s , equation (3.1.2) can be written for all f ∈ L2 (K) as D XX E −1 αr,s f, φ p x − u(r) φ p−1 x − u(s) = 0.

(3.1.5)

r∈N0 s∈N0

On the other side, identity (3.1.3) can be rewritten as L X m0 (ξ) 2 + m` (ξ) 2 = 1.

(3.1.6)

`=1

This may be further written as L X m` (ξ) m` ξ + pu(k) = 0, m0 (ξ) m0 ξ + pu(k) +

k = 1, 2, . . . , q − 1,

`=1

which is equivalent to m0 (ξ)

q−1 X

( q−1 ) L X X m0 ξ + pu(k) + m` ξ + pu(k) m` (ξ) = 1,

k=0

`=1

k=0

or ( m0 (ξ) m0 (ξ) −

q−1 X

) m0 ξ + pu(k)

k=1

+

L X `=1 (

m` (ξ) q−1 X

) m` ξ + pu(k)

=1 × m` (ξ) − k=1 ( q−1 ) L X X m0 (ξ) m0 ξ + pu(k) − 2 m0 ξ + pu(r) m` (ξ) k=0 ( `=1 ) q−1 X × m` ξ + pu(k) − 2 m` ξ + pu(r) = 1, r = 1, 2, . . . , q − 1. k=0

The above system is equivalent to

49


 L X X X     m (ξ) h χ (ξ) + m (ξ) hù(0)−qk χqk (ξ) = 1, 0 qk ` u(0)−qk     k∈N `=1 0 0  ( q−1 ) k∈N  L  XX X m0 (ξ) hu(r)−qk χqk−r (ξ) + m` (ξ)   r=1 k∈N0 `=1  ) ( q−1    XX    hù(r)−qk χqk−u(r) (ξ) = q − 1. ×   r=1 k∈N0

The above system can be further expressed as  L  X X X    m (ξ) hù(0)−qk χqk−u(0) (ξ) = 1, m (ξ) h χ (ξ) + ` 0 u(0)−qk qk−u(0)     k∈N0 k∈N0 `=1   L  X X X    m0 (ξ) m` (ξ) hù(1)−qk χqk−u(1) (ξ) = 1, hu(1)−qk χqk−u(1) (ξ) + k∈N0

.. .

             

m0 (ξ)

X

k∈N0

`=1

hu(q−1)−qk χqk−u(q−1) (ξ) +

k∈N0

L X

m` (ξ)

`=1

X

hù(q−1)−qk χqk−u(q−1) (ξ) = 1.

k∈N0

(3.1.7) ˆ We multiply the identities in (3.1.7) by φ(pξ)χ u(r) (ξ), r = 0, 1, . . . , q−1, respectively, and we get ( ˆ φ(pξ)χ u(r) (ξ) =

X

hu(r)−qk χqk (ξ)ψˆ0 (ξ) +

k∈N0

L X

) hù(r)−qk χqk (ξ)ψˆ` (ξ) .

(3.1.8)

`=1

Therefore, the system (3.1.7) can be written as  ( ) L  X X  `  φ(pξ)χ ˆ ˆ +  hu(0)−qk χqk (ξ)φ(ξ) hu(0)−qk χqk (ξ)ψˆ` (ξ) , u(0) (ξ) =     k∈N0 ( `=1  )  L  X X   ˆ ˆ +  φ(pξ)χ hu(1)−qk χqk (ξ)φ(ξ) hù(1)−qk χqk (ξ)ψˆ` (ξ) , u(1) (ξ) =              

.. .

k∈N0

`=1

( ˆ φ(pξ)χ u(q−1) (ξ) =

X

ˆ + hu(q−1)−qk χqk (ξ)φ(ξ)

k∈N0

L X `=1

50

) hù(q−1)−qk χqk (ξ)ψˆ` (ξ) ,


or  ( ) L  X X 1    φ p−1 x − u(0) = hu(0)−qk φ x − u(k) + hù(0)−qk ψ` x − u(k) ,   q   k∈N0 ( `=1  )  L  X X  1  `  φ p−1 x − u(1) = hu(1)−qk φ x − u(k) + hu(1)−qk ψ` x − u(k) , q k∈N `=1 0   ..   .   ( )   L  X X 1    hu(q−1)−qk φ x − u(k) + φ p−1 x − u(q − 1) = hù(q−1)−qk ψ` x − u(k) .   q k∈N `=1 0

On the reformulation of above system, we obtain for all r ∈ N0 ( ) L X X 1 φ p−1 x − u(r) = hu(r)−qk φ x − u(k) + hù(r)−qk ψ` x − u(k) . q k∈N `=1 0

(3.1.9) Using equations (1.3.2) and (1.3.6), we can rewrite formula (3.1.9) as X αr,s φ p−1 x − u(r) = 0, ∀ s ∈ N0 .

(3.1.10)

r∈N0

Thus, equation (3.1.3) is equivalent to identity (3.1.10). In conclusion, the proof of the theorem reduces to the proof of the equivalence of the equations (3.1.4), (3.1.5) and (3.1.10). It is obvious that equation (3.1.4) implies identity (3.1.10) which implies equation (3.1.5). In order to prove identity (3.1.5)=⇒ identity (3.1.4), we assume that f be a function of compact support, i.e., f ∈ Ω. By using the properties that for every fixed r, αr,s = 0 except for finitely many s, then the functional D X E βs (f ) = αr,s f, φ p−1 x − u(r) , s ∈ N0 , r∈N0

just has finite nonzero for s ∈ N0 .

ˆ Since φ(ξ) is nontrivial function, by tak-

ing the Fourier transform of equation (3.1.5), it follows that the polynomial P βs (f )χu(s) (ξ) is identically zero. Obviously, βs (f ) = 0, s ∈ N0 . In other words, s∈N0

we have

* f,

+ X

αr,s φ p−1 x − u(r)

r∈N0

51

= 0,

s ∈ N0 .


Thus the series in the above equation is a finite sum and hence represents a compactly supported function in L2 (K). By choosing f to be this function, it follows that X αr,s φ p−1 x − u(r) = 0, r∈N0

which implies that the polynomial

P

αr,s χ(ξ) is identically equal to 0 so that

r∈N0

αr,s = 0, r, s ∈ N0 . This completes the proof of the theorem. Theorem 3.1.2 gives the necessary and sufficient condition for the existence of the minimum-energy wavelet frames associated with refinable function φ. However it is not a good choice to use this theorem to construct the minimum-energy wavelet frames. For convenience, we need to present some conditions in terms of the framelet symbols. Theorem 3.1.3. Let φ ∈ L2 (K) be the refinable function with refinement mask ˆ m0 (ξ) such that φˆ is continuous at 0 and φ(0) = 1. If Ψ = {ψ1 , ψ2 , . . . , ψL } is the minimum-energy wavelet frame associated with φ, then q−1 X m0 ξ + pu(r) 2 ≤ 1,

for all ξ ∈ K.

(3.1.11)

r=0

Proof. Let Q(ξ) be the first column of the modulation matrix M(ξ). Then, M(ξ) = Q(ξ), R(ξ) , where   m1 ξ + pu(0) m1 ξ + pu(1) . . . m1 ξ + pu(q − 1)    m2 ξ + pu(0) m2 ξ + pu(1) . . . m2 ξ + pu(q − 1)    R(ξ) =   (3.1.12) . . . . .. .. .. ..     mL ξ + pu(0) mL ξ + pu(1) . . . mL ξ + pu(q − 1) and h i Q(ξ) = m0 ξ + pu(0) m0 ξ + pu(1) · · · m0 ξ + pu(q − 1) . Therefore, the condition (3.1.3) can be reformulated as Q(ξ)Q∗ (ξ) + R(ξ)R∗ (ξ) = Iq , 52

(3.1.13)


or equivalently, Iq − Q(ξ)Q∗ (ξ) = R(ξ)R∗ (ξ). Since R(ξ)R∗ (ξ) is a Hermitian matrix, the matrix Iq − Q(ξ)Q∗ (ξ) is positive semidefinite, so that

∗

det Iq − Q(ξ)Q (ξ) ≥ 0, and this gives q−1 X m0 ξ + pu(r) 2 ≤ 1,

∀ ξ ∈ K.

r=0

In fact, we have Iq

Q∗ (ξ)

Q(ξ)

1

!

det

det

Iq

−Q∗ (ξ)

−Q(ξ)

1

Iq

Q∗ (ξ)

Q(ξ)

1

! =

!

Iq

−Q∗ (ξ)

−Q(ξ)

1

= det ! = det

Iq − Q(ξ)Q∗ (ξ)

0

0

1 − Q(ξ)Q∗ (ξ)

Q∗ (ξ)

Iq

,

!

0 1 − Q(ξ)Q∗ (ξ) Iq

!

−Q∗ (ξ)

0 1 − Q(ξ)Q∗ (ξ)

, ! .

Therefore 2 ∗ ∗ det Iq − Q(ξ)Q (ξ) 1 − Q(ξ)Q (ξ) = 1 − Q(ξ)Q (ξ) ,

∗

and it gives 1 − Q(ξ)Q∗ (ξ) ≥ 0. This completes the proof. According to the Theorem 3.1.3, there may not exist minimum-energy wavelet frame associated with a given refinable function φ and in case if it exist, then the refinement mask must satisfy condition (3.1.11). Based on the polyphase representation of the framelet symbols m` (ξ), ` = 0, 1, . . . , L, we provide here a sufficient condition.

53

3.2 Polyphase Characterization of Wavelets

3.2

Polyphase Characterization of Wavelets

We first derive the polyphase representation of the refinement mask m0 (ξ) as 1 X hk χk (ξ) m0 (ξ) = √ q k∈N 0

q−1

1 XX = √ hu(r)+qk χu(r)+qk (ξ) q r=0 k∈N 0

q−1

X 1 X χu(r) (ξ) = √ hu(r)+qk χqk (ξ) q r=0 k∈N 0

q−1

1 X = √ χu(r) (ξ)fr0 χ(qξ) , q r=0 where fr0 (x) =

X

hu(r)+qk x(k),

r = 0, 1, . . . , q − 1, x ∈ K.

(3.2.1)

k∈N0

Similarly, the framelet symbols m` (ξ), ` = 1, 2, . . . , L, as defined in equation (1.3.10) can be splitted into polyphase components as q−1 1 X m` (ξ) = √ χu(r) (ξ)fr` χ(qξ) , q r=0

where fr` (x) =

X

hù(r)+qk x(k),

r = 0, 1, . . . , q − 1, x ∈ K.

(3.2.2)

k∈N0

With the polyphase components as defined in equations (3.2.1) and (3.2.2), we formulate the polyphase matrix P(ξ) as:  f00 χ(qξ) f01 χ(qξ)  0  f11 χ(qξ)  f1 χ(qξ) P χ(qξ) =  .. ..  . .  1 0 fq−1 χ(qξ) fq−1 χ(qξ)

... χ(qξ) ... χ(qξ) .. .. . . L . . . fq−1 χ(qξ) f0L f1L

    .  

Then, it is clear that M(ξ) = S χ(ξ) P χ(qξ) , 54

(3.2.3)


where   1   S χ(ξ) = √  q 

χu(o) ξ + u(0) χu(o) ξ + u(1) .. .

χu(1) ξ + u(0) χu(1) ξ + u(1) .. .

... ... ...

χu(o) ξ + u(q − 1) χu(1) ξ + u(q − 1) . . .

χu(q−1) ξ + u(0) χu(q−1) ξ + u(1) .. . χu(q−1) ξ + u(q − 1)

Since S χ(ξ) is unitary matrix, therefore condition (3.1.3) is equivalent to ∗ P χ(qξ) P χ(qξ) = Iq .

(3.2.4)

For convenience let χ(qξ) = ζ, then condition (3.1.11) can be rewritten as q−1 X 0 2 fr (ζ) ≤ 1.

(3.2.5)

r=0

Now, if there exists fq0 (ζ) such that q X 0 2 fr (ζ) = 1,

(3.2.6)

r=0

then, we have the following Theorem. Theorem 3.2.1. Let m0 be the refinement mask of a refinable function φ of an FMRA and satisfy inequality (3.2.5). Furthermore, if there exists fq0 of the form (3.2.6), then there exist a minimum-energy wavelet frame Ψ associated with φ(x). Proof. Under the given assumption, it is easy to verify that T 0 0 0 0 f = f0 (ζ), f1 (ζ), . . . , fq−1 (ζ), fq (ζ)

(3.2.7)

is a unit vector, where T stands for the transpose of a given vector. By multiplying the diagonal matrix D0 = diag(ζ t0 , ζ t1 , . . . , ζ tq ) to the left side of equation (3.2.7), we get n T X 0 (ζ), ζ tq fq0 (ζ) = bj ζ j ; t0 , . . . , tq ∈ N0 , f1 = ζ t0 f00 (ζ), ζ t1 f10 (ζ), . . . , ζ tq−1 fq−1 j=0

55

    .  


where bj ∈ K q+1 , with b0 6= 0 and bn 6= 0. It is also clear that f1 is a unit vector as !∗ ! n n X X ∗ j j bj ζ bj ζ = 1, for all ζ ∈ D f1 f1 = j=0

j=0

and consequently, bT0 bn = 0. Consider the (q + 1) × (q + 1) Householder matrix H1 = Iq+1 −

2 vvT , 2 |v|

(3.2.8)

where v = bn ± kbn ke1 , with e1 = (1, 0, . . . , 0)Tq+1 , and the + and − signs are so chosen that v 6= 0. Then H1 bn = ±kbn ke1 . By the orthogonal property of the Householder matrix, we have (H1 b0 )T (H1 bn ) = bT0 H1T H1 bn = bT0 bn = 0. Using previous equation, it follows that the first component of H1 b0 is 0. n P Since H1 f1 = (H1 bj )ζ j , therefore, we can construct a diagonal matrix D1 = j=0

diag(ζ

t(1)

, 1, . . . , 1), t(1) ∈ N0 such that f2 = D1 b1 f1 = D1

n X

j

(H1 bj )ζ =

j=0

n X

(1)

bj ζ j

j=0

(1)

(1)

is also a unit vector and n1 < n, b0 6= 0, bn1 6= 0. Similarly, we define the Householder matrix H2 = Iq+1 −

2 v ˜v ˜T , |˜ v|2

(3.2.9)

where v ˜ = bn1 ± kbn1 ke1 6= 0, and D2 = diag(ζ t(2) , 1, . . . , 1), t(2) ∈ N0 such that f3 = D2 H2 f2 = D2

n1 X

(1) H2 bj

j

ζ =

n1 X j=0

j=0

56

(2)

bj ζ j

3.2 Polyphase Characterization of Wavelets (2)

(2)

is also a unit vector and n2 < n1 , b0 6= 0, bn1 6= 0. Since every component of f is a finite sum, we repeat this procedure finite times to get some unitary matrices DN , HN , DN −1 , HN −1 , . . . , H2 , D1 , H1 such that DN HN DN −1 HN −1 . . . H2 D1 H1 f = e1 .

(3.2.10)

Therefore, it is clear that f is the first column of the unitary matrix ∗ ∗ ∗ ∗ H = D0∗ H0∗ D1∗ H1∗ · · · DN −1 HN −1 DN HN .

By setting,     H=  

f00 (ζ) f10 (ζ) .. .

f01 (ζ) f11 (ζ) .. .

... ... .. .

0 1 fq−1 (ζ) fq−1 (ζ) . . .

f0q (ζ) f1q (ζ) .. .

q fq−1 (ζ)

    ,  

it is immediate that H satisfies (3.2.4). Moreover, if we choose m` (ξ) of the form q−1 1 X m` (ξ) = √ χu(r) fr` χ(qξ) , q r=0

` = 1, 2, . . . , q.

in the modulation matrix, then we can obtain the unitary extension condition (3.1.3). Therefore, Theorem 3.1.2 implies that Ψ generates a minimum energy wavelet frame for L2 (K). This completes the proof of the Theorem 3.2.1. Corollary 3.2.2. Let φ(x) ∈ L2 (K) be the refinable function, with φˆ continuous at ˆ = 1, and its symbol function satisfies 0 and φ(0) q−1 X m0 ξ + pu(r) 2 ≤ 1,

ξ ∈ K.

(3.2.11)

r=0

If there exists fq0 (ζ), . . . , fr0 (ζ), r ≥ q such that r X 0 2 fi (ζ) = 1, i=0

then there exists a minimum-energy wavelet frame associated with φ(x).

57

(3.2.12)


Here, we continue the study based on the extension principles and give a polyphase matrix characterization of tight wavelet frames on local fields of positive characteristic. The polyphase matrix P(ξ) is called a unitary matrix if condition (3.2.4) holds which is equivalent to L X

fr` (ζ)fr`0 (ζ)

= δr,r0 ⇔

`=0

L X

fr`0 (ζ)fr` (ζ) = δr,r0 − fr0 (ζ)fr00 (ζ), 0 ≤ r, r0 ≤ q − 1.

`=1

(3.2.13) The following theorem, the main result of this section, shows that a unitary polyphase matrix leads to a tight wavelet frame on local fields of positive characteristic. Theorem 3.2.3. Suppose that the refinable function φ and the framelet symbols m0 , m1 , . . . , mL satisfy equations (1.3.3)-(1.3.10). Furthermore, if the polyphase matrix P(ζ) satisfy condition (3.2.4), then the wavelet system FΨ given by equation (1.3.11) constitutes a tight frame for L2 (K) i.e., L X X X

2 2 ` f, ψj,k = f , 2


(3.2.14)

`=1 j∈Z k∈N0

Proof. By Parseval’s formula, we have L X X L X X D X X

2 E 2 ` f, ψj,k = f, q j/2 ψ` p−j x − u(k) `=1 j∈Z k∈N0

`=1 j∈Z k∈N0 L X X D E 2 X ˆ j/2 ˆ −j = f , q ψ` p ξ χpj (ξ) `=1 j∈Z k∈N0

=

L X X

qj

`=1 j∈Z

=

L X X `=1 j∈Z

E 2 X D −j ` (ξ), χ(ξ) ˆ ˆ ψ f p ξ k∈N0

Z 2 ˆ −j 2 ˆ q f (p ξ) ψ` (ξ) dξ. j

(3.2.15)

K

Using the polyphase decomposition formula (3.2.2) of the framelet symbols 58


m` (ξ), ` = 1, . . . , L, we can write L L 2 X X 2 ˆ ˆ m` (pξ)φ(pξ) ψ` (ξ) = `=1

=

`=1 L X

ˆ ˆ m` (pξ) φ(pξ) m` (pξ)φ(pξ)

`=1

) q−1 X 1 ˆ = φ(pξ) χu(r) (pξ)fr` (ζ) √ q r=0 `=1 ) ( q−1 1 X ˆ χu(r0 ) (pξ)fr`0 (ζ) φ(pξ) × √ q r0 =0 ( L ) q−1 q−1 X X X 1 ˆ ˆ χu(r)−u(r0 ) (pξ) = φ(pξ) fr` (ζ) fr`0 (ζ) φ(pξ). q r=0 r0 =0 `=1 L X

(

Since the polyphase matrix P(ζ) is unitary, which is equivalent to condition (3.2.13), the above expression reduces to q−1 q−1 L 2 h i X 1 XX ˆ 0 0 ˆ ˆ 0 0 χu(r)−u(r ) (pξ) δr,r − fr (ζ) fr0 (ζ) φ(pξ) ψ` (ξ) = φ(pξ) q r=0 r0 =0 `=1

= = = =

q−1 q−1 1 XX ˆ ˆ ˆ ˆ φ(pξ) φ(pξ) − φ(pξ) χu(r)−u(r0 ) (pξ)fr0 (ζ) fr00 (ζ)φ(pξ) q r=0 r0 =0 2 ˆ ˆ ˆ m0 (pξ) m0 (pξ)φ(pξ) φ(pξ) − φ(pξ) 2 2 ˆ ˆ φ(pξ) − m (pξ) φ(pξ) 0 2 2 ˆ ˆ (3.2.16) φ(pξ) − φ(ξ) .

By substituting equation (3.2.16) in (3.2.15), we obtain L X X X X Z

2 ` f, ψj,k = qj `=1 j∈Z k∈N0

j∈Z

n 2 2 o ˆ −j 2 ˆ ˆ dξ f (p ξ) φ(pξ) − φ(ξ)

K

Z ˆ 2 X ˆ j+1 2 ˆ j 2 = (3.2.17) f (ξ) φ p ξ − φ p ξ dξ. j∈Z

K

Using equation (1.3.5), the summand in the above equation can be reformatted as 59

3.3 Decomposition and Reconstruction Algorithms of MEWF

X 2 2 j+1 j ˆ ˆ φ p ξ − φ p ξ dξ = j∈Z

2 2 lim φˆ pj+1 ξ − lim φˆ pj ξ

j→∞

j→−∞

2 2 lim φˆ pj ξ − lim φˆ p−j ξ j→∞ j→∞ 2 ˆ −j 2 ˆ = φ(0) − lim φ p ξ

=

j→∞

= 1. By using the above estimate in equation (3.2.17), we have Z L X X 2

2 X

2 2

` f, ψj,k = fˆ(ξ) dξ =

fˆ = f 2 . 2

`=1 j∈Z k∈N0

K

This completes the proof of the theorem.

3.3

Decomposition and Reconstruction Algorithms of MEWF

Suppose Ψ = {ψ1 , ψ2 , . . . , ψL } is the minimum-energy wavelet frame associated with the refinable function φ. Then, for each j ∈ Z, we consider Vj = span {φj,k : k ∈ N0 }

` and Wj = span ψj,k : k ∈ N0 , ` = 1, 2, . . . , L . (3.3.1)

Thus, Vj+1 = Vj + Wj ,

j ∈ Z.

(3.3.2)

Note that decomposition (3.3.2) is not a direct sum decomposition since in general Vj ∩ Wj 6= {0} . Thus, it follows from equations (3.3.1) and (3.3.2) that any f ∈ Vj+1 can be expressed as f (x) = Pj f (x) + Qj f (x),

60

(3.3.3)


where Pj f (x) =

X

hf, φj,k iφj,k (x),

(3.3.4)

k∈N0

Qj f (x) = Pj+1 f (x) − Pj f (x) =

L X X

` ` hf, ψj,k iψj,k (x),

(3.3.5)

`=1 k∈N0

are the projection and detailed operators defined on Vj and Wj , respectively. The importance of this frame expansion as compared to any other expansion Qj f =

L X X

` . cj,k ψj,k

(3.3.6)

`=1 k∈N0

of the same Qj f is that the energy in (3.3.5) is minimum in the sense that L X L X X

2 2 X ` f, ψj,k ≤ cj,k . `=1 k∈N0

(3.3.7)

`=1 k∈N0

Therefore, by using (3.3.5) and (3.3.6), we have

L X L X X

2 X ` ` f, ψj,k Qj f, f = = cj,k f, ψj,k ,

`=1 k∈N0

(3.3.8)

`=1 k∈N0

and this derives L X X

2 ` cj,k − f, ψj,k 0 ≤ `=1 k∈N0 L X L X L X X X

2

X 2 ` ` f, ψj,k cj,k − 2 + = cj,k f, ψj,k `=1 k∈N0

`=1 k∈N0

`=1 k∈N0

L X L X X 2 X

2 ` f, ψj,k . = cj,k − `=1 k∈N0

`=1 k∈N0

This inequality means that the coefficients of the error term Qj f in equation (3.3.5) have minimal l2 -norm among all sequences {cj,k } which satisfy relation (3.3.6). We now discuss the decomposition and reconstruction algorithms associated with minimum-energy wavelet frames on local fields of positive characteristic. For any f ∈ L2 (K), we consider cj,k = hf, φj,k i ;

` d`j,k = f, ψj,k , 61

` = 1, 2, . . . , L.

(3.3.9)


Then, by two scale relations (1.3.2) and (1.3.8), we obtain 1 X φj,i = √ hk−qi φj+1,k , q k∈N

1 X ` ` ψj,i =√ h ψ` , q k∈N k−qi j+1,k

0

` = 1, 2, . . . , L. (3.3.10)

0

By taking the inner products with f on both sides of the two equations in (3.3.10), we have a tight minimum energy wavelet frame decomposition: 1 X ` 1 X cj,i = √ hk−qi cj+1,k , d`j,i = √ h d` , ` = 1, 2, . . . , L, j ∈ N0 . q k∈N q k∈N k−qi j+1,k 0

0

(3.3.11) Using the fact that φj,k ∈ Vj and relations (1.3.2) and (1.3.8), from (3.3.3) we also have φj+1,i

1 X =√ q k∈N

( hi−qk φj,k +

L X

) ` hì−qk ψj,k

,

i ∈ N0 .

(3.3.12)

`=1

0

By taking the inner products with f on both sides of equation (3.3.12), we have a tight minimum energy wavelet frame reconstruction: ( ) L X 1 X hi−qk cj,k + hì−qk d`j,k , cj+1,i = √ q k∈N `=1 0

62

i ∈ N0 .

Chapter 4 Semi-orthogonal Wavelet Frames on LFPC The use of an MRA has come out to be a very effective tool in wavelet theory mainly because of its simplicity. It is therefore very interesting to generalize this notion as much as possible while preserving its connection with wavelet analysis. In this direction, the dyadic semi-orthogonal frame multiresolution analysis of L2 (R) with a single scaling function has been considered and successfully applied the theory in the analysis of narrow band signals. In this Chapter, we extend the notion of wavelet frames to semi-orthogonal wavelet frames(SWF) on local fields of positive characteristic using the classical Fourier transform. More precisely, we characterize all semi-orthogonal wavelets as a generalization of the orthonormal wavelets on local fields in terms of some basic equations in frequency domain. This Chapter is organised as follows. In Section 4.1, we provide the definitions and results required in the subsequent sections. In Section 4.2, we characterise semi-orthogonal wavelet frames on local fields of positive characteristic by virtue of the Fourier transform. Our results generalizes the characterization of wavelets on Euclidean spaces by means of two basic equations.

63

4.1 Preliminaries

4.1

Preliminaries

The characterization of wavelet frames on local fields of positive characteristic has been studied in detail by Behera and Jahan [17] by following the strategy of the proof of Bownik [26] based on affine and quasi-affine frames. In fact, for a given dilation p−1 , they characterized all orthonormal wavelets {ψ1 , ψ2 , . . . , ψL } in L2 (K) by means of two basic equations in the frequency domain as: Theorem 4.1.1. Let Ψ = {ψ1 , ψ2 , . . . , ψL } ⊂ L2 (K). The affine system FΨ given by (1.3.11) is a normalized tight frame for L2 (K) if and only if L X X ˆ −j 2 ψ (p ξ) = 1 for a.e. ξ ∈ K `

(4.1.1)

`=1 j∈Z L X ∞ X

ψˆ` (p−j ξ) ψˆ` p−j ξ + u(m)

= 0 for a.e. ξ ∈ K, m ∈ N0 \ qN0 .

(4.1.2)

`=1 j=0

In particular, Ψ is a set of basic wavelets of L2 (K) if and only if kψ` k2 = 1 for ` = 1, 2, . . . , L and equations (4.1.1) and (4.1.2) hold. Let us recall the definition of a frame multiresolution analysis on local fields of positive characteristic [162]. Definition 4.1.2. Let K be a local field of positive characteristic p > 0 and p be a prime element of K. A frame multiresolution analysis (FMRA) of L2 (K) is a sequence of closed subspaces {Vj : j ∈ Z} of L2 (K) satisfying the following properties: (a) Vj ⊂ Vj+1 for all j ∈ Z; (b)

S


j∈Z

(c)

T

Vj = {0};

j∈Z

(d) f (x) ∈ Vj if and only if f (p−1 x) ∈ Vj+1 for all j ∈ Z; 64

4.2 Construction of SWF on LFPC

(e) there is a function φ ∈ V0 , such that {Tk φ(x) : k ∈ N0 } forms a frame for V0 . Let Ψ = {ψ1 , ψ2 , . . . , ψL } be a set of basic frame wavelets of L2 (K). We define ` the spaces Wj , j ∈ Z, by Wj = span ψj,k : 1 ≤ ` ≤ L, k ∈ N0 . We also define L Vj = Wm , j ∈ Z. Then it follows that {Vj : j ∈ Z} satisfies the properties (a)-(e) m 0 to the level 0 with any combined mask M = [m0 , m1 , . . . , mL ] satisfying the unitary extension principle condition M(ξ)M∗ (ξ) = Iq , where the matrix M(ξ) is the modulation matrix.

5.1

Preliminaries

The construction of framelet systems often starts with the construction of MRA, which is built on refinable functions. A function φ ∈ L2 (K) is called refinable if it satisfies a refinement equation given by (1.3.2) The so-called unitary extension principle (UEP) provides a sufficient condition on Ψ = {ψ1 , . . . , ψL } such that the resulting wavelet system FΨ forms a tight frame of L2 (K). In this connection, Shah and Debnath [170] gave an explicit construction scheme for the construction of tight framelets on local fields of positive characteristic using unitary extension principles in the following way. Theorem 5.1.1. Suppose that the refinable function φ and the framelet symbols m0 , m1 , . . . , mL satisfy equations (1.3.2)-(1.3.10). Define ψ1 , . . . , ψL by equation (1.3.8). Let M(ξ) be the modulation matrix such that M(ξ)M∗ (ξ) = Iq ,

for

a.e. ξ ∈ σ(V0 )

(5.1.1)

where X σ(V0 ) := ξ ∈ D : |φˆ ξ + u(k) |2 6= 0 , k∈N0

then the wavelet system FΨ given by equation (1.3.11) constitutes a normalized tight wavelet frame for L2 (K). Moreover, if the framelet symbols m` , ` = 0, 1, . . . , L, satisfy the UEP condition (5.1.1). Then, for any ξ ∈ K, we have q−1 X m` pξ + pu(k) 2 ≤ 1, k=0

74

(5.1.2)

5.2 TFP via Wavelet Spaces Wj,`

and L X

m` pξ + pu(r) m` pξ + pu(s) = δr,s ,

0 ≤ r, s ≤ q − 1.

(5.1.3)

`=0

For each j ∈ Z, we define Vj = span φj,k : k ∈ N0 , and ` : k ∈ N0 , Wj,` = span ψj,k

` = 0, 1, . . . , L.

Therefore, in view of tight frame decomposition, we have Vj = Vj−1 +

L X

Wj−1,` .

(5.1.4)

`=1

It is immediate from the above decomposition that these L + 1 spaces are in general not orthogonal. Therefore, by the repeated applications of (5.1.4), we can further split the Vj spaces as: j−1 j−1 L j−1 L L L X X X X X X X Vj = Vj−1 + Wj−1,` = Vj−2 + Wr,` = · · · = Vj0 + Wr,` = Wr,` . r=j−2 `=1

`=1

5.2

r=j0 `=1

r=−∞ `=1

TFP via Wavelet Spaces Wj,`

We start this section with a splitting lemma which plays a crucial role in the construction of tight framelet packets on local field K of positive characteristic. We split the wavelet spaces Wj,` by framelet symbols m` , ` = 0, 1, . . . , L and then by selecting and recursively decomposition, we will obtain various tight framelet packets of L2 (K). Lemma 5.2.1 (Splitting Lemma). Let g ∈ L2 (K) and {gj,k : k ∈ N0 } be a Bessel’s sequence in L2 (K) i.e., X gˆ ξ + u(k) 2 ≤ B, k∈N0

75

ξ∈K

(5.2.1)


for any fixed j ∈ Z. Let m` , 0 ≤ ` ≤ L be the framelet masks associated with the refinable function φ and the tight framelets ψ` , 1 ≤ ` ≤ L satisfying the UEP condition (5.1.1). Suppose g ` (x) = q

X

m` u(n) g p−1 x − u(n) ,

(5.2.2)

k∈N0

` G` = span gj−1,k : k ∈ N0 ,

(5.2.3)

and G = span {gj,k : k ∈ N0 }, for 0 ≤ ` ≤ L. Then ` : k ∈ N0 forms a Bessel’s sequence with (a) For ` = 0, 1, . . . , L, each set gj−1,k

` 2

g ≤ B and g 2 ≤ B. 2

2

L (b) For any sequence z ∈ l2 (Z), there exists L + 1 sequences z` `=0 defined by √ X z` u(k) = q m` u(n) − p−1 u(k) z u(n) ,

k ∈ N0

(5.2.4)

k∈N0

such that

L X

2

2

z 2 =

z` , l (Z)

(5.2.5)

`=0

and X

L X X ` z u(k) gj,k = z` u(k) gj−1,k .

k∈N0

`=0 k∈N0

(5.2.6)

(c) In particular for any f ∈ L2 (K), let z u(k) = hf, gj,k i, k ∈ N0 , then z ∈ l2 (Z) and equations (5.2.4)-(5.2.6) gives ` , z` u(k) = f, gj−1,k

k ∈ N0 , ` = 0, 1, . . . , L,

L X X X 2 ` hf, gj,k i 2 = hf, gj−1,k i , k∈N0

k∈N0

(5.2.8)

`=0 k∈N0

and X

(5.2.7)

hf, gj,k igj,k =

L X X

`=0 k∈N0

respectively. 76

` ` f, gj−1,k gj−1,k ,

(5.2.9)


(d) G has the decomposition G = G0 + G1 + · · · + GL . Proof. (a) By Plancherel’s formula, we have

2

g = gˆ 2 2 Z 2 gˆ(ξ)χk (ξ) 2 dξ = K

Z X gˆ ξ + u(k) 2 χk (ξ) 2 dξ. = D k∈N0

Using equation (5.2.1) and the fact that the set

2 orthonormal system on D, we obtain g ≤ B.

χu(n) : n ∈ N0

is a complete

2

On taking Fourier transform of equation (5.2.2), we obtain gˆ` (ξ) = m` (pξ) gˆ(pξ).

(5.2.10)

Using equations (5.1.2) and (5.2.1), we have X X gˆ` ξ + u(k) 2 = m` pξ + pu(k) 2 gˆ pξ + pu(k) 2 k∈N0

k∈N0 q−1 X X m` pξ + pu(qk + s) 2 gˆ pξ + pu(qk + s) 2 = s=0 k∈N0 q−1 X X m` pξ + pu(s) 2 gˆ pξ + pu(qk + s) 2 = s=0

k∈N0

q−1

=

X X gˆ pξ + pu(s) + u(k) 2 m` pξ + pu(s) 2 s=0

k∈N0

q−1

≤ B

X m` pξ + pu(s) 2 s=0

≤ B,

for ` = 0, 1, . . . , L.

(b) For each 0 ≤ ` ≤ L, the Fourier transform of equation (5.2.4) gives zˆ` (ξ) = q

−1/2

q−1 X

m` pξ + pu(r) zˆ pξ + pu(r) .

s=0

77

(5.2.11)


By summing equation (5.2.11) over ` = 0 to L and using (5.1.3), we obtain q−1 L X L X X zˆ` (ξ) 2 = q −1 m` pξ + pu(r) zˆ pξ + pu(r) m` pξ + pu(s) `=0 r,s=0

`=0

×ˆ z pξ + pu(r) = q

−1

q−1 X

L X zˆ pξ + pu(r) zˆ pξ + pu(r) m` pξ + pu(s)

r,s=0

`=0

×m` pξ + pu(r) = q

−1

q−1 X

zˆ pξ + pu(r) zˆ pξ + pu(r) δr,s

r,s=0

= q

−1

q−1 X zˆ pξ + pu(r) 2 . r=0

Therefore L L X X X

2

zˆ` 2 zˆ` u(k) 2 = ` (Z) `=0

`=0 k∈N0 L Z X zˆ` u(k) 2 dξ = `=0 D

Z X L zˆ` u(k) 2 dξ = D

= q

`=0

−1

Z X q−1 zˆ pξ + pu(r) 2 dξ D

Z =

r=0

zˆ(ξ) 2 dξ

D

2 Z X = z u(n) χu(n) (ξ) dξ D n∈N0 X z u(n) 2 = =

n∈N0

2

zˆ 2 . ` (Z)

78


Equation (5.2.6) can be recast in the frequency domain as: q

−j/2

j

j

zˆ(p ξ) gˆ(p ξ) = q

L X

1−j 2

zˆ` (pj−1 ξ) gˆ` (pj−1 ξ).

(5.2.12)

`=0

Thus, in order to show that (5.2.6) holds, it suffices to verify only the equality (5.2.12). R.H.S. = q

1−j 2

L X

zˆ` (pj−1 ξ) gˆ` (pj−1 ξ)

`=0

= q

1−j 2

L X

zˆ` (pj−1 ξ) m` (pj ξ)ˆ g` (pj ξ)

`=0

= q

1−j 2

gˆ(pj ξ)

"

L X

q −1

`=0 q−1

= q

−j/2

j

gˆ(p ξ)

= q −j/2 gˆ(pj ξ) = q

−j/2

j

X

q−1 X

# zˆ pξ + pu(r) m` pξ + pu(r) m` (pj ξ)

r=0 L X zˆ pξ + pu(r) m` pξ + pu(r) m` (pj ξ)

r=0 q−1

`=0

X

zˆ pξ + pu(r) δr,0

r=0 j

gˆ(p ξ)ˆ z p ξ = L.H.S.

(c). For the proof of the part (c) of the Splitting lemma, it is sufficient to verify equation (5.2.7) only. The equations (5.2.8) and (5.2.9) are direct consequences of equations (5.2.5) and (5.2.6) which have been proved. Moreover, from equation (5.2.4), we have X z` u(k) = q 1/2 m` u(n) − p−1 u(k) z u(n) n∈N0

= q 1/2

X

m` u(n) − p−1 u(k) hf, gj,n i

n∈N0

* f, q 1/2

= =

+ X

m` u(n) − p−1 u(k) gj,n

n∈N0 ` f, gj−1,k ,

` = 0, 1, . . . , L.

(d). This is immediate from equations (5.2.2) and (5.2.3). 79

5.3 TFP for L2 (K)

5.3

TFP for L2(K)

In the following section, we construct tight framelet packets for L2 (K) via MRA generated by the framelet symbols . To do this, let {ψ` , m` }L`=0 satisfy the conditions of the unitary extension principle and ω0 = φ. Define the functions ωn (x), n = 0, 1, 2, . . . , associated with the refinable function φ recursively by ` = 0, 1, . . . , L, r ∈ N0 .

ω ˆ n (ξ) = ω ˆ (L+1)r+` (ξ) = m` (pξ) ωr (pξ),

(5.3.1)

Note that for r = 0 and ` = 0, 1, . . . , L, we have ω ˆ ` (ξ) = m` (pξ) ω0 (pξ) = m` (pξ)φ(pξ),

(5.3.2)

which shows that ω` (x) = ψ` (x), ` = 0, 1, . . . , L. For n ∈ N0 , define a family of subspaces of L2 (K) by Un = span ωn,0,k : k ∈ N0 .

(5.3.3)

Clearly U0 = V0 and U` = W0,` , for ` = 1, . . . , L. Since FΨ is a tight wavelet frame constructed via UEP in an MRA generated by φ. Therefore, we have X 2 ω ˆ 0 ξ + u(k) ≤ 1, ξ ∈ K. n∈N0

By invoking Lemma 5.2.1, for n = 1, 2, . . . , we obtain (L+1)(n+1)−1

X 2 ω ˆ n ξ + u(k) ≤ 1,

Un1

=

k∈N0

X

Ut ,

t=(L+1)n

and for any f ∈ L2 (K), (L+1)(n+1)−1 X X X hf, ωn,1,k i 2 = hf, ωt,0,k i 2 . k∈N0

t=(L+1)n

k∈N0

A repeated application of the Splitting Lemma 5.2.1 for j = 1, 2, . . . , yields (L+1)j (n+1)−1

Unj

=

X t=(L+1)j n

80

Ut

(5.3.4)

5.3 TFP for L2 (K)

and for any f ∈ L2 (K) j

(n+1)−1 X X 2 (L+1)X hf, ωn,j,k i = hf, ωt,0,k i 2 . t=(L+1)j n

k∈N0

(5.3.5)

k∈N0

Substituting n = 0 in equations (5.3.4) and (5.3.5), we get (L+1)j −1

X

Vj =

Ut

(5.3.6)

t=0

and

j

(L+1) −1 X X X hf, ωt,0,k i 2 hf, φj,k i 2 = t=0

k∈N0

(5.3.7)

k∈N0

for any f ∈ L2 (K), respectively. Moreover, for n = `, where ` = 1, . . . , L, equations (5.3.4) and (5.3.5) yield (L+1)j (`+1)−1

Wj,` =

j W0,`

=

U`j

X

=

Ut ,

(5.3.8)

t=(L+1)j `

and for any f ∈ L2 (K) j

(`+1)−1 X X X 2 (L+1)X ` 2 hf, ψj,k hf, ωt,0,k i 2 . i = hf, ω`,j,k i = k∈N0

t=(L+1)j `

k∈N0

(5.3.9)

k∈N0

From equation (5.3.9), it follows that each wavelet space Wj,` , j ∈ N0 , ` = 1, . . . , L can be further splitted into (L + 1)j subspaces Ut , t ∈ [(L + 1)j `, (L + 1)j (` + 1) − 1]. If we keep the parameter j fixed, say J > 0, we will obtain (L+1)J −1 2

L (K) =

X

Ut +

t=0

L X X

Wj,` .

(5.3.10)

`=1 j≥J

Theorem 5.3.1. Let FΨ be a tight wavelet frame constructed via UEP in an MRA and m1 , m2 , . . . , mL are the framelet symbols satisfying the UEP condition (5.1.1). Let {ωn : n ∈ N0 } be defined as in (5.3.1). Then for any fixed J > 0, the family of functions n o[n o J ` F = ωn,0,k : 0 ≤ n ≤ (L + 1) − 1, k ∈ N0 ψj,k : 1 ≤ ` ≤ L, j ≥ J, k ∈ N0 forms a tight frame for L2 (K). 81

5.3 TFP for L2 (K)

Proof. By Theorem 5.1.1, the wavelet system FΨ constitutes a tight wavelet frame for L2 (K). Therefore by equation (5.3.6), we have for any f ∈ L2 (K) L X X X X

2

2 `

f = hf, φ0,k i 2 + f, ψj,k 2 `=1 j∈Z k∈N0

k∈N0

=

L X X X X

2 ` hf, φJ,k i 2 + f, ψj,k `=1 j≥J k∈N0

k∈N0 (L+1)J −1

=

X

L X X X X

2 ` . hf, ωn,0,k i 2 + f, ψj,k

n=0

n∈N0

`=1 j≥J k∈N0

This completes the proof. Definition 5.3.2. The functions {ωn : n ∈ N0 } are called as the basic framelet packets on the local field K of positive characteristic associated with the refinable function φ. With the help of basic framelet packets, we are now in a position to construct a class of tight frames for L2 (K) by choosing other L2 (K) space decompositions. For simplicity, let us consider a disjoint partition ΥJ of a finite set of non-negative integers n o ΩJ = r ∈ N0 : 0 ≤ r ≤ (L + 1)J − 1 into disjoint of the form n o Λj,n = (L + 1)j n, . . . , (L + 1)j (n + 1) − 1 ,

(5.3.11)

j, n ∈ N0 ,

i.e., n

ΥJ = Λj,n :

[

o

Λj,n = ΩJ ,

(5.3.12)

Then, it follows from equations (5.3.4) and (5.3.9) that (L+1)J −1 2

L (K) =

X

Ut +

L X X

t=0

Wj,`

`=1 j≥J (L+1)J (n+1)−1

=

=

X

X

ΛJ,n ∈ΥJ

t=(L+1)j n

X ΛJ,n ∈ΥJ

Utj

+

Ut +

L X X `=1 j≥J

82

L X X `=1 j≥J

Wj,` .

Wj,`

5.4 TFP on LFPC via MRA Space VJ

Theorem 5.3.3. Suppose FΨ is a tight wavelet frame constructed via UEP in an MRA and m1 , m2 , . . . , mL are the framelet symbols satisfying the UEP condition (5.1.1). Let {ωn : n ∈ N0 } be defined as in equation (5.3.1). For any fixed J > 0, ΥJ is a partition of ΩJ , where ΩJ and ΥJ are defined in equations(5.3.11) and (5.3.12), respectively. Then the family of functions n o[n o ` FΥJ = ωn,0,k : ΛJ,n ∈ ΥJ , k ∈ N0 ψj,k : 1 ≤ ` ≤ L, j ≥ J, k ∈ N0 constitutes a tight frame for L2 (K). Proof. For any arbitrary f ∈ L2 (K), we have (L+1)j (n+1)−1

X

X hf, ωn,j,k i 2 =

ΛJ,n ∈ΥJ k∈N0

X

X

ΛJ,n ∈ΥJ

n=(L+1)j n

X hf, ωn,0,k i 2 k∈N0

(L+1)J −1

=

X

hf, ωn,0,k i 2 .

n=0

By invoking Theorem 5.3.1, we get the desired result.

5.4

TFP on LFPC via MRA Space VJ

Besides the recursive derivation of tight framelet packets introduced in above section, tight framelet packets can also be constructed by decomposing the MRA space VJ directly for a fixed level J > 0 to the level 0. At the first level of decomposition, by Lemma 5.2.1, VJ is decomposed into the L + 1 spaces WJ−1,r , r ∈ ∆1 where n o ∆1 = r = (rJ , rJ−1 , . . . , r1 ) : 0 ≤ rJ ≤ L, rJ−1 = · · · = r1 = 0 . For this choice of r = (rJ , rJ−1 , . . . , r1 ), we define r(n) = rn , ωr (x) = q 1/2

n = 1, 2, . . . , J, X hr(1) φ p−1 x − u(n) , n n∈N0

83


and WJ−1,r := span ωr,J−1,k : k ∈ N0 . Therefore, for any f ∈ L2 (K), we have X

|hf, φJ,k i|2 =

X X

f, ωr,J−1,k 2 . r∈∆1 k∈N0

k∈N0

At the second level of decomposition, by Lemma 5.2.1, each space WJ−1,r , r ∈ ∆1 is decomposed with the constructed mask M into spaces WJ−2,r0 , r0 ∈ ∆r2 , where ∆r2 is a subset of ∆2 defined by ∆r2 = r0 ∈ ∆2 : r0 (1) = r(1) and ∆2 is a J-tuple index set defined by n o ∆2 = r = (rJ , rJ−1 , . . . , r1 ) : 0 ≤ rJ−1 , rJ ≤ L, rJ−2 = · · · = r1 = 0 , ωr0 (x) = q 1/2

X

0 hrn (2) φ p−1 x − u(n) ,

n∈N0

WJ−2,r0 := span ωr0 ,J−2,k : k ∈ N0 . Thus, for any f ∈ L2 (K), we have X X X hf, ωr0 ,J−2,k i 2 . hf, ωr,J−1,k i 2 = r0 ∈∆r2 k∈N0

k∈N0

Finally, at the m-th level (2 ≤ m ≤ J) of decomposition, by Lemma 5.2.1, each space WJ−m+1,r , r ∈ ∆m−1 is decomposed with the constructed mask M into spaces WJ−m,r0 , r0 ∈ ∆rm , where ∆rm is a subset of ∆m defined by n o r 0 0 ∆m = r ∈ Λm : r (n) = r(n), for 1 ≤ n ≤ m − 1 and ∆m is a J-tuple index set defined by n o ∆m = r = (rJ , rJ−1 , . . . , r1 ) : 0 ≤ rJ−m ≤ L, rJ−m = · · · = r1 = 0 , ωr0 (x) = q 1/2

X

0 hrn (m) φ p−1 x − u(n) ,

n∈N0

84


WJ−m,r0 := span ωr0 ,J−m,k : k ∈ N0 . Therefore for any f ∈ L2 (K), we have X

X X

f, ωr0 ,J−m,k 2 . f, ωr,J−m+1,k 2 = r0 ∈∆rm k∈N0

k∈N0

In particular, at the J-th level of decomposition, by Lemma 5.2.1, each space W1,r , r ∈ ∆J−1 is decomposed with M into spaces W0,r0 , r0 ∈ ∆rJ , where ∆rJ is a subset of ∆J defined by n o ∆rJ = r0 ∈ ∆J : r0 (n) = r(n), for 1 ≤ n ≤ J − 1 and ∆J is a J-tuple index set defined by n o ∆J = r = (rJ , rJ−1 , . . . , r1 ) : 0 ≤ rt ≤ L, 1 ≤ t ≤ J , ωr0 (x) = q 1/2

X

(5.4.1)

0 hrn (J) φ p−1 x − u(n) ,

n∈N0

W0,r0

:= span ωr0 ,0,k : k ∈ N0 .

Thus, for any f ∈ L2 (K), we have X

X X

f, ωr,1,k 2 = f, ωr0 ,0,k 2 . r0 ∈ΛrJ k∈N0

k∈N0

Combining all the inner product equations in the above construction, we get X X X hf, φJ,k i 2 = hf, ωr,0,k i 2 , for any f ∈ L2 (K). (5.4.2) k∈N0

r∈ΛJ k∈N0

In other words, we obtain another representation of VJ as n o VJ := span ωr,0,k : r ∈ ∆J , k ∈ N0 . Theorem 5.4.1. Suppose FΨ is a tight wavelet frame constructed via UEP in an MRA and M = [m0 , m1 , . . . , mL ] is the combined mask satisfying the UEP condition (5.1.1). Then for any fixed J > 0, the family of functions n o[n o ` F = ωr,0,k : r ∈ ∆J ψj,k : ` = 1, . . . , L, j ≥ J, k ∈ N0 forms a tight frame for L2 (K), where ∆J is a index set defined in (5.4.1). 85


Proof. Since FΨ is a tight wavelet frame of L2 (K), then by (5.4.2), we have L X X X X

2

2 `

f = hf, φJ,k i 2 + f, ψj,k 2 `=1 j≥J k∈N0

k∈N0

=

L X X X X X

2 ` hf, ωr,0,k i 2 + f, ψj,k r∈∆J k∈N0

`=1 j≥J k∈N0

for any f ∈ L2 (K). This completes the proof. Similar to the recursive construction of tight framelet packets on local fields of positive characteristic, we can obtain tight framelet packets by performing various disjoint partitions ΓJ of ∆J with each partition separating ∆J into disjoint subsets of the form n o Ij,r = (rJ , . . . , rj+1 , rj0 , . . . , r10 ) ∈ ∆J : r = (rJ , . . . , rj+1 , 0, . . . , 0) ∈ ∆J−j , i.e., n o [ ΓJ = Ij,r : Ij,r = ∆J .

(5.4.3)

Corollary 5.4.2. Suppose FΨ is a tight wavelet frame constructed via UEP in an MRA and M = [m0 , m1 , . . . , mL ] is the combined mask satisfying the UEP condition (5.1.1). Let ΓJ be a disjoint partition of ∆J , where ∆J and ΓJ are defined in (5.4.1) and (5.4.3), respectively. Then the collection n o[n o ` FΓJ = ωr,j,k : Ij,r ∈ ΓJ , k ∈ N0 ψj,k : ` = 1, . . . , L, j ≥ J ∈ Z, k ∈ N0 generates a tight frame for L2 (K). Proof. Since ΓJ is a disjoint partition of ∆J , for any f ∈ L2 (K), we have X X hf, ωr,j,k i 2 =

X X X

f, ωr0 ,0,k 2 Ij,r ∈ΓJ r0 ∈Ij,r k∈N0

Ij,r ∈ΓJ k∈N0

=

X X hf, ωr,0,k i 2 . r∈∆J k∈N0

By applying Theorem 5.4.1, Corollary 5.4.2 is proved.

86

Chapter 6 Nonuniform Wavelets/Vector-valued Wavelets Multiresolution analysis is an important mathematical tool since it provides a natural framework for understanding and constructing discrete wavelet systems. The concept of MRA has been extended in various ways in recent years. These concepts are generalized to L2 Rd , to lattices different from Zd , allowing the subspaces of MRA to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer M ≥ 2 or by an expansive matrix A ∈ GLd (R) as long as A ⊂ AZd . All these concepts are developed on regular lattices, that is the translation set is always a group. Recently, Gabardo and Nashed [81, 82, 83, 84] considered a generalization of Mallat’s [140] celebrated theory of MRA based on spectral pairs, in which the translation set acting on the scaling function associated with the MRA to generate the subspace V0 is no longer a group, but is the union of Z and a translate of Z. More results in this direction can be found in Refs. [143, 164, 174]. Recently, Shah and Abdullah [164] have generalized the concept of multiresolution analysis on Euclidean spaces Rn to nonuniform multiresolution analysis on local fields of positive characteristic, in which the translation set acting on the

87

Nonuniform Wavelets/Vector-valued Wavelets

scaling function associated with the multiresolution analysis to generate the subspace V0 is no longer a group, but is the union of Z and a translate of Z, where Z = {u(n) : n ∈ N0 } is a complete list of (distinct) coset representation of the unit disc D in the locally compact Abelian group K + . More precisely, this set is of the form Λ = {0, r/N } + Z, where N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime. They call this a nonuniform multiresolution analysis on local fields of positive characteristic. On the other hand, Xiang-Gen Xia and Suter [194] introduced the concept of vector-valued multiresolution analysis and orthogonal vector-valued wavelet basis and showed that vector-valued wavelets are a class of generalized multiwavelets. Chen and Cheng [33] presented the construction of a class of compactly supported orthogonal vector-valued wavelets and investigated the properties of vector-valued wavelet packets. Vector-valued wavelets are a class of generalized multiwavelets and multiwavelets can be generated from the component function in vector-valued wavelets. Vector-valued wavelets and multiwavelets are different in the following sense. Vector-valued wavelets can be used to decorrelate a vector-valued signal not only in the time domain but also between components for a fixed time where as multiwavelets focuses only on the decorrelation of signals in time domain. Moreover, prefiltering is usually required for discrete multiwavelet transform but not necessary for discrete vector-valued wavelet transforms. We generalize the concept of vector-valued MRA on Euclidean spaces Rn to vector-valued nonuniform multiresolution analysis (VNUMRA) on local fields of positive characteristic, in which the translation set acting on the scaling vector associated with the MRA to generate the subspace V0 is no longer a group, but is the union of Z and a translate of Z, where Z = {u(n) : n ∈ N0 } is a complete list of (distinct) coset representation of the unit disc D in K + . We call this a VNUMRA on local fields of positive characteristic. As a consequence of this generalization, we obtain a necessary and sufficient condition for the existence of associated wavelets. Since the Cantor dyadic group and the Vilenkin groups are examples of local fields of positive characteristic, these results also hold for these groups.

88

6.1 Nonuniform Multiresolution Analysis on LFPC

This Chapter is organised as follows. In Section 6.1, we introduce the notion of nonuniform wavelet packets on local field K and prove that they generate an orthonormal basis for L2 (K). Here, we also examine their properties by means of the Fourier transform. In Section 6.3, we introduce the notion of VNUMRA on local field K of positive characteristic p > 0 and establish a necessary and sufficient condition for the existence of associated wavelet. In Section 6.4, we construct a VNUMRA on local field K of positive characteristic starting from a vector refinement mask G(ξ) satisfying appropriate conditions.

6.1

Nonuniform

Multiresolution

Analysis

on

LFPC Let Z = {u(n) : n ∈ N0 }, where {u(n) : n ∈ N0 } is a complete list of (distinct) coset representation of D in K + . For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ qN − 1 such that r and N are relatively prime, we define n ro Λ = 0, + Z. N

(6.1.1)

It is easy to verify that Λ is not a group on local field K, but is the union of Z and a translate of Z. In this set up, Shah and Abdullah [164] formulated the notion of multiresolution analysis on local field of positive characteristic, which is called nonuniform multiresolution analysis (NUMRA) and is based on the theory of spectral pairs. We first recall the definition of a NUMRA on local fields of positive characteristic (as defined in [164]) and associated set of wavelets: Definition 6.1.1. For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ qN − 1 such that r and N are relatively prime, an associated nonuniform multiresolution analysis on local field K of positive characteristic is a sequence of closed subspaces {Vj : j ∈ Z} of L2 (K) such that the following properties hold: (a) Vj ⊂ Vj+1 for all j ∈ Z; 89


(b)

S


j∈Z

(c)

T

Vj = {0};

j∈Z

(d) f (x) ∈ Vj if and only if f (p−1 N x) ∈ Vj+1 for all j ∈ Z; (e) There exists a function φ in V0 such that {ϕ(x − λ) : λ ∈ Λ}, is a complete orthonormal basis for V0 . It is worth noticing that, when N = 1, one recovers from the definition above the definition of a multiresolution analysis on local fields of positive characteristic p > 0. When, N > 1, the dilation is induced by p−1 N and |p−1 | = q ensures that qN Λ ⊂ Z ⊂ Λ. For every j ∈ Z, define Wj to be the orthogonal complement of Vj in Vj+1 . Then we have Vj+1 = Vj ⊕ Wj

and W` ⊥ W`0

if ` 6= `0 .

(6.1.2)

It follows that for j > J, j−J−1

Vj = VJ ⊕

M

Wj−`

(6.1.3)

`=0

where all these subspaces are orthogonal. By virtue of condition (b) in the Definition 6.1.1, this implies L2 (K) =

M

Wj ,

(6.1.4)

j∈Z

a decomposition of L2 (K) into mutually orthogonal subspaces. As in the standard case, one expects the existence of qN −1 number of functions so that their translation by elements of λ and dilations by the integral powers of p−1 N form an orthonormal basis for L2 (K). Definition 6.1.2. A set of functions {ψ1 , ψ1 , . . . , ψqN −1 } in L2 (K) is said to be a set of basic wavelets associated with the nonuniform multiresolution analysis {Vj : j ∈ Z} if the family of functions {ψ` (x − λ) : 1 ≤ ` ≤ qN − 1, λ ∈ Λ} forms an orthonormal basis for W0 . 90


We denote ψ0 = φ, the scaling function, and consider qN − 1 functions ψ` , 1 ≤ ` ≤ qN − 1, in W0 as possible candidates for wavelets. Since (1/qN ) ψ` (p/N x) ∈ V−1 ⊂ V0 , it follows from property (d) of Definition 6.1.1 that for each `, 0 ≤ ` ≤ X 2 h`λ < ∞ such that qN − 1, there exists a sequence h`λ : λ ∈ Λ with λ∈Λ

px X 1 ψ` = h`λ φ(x − λ). qN N λ∈Λ

(6.1.5)

On taking Fourier transform, we obtain ˆ ψˆ` p−1 N ξ = m` (ξ) φ(ξ),

(6.1.6)

where m` (ξ) =

X

h`λ χ (λ, ξ),

(6.1.7)

λ∈Λ

are the integral periodic functions in L2 (D) and are called the wavelet symbols. In view of the specific form of Λ, we observe that r m` (ξ) = m1` (ξ) + χ , ξ m2` (ξ), N

0 ≤ ` ≤ qN − 1,

(6.1.8)

where m1` and m2` are locally L2 -periodic functions. Therefore, by the scaling property of the wavelet spaces Wj ’s and (6.1.4), it is clear that the family of functions

(qN )j/2 ψ` (p−1 N )j x − λ : 1 ≤ ` ≤ qN − 1, λ ∈ Λ

(6.1.9)

forms an orthonormal basis for L2 (K). In fact, it was shown in [[164], Lemma 3.3] that the orthonormality of the system ψk x − λ : 1 ≤ k ≤ qN − 1, λ ∈ Λ is equivalent to the following two conditions: qN −1

X s=0

p 2 ξ + pu(s) ξ + pu(s) + mk ξ + pu(s) N N N p 2 × m` ξ + pu(s) = δk,` , N qN −1 p p X r χ , pu(s) m1k ξ + pu(s) m1` ξ + pu(s) N N N s=0 p 2 p 2 + mk ξ + pu(s) m` ξ + pu(s) = 0, N N

m1k

p

m1`

p

91

6.2 Nonuniform Wavelet Packets on LFPC

for 0 ≤ k, ` ≤ qN − 1. As we know that the classic technique involved for the construction of wavelet packets is through splitting the wavelet spaces Wj successively into finite number of orthogonal sub-spaces. This splitting is carried out by the following lemma, whose proof is similar to that of [[164] Lemma 3.3]. Lemma 6.1.3. Let φ ∈ L2 (K) be such that {φ(x − λ) : λ ∈ Λ} is an orthonor mal system in L2 (K) and let V = span (qN )1/2 φ (p−1 N )x − λ : λ ∈ Λ . Let ψ` and m` (ξ) be the functions defined by (6.1.6) and (6.1.8), respectively. Then, {ψ` (x − λ) : 1 ≤ ` ≤ qN − 1, λ ∈ Λ} is an orthonormal system if and only if m1` and m2` satisfy (6.1.10) and (6.1.10). Furthermore, this system is an orthonormal basis for V if and only if it is orthonormal. Corollary 6.1.4. Let {Eλ : λ ∈ Λ} be an orthonormal basis of a separable Hilbert space H, and m` , 0 ≤ ` ≤ qN − 1, be as in Lemma 6.1.3 satisfying (6.1.10) and (6.1.10). Define Fσ` = (qN )1/2

X

hλ−qN σ Eλ ,

0 ≤ ` ≤ qN − 1, σ ∈ Λ.

λ∈Λ

Then, Fσ` : σ ∈ Λ is an orthonormal basis for its closed linear span H` and H = LqN −1 `=0 H` .

6.2

Nonuniform Wavelet Packets on LFPC

It is well known that the classical orthonormal wavelet bases have poor frequency localization. For example, if the wavelet ψ is band limited, then the measure of the ˆ To overcome this disadvantage, Coifman supp of (ψj,k )∧ is 2j -times that of supp ψ. et al. [51] introduced the notion of orthogonal univariate wavelet packets. Well known Daubechies orthogonal wavelets are a special of wavelet packets. Chui and Li [38] generalized the concept of orthogonal wavelet packets to the case of nonorthogonal wavelet packets so that they can be employed to the spline wavelets and so on. Shen [182] generalized the notion of univariate orthogonal wavelet packets to 92


the case of multivariate wavelet packets. The construction of wavelet packets and wavelet frame packets on local fields of positive characteristic were recently reported by Behera and Jahan in [14]. They proved lemma on the so-called splitting trick and several theorems concerning the Fourier transform of the wavelet packets and the construction of wavelet packets to show that their translates form an orthonormal basis of L2(K). Other notable generalizations are the vector-valued wavelet packets [34], wavelet packets and framelet packets related to the Walsh polynomials [159, 160, 163] and M-band framelet packets [169, 177]. Motivated and inspired by the concept of nonuniform multiresolution analysis on local fields of positive characteristic, we construct the associated orthogonal wavelet packets for such an MRA. More precisely, we show that the collection of all dilations and translations of the wavelet packets is an overcomplete system in L2 (K). Finally, we investigate certain properties of the nonuniform wavelet packets on local fields by virtue of the Fourier transform. Let {Vj : j ∈ Z} be an NUMRA with the scaling function φ. Then there exˆ ˆ ists a function m0 such that φ(ξ) = m0 (pξ/N )φ(pξ/N ), where m0 (ξ) = m10 (ξ) + χ (r/N, ξ) m20 (ξ). Applying the splitting Lemma 6.1.3 to the space V1 , we get functions Γ` , ` = 0, 1, . . . , qN − 1, where ˆ ` (ξ) = m` Γ

pξ N

pξ φˆ , N

(6.2.1)

such that {Γ` (x − λ) : 1 ≤ ` ≤ qN − 1, λ ∈ Λ} forms an orthonormal basis for V1 . For ` = 0, we obtain the scaling function i.e., Γ0 = φ and for ` = 1, . . . , qN − 1, we have the basic wavelets Γ` = ψ` . For n ≥ 0, the basic nonuniform wavelet packets associated with a scaling function φ on a local fields of positive characteristic are defined recursively by Γn (x) = ΓqN γ+σ (x) = (qN )1/2

X

hσλ Γγ p−1 N x − λ ,

0 ≤ σ ≤ qN − 1, (6.2.2)

λ∈Λ

where γ ∈ N0 is the unique element such that n = (qN )γ + σ, 0 ≤ σ ≤ qN − 1 holds.

93


By implementing the Fourier transform for the both sides of (6.2.2), we have pξ ˆ pξ ∧ (ΓqN γ+σ ) (ξ) = mσ , 0 ≤ σ ≤ qN − 1. (6.2.3) Γγ N N Next, we obtain an expression for the Fourier transform of the nonuniform wavelet packets in terms of the wavelet masks m` as: Proposition 6.2.1. Let {Γn : n ≥ 0} be the basic nonuniform wavelet packets constructed above and n=

j−1 X

εk (qN )k ,

0 ≤ εk ≤ qN − 1, εj 6= 0

(6.2.4)

k=0

be the unique expansion of the integer n in the base qN . Then j j 2 p p p p ˆ Γn (ξ) = mε1 ξ mε2 ξ . . . mεj ξ φˆ ξ . N N N N

(6.2.5)

Proof. If an integer n has an expansion of the form (6.2.4), then we say that it is of length j. We use induction on the length of n to prove the proposition. Since Γ0 = φ is the scaling function and Γσ = ψσ , 1 ≤ σ ≤ qN −1, are the basic nonuniform wavelets, it follows from (6.2.1) that the claim is true for all n of length 1. Assume that it holds for all integers of length j. Then an integer m of length j + 1 is of the form m = σ + (qN )n, where 0 ≤ σ ≤ qN − 1, and n has length j. Therefore, we have m = σ + (qN )n = σ +

j X

εk (qN )k .

k=1

Suppose n has the expansion of the form (6.2.4). Then, from (6.2.3) and (6.2.5), we have ∧ Γσ+(qN )n (ξ) p p ˆn = mσ ξ Γ ξ N N j+1 j+1 p p 2 p p = mσ ξ mε1 ξ . . . mεj ξ φˆ ξ . N N N N

ˆ m (ξ) = Γ

This completes the proof. 94


For the construction of the wavelet packets, it is necessary to show that their translates form an orthonormal basis for L2 (K). This is evident from the following theorem. Theorem 6.2.2. Let {Γn : n ≥ 0} be the basic nonuniform wavelet packets associated with the nonuniform multiresolution analysis {Vj : j ∈ Z}. Then, (i)

Γn x − λ : (qN )j ≤ n ≤ (qN )j+1 − 1, λ ∈ Λ is an orthonormal basis of

Wj , j ≥ 0. (ii)

Γn x − λ : 0 ≤ n ≤ (qN )j − 1, λ ∈ Λ is an orthonormal basis of Vj , j ≥ 0.

(iii)

Γn x − λ : n ≥ 0, λ ∈ Λ is an orthonormal basis of L2 (K).

Proof. We prove the theorem by induction on j. Since {Γn : 1 ≤ n ≤ qN − 1} is the basic set of wavelets in W0 , so (i) is true for j = 0. Let us assume that it holds for j. We will prove it for j + 1. By our assumption, the family of func tions (qN )1/2 Γn (p−1 N )x − λ : (qN )j ≤ n ≤ (qN )j+1 − 1, λ ∈ Λ is an orthonormal basis of Wj+1 . Set n o En = span (qN )1/2 Γn (p−1 N )x − λ : λ ∈ Λ , so that

(qN )j+1 −1

Wj+1 =

M

En .

(6.2.6)

n=(qN )j

By applying the splitting Lemma 6.1.3 to En , we obtain p p ∧ ˆn ξ Γ ξ , 0 ≤ ` ≤ qN − 1, g`n (ξ) = m` N N

(6.2.7)

such that {g`n (x − λ) : 0 ≤ ` ≤ qN − 1, λ ∈ Λ} is an orthonormal basis of En . Now, if n has the expansion as in (6.2.4). Then, with the help of (6.2.5), we obtain 2 j+1 j+1 p ∧ p p p n g` (ξ) = m` ξ mε1 ξ . . . mεj ξ φˆ ξ . N N N N (6.2.8)

95

6.2 Nonuniform Wavelet Packets on LFPC ˆ m (ξ), where But the expression on the right-hand side of (6.2.8) is precisely Γ m = ` + (qN )ε1 + (qN )2 ε2 + · · · + (qN )j εj = ` + (qN )n. Hence, we get g`n = Γ`+(qN )n . Application of this fact together with equation (6.2.6) shows that n o Γ`+(qN )n (x − λ) : 0 ≤ ` ≤ qN − 1, (qN )j ≤ n ≤ (qN )j+1 − 1, λ ∈ Λ n o j+1 j+2 = Γn (x − λ) : (qN ) ≤ n ≤ (qN ) − 1, λ ∈ Λ , is an orthonormal basis of Wj+1 . Thus we have proved (i) for j +1 and the induction is complete. Part (ii) follows from the fact that Vj = V0 ⊕ W0 ⊕ · · · ⊕ Wj−1 , and (iii) from the decomposition (6.1.4). Definition 6.2.3. Let {Γn : n ≥ 0} be the basic wavelet packets associated with the nonuniform multiresolution analysis {Vj : j ∈ Z} of L2 (K). The family of functions o n j −1 j/2 (6.2.9) F = (qN ) Γn p N x − λ : n ≥ 0, j ∈ Z, λ ∈ Λ , will be called the general nonuniform wavelet packets corresponding to the nonuniform multiresolution analysis {Vj : j ∈ Z} of L2 (K). Next, we prove several decompositions of the wavelet subspaces Wj by virtue of a series of subspaces of nonuniform wavelets packets on local fields of positive characteristic. For n ∈ N0 and j ∈ Z, we define o n j Ujn = span (qN )j/2 Γn p−1 N x − λ : λ ∈ Λ ,

(6.2.10)

Since Γ0 is the scaling function and Γn , 1 ≤ n ≤ qN − 1 are the basic nonuniform wavelets, we observe that qN −1

Uj0

M

= Vj ,

Uj` = Wj ,

j ∈ Z,

`=1

so that the orthogonal decomposition Vj+1 = Vj ⊕ Wj can be written as qN −1 0 Uj+1

=

M

Uj` .

`=0

The following theorem decomposes

n Uj+1 ,

into qN orthogonal subspaces.

96

(6.2.11)


Theorem 6.2.4. For n ∈ N0 and j ∈ Z, we have qN −1 n Uj+1

=

M

`+(qN )n

Uj

.

(6.2.12)

`=0

Proof. By definition n Uj+1

n o j+1 (j+1)/2 −1 = span (qN ) Γn p N x−λ :λ∈Λ .

Let bλ (x) = (qN )(j+1)/2 Γn (p−1 N )j+1 x − λ , λ ∈ Λ. Then {bλ : λ ∈ Λ} forms an n orthonormal basis for the Hilbert space Uj+1 . For 0 ≤ ` ≤ qN − 1, we define X Fσ` (x) = (qN )1/2 h`λ−qN σ bλ (x), σ ∈ Λ, (6.2.13) λ∈Λ

and P` = span Fσ` : σ ∈ Λ . Then, by Corollary 6.1.4, we have qN −1 n Uj+1

=

M

P` .

`=0

Therefore, Eq. (6.2.13) becomes X Fσ` (x) = (qN )1/2 h`λ−qN σ bλ (x) λ∈Λ

X (qN )1/2 h`λ bλ+qN σ (x) = λ∈Λ

X j+1 = (qN )(j+2)/2 h`λ Γn p−1 N x − λ − p−1 N σ λ∈Λ

= (qN )j/2

X

(qN )h`λ Γn

p−1 N

p−1 N

j

x−σ −λ

λ∈Λ

= (qN )j/2 Γ`+qN n

p−1 N

j

x−σ .

Hence

qN −1

P` =

`+(qN )n Uj

and

n Uj+1

=

M `=0

This completes the proof. 97

`+(qN )n

Uj

.


The above decomposition can be used to obtain various decompositions of the wavelet subspaces Wj , j ≥ 0 as follows: Wj =

qN −1

(qN )2 −1

M

M

Uj` =

`=1

(qN )j+1 −1 ` Uj−1 = ··· =

M

U0` .

(6.2.14)

`=(qN )j

`=qN

Note that one can construct various orthonormal basis of L2 (K) by using (6.2.14). As we know L2 (K) = V0 ⊕ W0 ⊕ W1 ⊕ W2 ⊕ . . . . Therefore, for each j ≥ 0, we can choose any of the decomposition of Wj obtained above. For example, if we do not want to decompose any Wj , then we have the usual wavelet decomposition. On the other hand, if we prefer the last decomposition in (6.2.14) for each Wj , then we get the non-uniform wavelet packet decomposition. Let S ⊂ N0 × Z. We want to characterize the sets S such that the collection n o j FS = (qN )j/2 Γn p−1 N x − λ : λ ∈ Λ, (n, j) ∈ S , will form an orthonormal basis of L2 (K). In other words, we are searching those subsets S of N0 × Z for which M

Ujn = L2 (K).

(6.2.15)

(n,j)∈S

Theorem 6.2.5. Let {Γn : n ≥ 0} be the basic wavelet packets associated with the NUMRA {Vj : j ∈ Z} of L2 (K) and S ⊂ N0 × Z. Then FS is an orthonormal basis of L2 (K) if and only if {In,j : (n, j) ∈ S} is a partition of N0 , where In,j = ` ∈ N0 : (qN )j n ≤ ` ≤ (qN )j (n + 1) − 1 . Proof. Using decomposition (6.2.12) repeatedly, we obtain (qN )−1

Ujn

=

M

`+(qN )n

Uj−1

`=0 (qN )(n+1)−1

=

M `=(qN )n

98

` Uj−1


(qN )(n+1)−1

=





(qN )−1

M

M 

k+(qN )` 

Uj−2

k=0

`=(qN )n (qN )2 (n+1)−1

=

M

(qN )j (n+1)−1 ` Uj−2

M

= ··· =

`=(qN )2 n

U0` =

`=(qN )j n

M

U0` .

`∈In,j

Therefore, we have M

Ujn =

M

M

U0` .

(n,j)∈S `∈I(n,j)

(n,j)∈S

By Theorem 6.2.2 (iii), we get L2 (K) =

L

`∈N0

U0` . Hence (6.2.15) holds if and only

if {In,j : (n, j) ∈ S} is a partition of N0 , which completes the proof.

6.2.1

Orthogonal Properties of Nonuniform Wavelet Packets

In this Section, we investigate certain orthonormal properties of the nonuniform wavelet packets on LFPC by virtue of the Fourier transform. Lemma 6.2.6. Let f (x) be any function in L2 (K). Then, the system f (x − λ) : λ ∈ Λ is orthonormal if and only if X 2 ˆ f ξ + u(k) = 1

(6.2.16)

k∈N0

and X k∈N0

χ

r 2 , u(k) fˆ ξ + u(k) = 0. N

Proof. We have

f (x), f (x − λ)

Z =

fˆ(ξ)fˆ(ξ) χ(λ, ξ) dξ

K

X Z 2 ˆ = f ξ + u(k) χ λ, ξ + u(k) dξ. k∈N0 D

99

(6.2.17)


If λ ∈ N0 , we have

f (x), f (x − λ) =

# Z "X 2 ˆ f ξ + u(k) χ(λ, ξ) dξ. k∈N0

D

On taking λ =

r + u(m), m ∈ N0 , we obtain N

XZ

f (x), f (x − λ) =

2 r ˆ f ξ + u(k) χ + u(m), ξ + u(k) dξ N k∈N0 D # Z "X 2 r r = χ , u(k) fˆ ξ + u(k) χ , ξ χ u(m), ξ dξ. N N k∈N 0

D

Therefore, the system f (x − λ) : λ ∈ Λ is orthonormal if and only if the equalities (6.2.16) and (6.2.17) hold. This completes the proof. Lemma 6.2.7. If {ψ` (x − λ) : 1 ≤ ` ≤ qN − 1, λ ∈ Λ} are the basic orthonormal wavelets associated with a NUMRA {Vj : j ∈ Z}. Then X λ∈Λ

hkλ h`λqN −σ =

1 δk,` δ0,σ , qN

1 ≤ k, ` ≤ qN − 1, σ ∈ Λ.

(6.2.18)

Proof. By wavelet equation (6.1.5), we have Z X

2 ψk (x), ψ` (x − σ) = (qN ) hkλ φ p−1 N x − λ K λ∈Λ

×

X

h`ω φ (p−1 N x − p−1 N σ − ω) dx

ω∈Λ

= (qN )2

Z X

hkλ φ p−1 N x − λ

= (qN ) = (qN )

X

h`ω−qN σ φ (p−1 N x − ω) dx

ω∈Λ

K λ∈Λ

XX

X

hkλ h`ω−qN σ φ(x − λ), φ(x − ω)

λ∈Λ ω∈Λ

hkλ h`λ−qN σ .

λ∈Λ

which implies that (6.2.18) follows, hence completes the proof. 100


We are now in a position to investigate the orthogonal properties of the nonuniform wavelet packets on local fields of positive characteristic. Theorem 6.2.8. Let {Γn : n ∈ N0 } be the basic nonuniform wavelet packets associated with a NUMRA {Vj : j ∈ Z} of L2 (K). Then, we have

Γn (x), Γn (x − λ) = δ0,λ ,

λ ∈ Λ.

(6.2.19)

Proof. We prove this result by using induction on n. Since

Γ0 (x), Γ0 (x − λ) = φ(x), φ(x − λ) = δ0,λ , and hence, the claim is true for n = 0. Assume that (4.4) follows if 0 ≤ n ≤ (qN )r , r is a fixed positive integer. Then, for (qN )r ≤ n ≤ (qN )r+1 , we have (qN )r−1 ≤ [n/N ] ≤ (qN )r . Let n = N [n/N ] + s, s = 0, 1, . . . , qN − 1. In view of induction hypothesis and Lemma 6.2.6, we have X

2 ˆ [n/N ] ξ + u(k) = Γ[n/N ] (x), Γ[n/N ] (x − λ) = δ0,λ ⇔ Γ k∈N0

X k∈N0

 1,      

(6.2.20)

 r  2  ˆ  , u(k) Γ[n/N ] ξ + u(k) = 0.  χ  N

By virtue of (6.2.3), we obtain 2 Z 2

pξ ˆ [n/N ] pξ χ(λ, ξ) dξ mσ Γ Γn (x), Γn (x − λ) = N N K 2 2 X Z pξ pξ ˆ χ(λ, ξ) dξ = m Γ σ [n/N ] N N k∈N0 qN (D+k) p X Z p 2 2 ˆ = m ξ + u(k) Γ ξ + u(k) σ [n/N ] N N k∈N 0qN D

p ×χ λ, ξ + u(k) dξ. N

In view of the specific form of Λ, we can write

101


mσ

p N

2 ξ + u(k) (

) pξ p r = m1σ , + u(k) m2σ ξ + u(k) N N N N p p r pξ 1 2 × mσ ξ + u(k) + χ , + u(k) mσ ξ + u(k) N N N N 2 2 2 pξ 1 pξ pξ pξ r pξ r 2 1 + m + m m χ , χ , u(k) = mσ σ σ σ N N N N N N N pξ pξ r pξ r + m1σ m2σ χ , χ , u(k) . N N N N N p

ξ + u(k) + χ

If λ ∈ N0 , then by using (6.2.20), we obtain Z ( 2 2 ) 1 pξ

mσ + m2σ pξ χ(λ, ξ) Γn (x), Γn (x − λ) = N N qN D

p X 2 ˆ × ξ + u(k) dξ Γ[n/N ] N k∈N0 Z " pξ pξ r pξ 2 1 mσ mσ χ , χ(λ, ξ) + N N N N qN D

# r p 2 ˆ × χ , u(k) Γ ξ + u(k) dξ [n/N ] N N k∈N0 Z " pξ r pξ pξ 1 m2σ χ , χ(λ, ξ) + mσ N N N N qN D # p X r 2 ˆ χ × , u(k) Γ[n/N ] ξ + u(k) dξ N N k∈N0 ( ) qN −1 Z X 1 pξ 2 2 pξ 2 mσ + mσ χ(λ, ξ)dξ = N N s=0 X

sD

Z qN −1 X 2 2 p 2 1 p = ξ + u(s) + mσ ξ + u(s) mσ N N s=0 D

×χ(λ, ξ)dξ 102

6.2 Nonuniform Wavelet Packets on LFPC Z =

χ(λ, ξ)dξ D

= δ0,λ . Similarly, let λ =

r + u(m), m ∈ N0 , we obtain N Z qN −1 p X 2 r 1 χ , u(s) ξ + u(s) mσ N N s=0 D p 2 r 2 ξ + u(s) χ , ξ χ u(s), ξ dξ + mσ N N = 0.

Γn (x), Γn (x − λ) =

This completes the proof. Theorem 6.2.9. Let {Γn : n ∈ N0 } be the basic nonuniform wavelet packets associated with a NUMRA {Vj : j ∈ Z} of L2 (K). Then for every γ ∈ N0 , 0 ≤ k, ` ≤ qN − 1, we have

ΓqN γ+k (x), ΓqN γ+` (x − λ) = δk,` δ0,λ ,

λ ∈ Λ.

(6.2.21)

Proof. By Lemma 6.2.7 and Theorem 6.2.8, we have Z X

2 ΓqN γ+k (x), ΓqN γ+` (x − λ) = (qN ) hkλ Γγ p−1 N x − λ K λ∈Λ

×

X

h`ω Γγ (p−1 N x − p−1 N λ − ω) dx

ω∈Λ

= (qN )

XX

hkλ

h`ω−qN λ

λ∈Λ ω∈Λ

=

XX

Z Γγ (x − λ) Γγ (x − ω) dx K

D

E hkλ h`ω−qN λ Γγ (x − λ), Γγ (x − ω)

λ∈Λ ω∈Λ

= δk,` δ0,λ . Thus, the system Γn (x − λ) : n ∈ N0 forms an orthogonal system in L2 (K). This completes the proof.

103

6.3 Vector-valued NUMRA on LFPC

6.3

Vector-valued NUMRA on LFPC

In this section, we introduce the notion of vector-valued nonuniform multiresolution analysis(VNUMRA) on local field K of positive characteristic and establish a necessary and sufficient condition for the existence of associated wavelets. Let M be a constant and 2 ≤ M ∈ Z. By L2 K, CM , we denote the set of all vector-valued functions f i.e., o n T L2 K, CM = f (x) = f1 (x), f2 (x), . . . , fM (x) : x ∈ K, ft (x) ∈ L2 (K) , where t = 1, 2, . . . , M and T means the transpose of a vector. The space L2 K, CM

is called vector-valued function space on local field K of positive characteristic. For f ∈ L2 K, CM , f denotes the norm of vector-valued function f and is defined as: 1/2  Z M X

ft (x) 2 dx .

f =  2

(6.3.1)

t=1 K

For a vector-valued function f ∈ L2 K, CM , the integration of f is defined as: T

 Z

Z f (x)dx =  K

Z f1 (x)dx,

K

Z f2 (x)dx, . . . ,

K

fM (x)dx . K

Moreover, the Fourier transform of f is defined by Z ˆf (ξ) = f (x)χξ (x) dx. K

For any two vector-valued functions f , g ∈ L2 K, CM , their vector-valued inner product hf , gi is defined as: Z hf , gi =

f (x)g(x) dx.

(6.3.2)

K

With Λ = {0, r/N } + Z as defined above, we define the vector-valued nonuniform multiresolution analysis(VNUMRA) on local fields of positive characteristic as follows: 104


Definition 6.3.1. For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ qN − 1 such that r and N are relatively prime, a VNUMRA on local field K of positive characteristic is a sequence of closed subspaces {Vj : j ∈ Z} of L2 K, CM such that the following properties hold: (a) Vj ⊂ Vj+1 for all j ∈ Z; (b)

S

Vj is dense in L2 K, CM ;

j∈Z

(c)

T

Vj = {0}, where 0 is the zero vector of L2 K, CM ;

j∈Z

(d) Φ(x) ∈ Vj if and only if Φ(p−1 N x) ∈ Vj+1 for all j ∈ Z; (e) There exists a function Φ in V0 such that {Φ(x − λ) : λ ∈ Λ}, is a complete orthonormal basis for V0 . The vector valued function Φ is called a vector-valued scaling function of the VNUMRA. For every j ∈ Z, define Wj to be the orthogonal complement of Vj in Vj+1 . Then we have Vj+1 = Vj ⊕ Wj

if ` 6= `0 .

and W` ⊥ W`0

(6.3.3)

It follows that for j > J, j−J−1

Vj = VJ

M

M

! Wj−`

(6.3.4)

`=0

where all these subspaces are orthogonal. By virtue of condition (b) in the Definition 6.3.1, this implies M L2 K, CM = Wj ,

(6.3.5)

j∈Z

a decomposition of L2 K, CM into mutually orthogonal subspaces. As in the standard case, one expects the existence of qN −1 number of functions so that their translation by elements of Λ and dilations by the integral powers of p−1 N form an orthonormal basis for L2 K, CM . 105


Definition 6.3.2. A set of functions {Ψ1 , Ψ2 , . . . , ΨqN −1 } in L2 K, CM will be called a set of basic wavelets associated with a given VNUMRA if the family of functions {Ψ` (x − λ) : 1 ≤ ` ≤ qN − 1, λ ∈ Λ} forms an orthonormal basis for W0 . In

the

following, we want to seek a set of wavelet functions Ψ1 , Ψ2 , . . . , ΨqN −1 in W0 such that (qN )j/2 Ψ` (p−1 N )j x − λ : 1 ≤ ` ≤ qN − 1, λ ∈ Λ form an orthonormal basis of Wj . By the nested structure of VNUMRA, this task can be reduce to find Ψ` ∈ W0 such that Ψ` x − λ : 1 ≤ ` ≤ qN − 1, λ ∈ Λ constitutes an orthonormal basis of W0 .

Let Φ = φ1 , φ2 , . . . , φM

T

be a scaling vector of the given VNUMRA. Since

Φ ∈ V0 ⊂ V1 , there exist M × M constant matrix sequence {Gλ }λ∈Λ such that Φ(x) = (qN )1/2

X

Gλ Φ (p−1 N )x − λ .

(6.3.6)

λ∈Λ

Taking Fourier transform on both sides of equation (6.3.6), we obtain ˆ (ξ) = G pξ Φ ˆ pξ , Φ N N

(6.3.7)

where G(ξ) =

X 1 Gλ χ(λξ), (qN )1/2 λ∈Λ

is called symbol or vector refinement mask of the scaling function Φ. By replacing ξ by pξ/N in relation (6.3.7), we obtain 2 2 pξ p p ˆ ˆ Φ =G ξ Φ ξ , N N N and then ˆ Φ(ξ) =G

pξ N

2 2 p p ˆ G ξ Φ ξ . N N

We can continue this and obtain, for any n ∈ N, 2 p n p n pξ p ˆ ˆ Φ(ξ) = G G ξ ...G ξ Φ ξ N N N N n p n Y p m ˆ = Φ ξ G ξ . N N m=1 106

6.3 Vector-valued NUMRA on LFPC p n 1 By taking n → ∞ and noting that → 0 as n → ∞, the above = N (qN )n relation reduces to ∞ p m Y ˆ ˆ ξ . (6.3.8) Φ(ξ) = Φ(0) G N m=1 ˆ ˆ As usual, we assume Φ(ξ) is continuous at zero, and Φ(0) = IM , where IM denotes the identity matrix of order M × M . Therefore, equation (6.3.8) becomes ˆ Φ(ξ) =

∞ Y

G

m=1

p m ξ N

(6.3.9)

Moreover, it is immediate from equation (6.3.7) that G(0) = IM , which is essential ∞ m Q for convergence of the infinite product G Np ξ . m=1

We now investigate the orthogonal property of the scaling function Φ by means of the vector refinement mask G(ξ). Lemma 6.3.3. If Φ ∈ L2 K, CM defined by equation (6.3.6) is an orthogonal scaling function, then we have X Gu(m) Gp−1 N (λ−λ0 )+u(m) = qN δλ,λ0 IM ,

∀ λ, λ0 ∈ Λ,

(6.3.10)

m∈N0

where δλ,λ0 denotes the Kronecker’s delta. Proof. Since the scaling function is orthogonal vector-valued, we have Z δλ,λ0 IM = Φ(x − λ)Φ(x − λ0 )dx K

=

XZ

Gσ Φ p−1 N x − p−1 N λ − σ

σ∈Λ K

=

XX σ∈Λ σ∈Λ

X

Gσ Φ p−1 N x − p−1 N λ0 − σ dx

σ∈Λ

Gσ

 Z

Φ p−1 N x − p−1 N λ − σ



K

×Φ p−1 N x − p−1 N λ0 − σ

=

 Z

1 XX Gσ  qN σ∈Λ σ∈Λ

K

o

dxGσ   Φ x − p−1 N λ − σ Φ x − p−1 N λ0 − σ dxGσ .  107


Taking σ = u(m) and σ = u(n), where m, n ∈ N0 , we have D E 1 XX Gσ Φ x − p−1 N λ − σ , Φ x − p−1 N λ0 − σ Gσ qN σ∈Λ σ∈Λ 1 X X = Gu(m) qN m∈N n∈N 0 D0 E −1 −1 0 × Φ x − p N λ − u(m) , Φ x − p N λ − u(n) Gu(n) 1 X Gu(m) Gp−1 N (λ−λ0 )+u(m) . = qN m∈N

δλ,λ0 IM =

0

Therefore, identity (6.3.10) follows. Taking σ = m, n ∈ N0 , we have δ

λ,λ0

IM

r + u(m) and σ = u(n), where N

D E 1 XX −1 −1 0 Gσ Φ x − p N λ − σ , Φ x − p N λ − σ Gσ = qN σ∈Λ σ∈Λ 1 X X = G r +u(m) qN m∈N n∈N N 0 0 D E r − u(m) , Φ x − p−1 N λ0 − u(n) Gu(n) × Φ x − p−1 N λ − N 1 X = Gu(m) Gp−1 N (λ−λ0 )+u(m) . qN m∈N 0

Thus, in both the cases, we get the desired result. We denote Ψ0 = Φ, the scaling function, and consider qN −1 functions Ψ` , 1 ≤ ` ≤ qN − 1, in W0 as possible candidates for wavelets. Since (1/(qN )) Ψ` (p/N x) ∈ V−1 ⊂ V0 , it follows from property (d) of Definition 6.3.1 that for each `, 0 ≤ ` ≤ qN − 1, there exists a uniquely supported sequence Hλ` : λ ∈ Λ, 1 ≤ ` ≤ qN − 1 of M × M constant matrices such that Ψ` (x) = (qN )1/2

X

Hλ` Φ p−1 N x − λ .

(6.3.11)

λ∈Λ

On taking the Fourier transform on both sides of equation (6.3.11), we have ˆ ` p−1 N ξ = H` (ξ)Φ(ξ), ˆ Ψ

108

(6.3.12)


where X 1 (6.3.13) H ` χ(λξ). (qN )1/2 λ∈Λ λ n ro In view of the specific form of Λ = 0, + Z,, we observe that N r `,1 ξ Hλ`,2 (ξ), , 0 ≤ ` ≤ qN − 1, (6.3.14) H` (ξ) = Hλ (ξ) + χ N where Hλ`,1 : λ ∈ Λ and Hλ`,2 : λ ∈ Λ are M × M constant symmetric matrix H` (ξ) =

sequences. Lemma 6.3.4. Consider a vector-valued nonuniform multiresolution analysis on local field K as in Definition 6.3.1. Suppose that there exist qN −1 functions Ψk , k = 1, 2, . . . , qN − 1 in V1 . Then the family of functions

Ψk (x − λ) : λ ∈ Λ, k = 0, 1, . . . , qN − 1

(6.3.15)

forms an orthonormal system in V1 if and only if qN −1

X r=0

Hk

p

p ξ + pu(r) H` ξ + pu(r) = δk,` IM , N N

0 ≤ k, ` ≤ qN − 1. (6.3.16)

Proof. Firstly, we will prove the necessary condition. By the orthonormality of the system {Ψk (x − λ) : λ ∈ Λ, k = 0, 1, . . . , qN − 1}, we have D E Z Ψk (x − λ), Ψk0 (x − σ) = Ψk (x − λ) Ψ` (x − σ) dx = δk,` δλ,σ IM , K

where λ, σ ∈ Λ and k, ` ∈ {0, 1, 2, . . . , qN − 1}. Above relation can be recast in frequency domain as Z δk,` δλ,σ IM =

ˆ k (ξ) χ(λξ) Ψ ˆ ` (ξ) χ(σξ) dξ Ψ

K

Z =

ˆ k (ξ) Ψ ˆ ` (ξ) χ (λ − σ)ξ dξ. Ψ

K

109


Taking λ = u(m) and σ = u(n), where m, n ∈ N0 , we have Z ˆ k (ξ) Ψ ˆ ` (ξ)dξ δk,` δm,n IM = χ u(m) − u(n) ξ Ψ K

Z =

χ

X ˆ ` ξ + N u(j) dξ. ˆ k ξ + N u(j) Ψ u(m) − u(n) ξ Ψ j∈N0

ND

Define Fk,` (ξ) =

X

ˆ k ξ + N u(j) Ψ ˆ ` ξ + N u(j) , Ψ

0 ≤ k, ` ≤ qN − 1.

j∈N0

Then, we have Z δk,` δm,n IM =

χ u(m) − u(n) ξ Fk,` (ξ) dξ

ND

Z =

χ

u(m) − u(n)

(qN −1 X

Fk,` ξ + pu(s)

)

dξ,

s=0

pD

and

qN −1

X

Fk,` ξ + pu(s) = qδk,` IM .

(6.3.17)

s=0

On taking λ = Z 0 =

r + u(m) and σ = u(n), where m, n ∈ N0 , we obtain N

ˆ k (ξ) Ψ ˆ ` (ξ) dξ χ(λξ) χ(σξ) Ψ

K

Z

= = =

=

r ˆ k (ξ) Ψ ˆ ` (ξ) dξ χ u(m) + ξ χ u(n)ξ Ψ N K Z r ˆ k (ξ) Ψ ˆ ` (ξ) dξ χ u(m)ξ χ u(n)ξ χ ξ Ψ N K Z r χ u(m) − u(n) ξ χ ξ Fk,` (ξ)dξ N ND ( −1 ) Z X r r qN χ u(m) − u(n) ξ χ ξ χ p u(s) Fk,` ξ + pu(s) dξ. N N s=0

pD

110


We conclude that

qN −1

X s=0

χ

r p u(s) Fk,` ξ + pu(s) = 0. N

(6.3.18)

Also we have qN −1

X

X ˆ k ξ + pu(j) Ψ ˆ ` ξ + pu(j) . Fk,` ξ + pu(s) = Ψ

s=0

j∈N0

Therefore, equations (6.3.17) reduces to X

ˆ k ξ + pu(j) Ψ ˆ ` ξ + pu(j) = qδk,` IM . Ψ

(6.3.19)

j∈N0

Moreover, we have Fk,` p−1 N ξ

=

X

ˆ ` p−1 N ξ + N u(j) ˆ k p−1 N ξ + N u(j) Ψ Ψ

j∈N0

=

X

ˆ k p−1 N ξ + pu(j) Ψ ˆ ` p−1 N ξ + pu(j) Ψ

j∈N0

=

X

ˆ ξ + pu(j) H` ξ + pu(j) ˆ ξ + pu(j) Φ Hk ξ + pu(j) Φ

j∈N0

=

X

ˆ ξ + u(n)N + pu(0) Hk ξ + u(n)N + pu(0) Φ

j=n.qN +0

+

ˆ ξ + u(n)N + pu(0) H` ξ + u(n)N + pu(0) ×Φ X ˆ ξ + u(n)N + pu(1) Hk ξ + u(n)N + pu(1) Φ j=n.qN +1

ˆ ξ + u(n)N + pu(1) H` ξ + u(n)N + pu(1) ×Φ X +··· + Hk ξ + u(n)N + pu(qN − 1) j=n.qN +(qN −1)

ˆ ξ + u(n)N + pu(qN − 1) Φ ˆ ξ + u(n)N + pu(qN − 1) ×Φ ×H` ξ + u(n)N + pu(qN − 1) ( ) X ˆ ξ + u(n)N Φ ˆ ξ + u(n)N H` (ξ) = Hk (ξ) Φ j=n.qN

111

6.3 Vector-valued NUMRA on LFPC ( X

+Hk ξ + pu(1)

ˆ ξ + u(n)N + pu(1) Φ

j=n.qN +1

o ˆ × Φ ξ + u(n)N + pu(1) H` ξ + pu(1) +··· + Hk ξ + pu(qN − 1)

  

X

ˆ ξ + u(n)N + pu(qN − 1) Φ

j=n.qN +(qN −1)

o ˆ ξ + u(n)N + pu(qN − 1) H` ξ + pu(qN − 1) ×Φ n = q Hk (ξ)H` (ξ) + Hk ξ + pu(1) H` ξ + pu(1) + · · · + o Hk ξ + pu(qN − 1) H` ξ + pu(qN − 1) qN −1

= q

X

Hk ξ + pu(j) H` ξ + pu(j) .

j=0

Therefore, we have X j∈N0

qN −1 p X p ˆ k ξ + pu(j) Ψ ˆ ` ξ + pu(j) = q Ψ ξ + pu(j) H` ξ + pu(j) . Hk N N j=0

On using equation (6.3.19), we conclude that qN −1

X

Hk

j=0

p

p ξ + pu(j) H` ξ + pu(j) = δk,` IM . N N

In other words, we can say p p p p Hk ξ H` ξ + Hk ξ + pu(1) H` ξ + pu(1) + · · · + N N N N p p Hk ξ + pu(qN − 1) H` ξ + pu(qN − 1) = δk,` IM . N N Now we will prove the sufficiency. By equations (6.3.12), we have X ˆ k ξ + pu(j) Ψ ˆ ` ξ + pu(j) Ψ j∈N0

=

X j∈N0

Hk

p N

ξ + pu(j)

ˆ Φ

p

p ξ + pu(j) H` ξ + pu(j) N N ˆ ×Φ 112

p N

ξ + pu(j)

6.3 Vector-valued NUMRA on LFPC ) p ˆ ˆ Φ = Hk ξ + pu(1) Φ ξ + pu(1) N N N j=n.qN +1 p p ξ + pu(1) + · · · + Hk ξ + pu(qN − 1) × H`  N N   X  p p ˆ ˆ Φ ξ + pu(qN − 1) Φ ξ + pu(qN − 1) ×   N N j=n.qN +(qN −1) p × H` ξ + pu(qN − 1) N p p p p ξ H` ξ + Hk ξ + pu(1) H` ξ + pu(1) = q Hk N N N N p p + · · · + Hk ξ + pu(qN − 1) H` ξ + pu(qN − 1) N N = qδk,` IM . It proves the orthonormality of the system Ψk (x − λ) : λ ∈ Λ, k = 0, 1, . . . , qN − 1 . p

ξ + pu(1)

(

X

p

Theorem 6.3.5. Suppose {Ψk (x − λ) : λ ∈ Λ, k = 0, 1, . . . , qN − 1} is the system as defined in Lemma 6.3.4 and orthonormal in V1 . Then this system is complete in W0 ≡ V1 V0 . Proof. Since the system (6.3.15) is orthonormal in V1 . By Lemma 6.3.4 we have p p p p Hk ξ H` ξ + Hk ξ + pu(1) H` ξ + pu(1) + · · · + N N N N p p Hk ξ + pu(qN − 1) H` ξ + pu(qN − 1) = δk,` IM . N N We will now prove its completeness. For fk ∈ W0 , there exists constant matrices Pλk such that X fk (x) = (qN )1/2 Pλk Φ (p−1 N )x − λ , 0 ≤ k ≤ qN − 1. λ∈Λ

Above relation can be written in the frequency domain as p ˆfk (ξ) = Pk p ξ Φ ˆ ξ , N N where Pk (ξ) =

X 1 P k χ(λξ). (qN )1/2 λ∈Λ λ 113

(6.3.20)


On the other hand, fk ∈ / V0 and fk ∈ W0 implies Z fk (x)Φ(x − λ) dx = 0,

λ ∈ Λ.

K

This condition is equivalent to X

ˆfk ξ + pu(n) Φ ˆ ξ + pu(n) = 0,

ξ ∈ K.

n∈N0

Therefore, the identities (6.3.7) and (6.3.20) give for all ξ ∈ K, X

Pk

p

n∈N0

p p p ˆ ˆ ξ + pu(n) Φ ξ + pu(n) G ξ + pu(n) Φ ξ + pu(n) = 0. N N N N

As similar to the identity (6.3.16) in Lemma 6.3.4, we have for 0 ≤ k ≤ qN − 1 Pk

p p p p ξ G ξ + Pk ξ + pu(1) G ξ + pu(1) + · · · + N N N N p p (6.3.21) Pk ξ + pu(qN − 1) G ξ + pu(qN − 1) = 0. N N

Let p p p Pk 0 ξ = Pk ξ + pu(0) , . . . , Pk ξ + pu(qN − 1) , N N N ˜ p ξ = G p ξ + pu(0) , . . . , G p ξ + pu(qN − 1) , G N N N p p p Hk0 ξ = Hk ξ + pu(0) , . . . , Hk ξ + pu(qN − 1) . N N N Then, equation (6.3.16) implies that for any ξ ∈ K, the column vectors in qN M ×M ˜ and the column vectors in qN M × M matrix Hk0 are orthogonal for matrix G k = 0, 1, . . . , qN −1 and these vectors form an orthogonal basis of qN M dimensional complex Euclidean space CqN M . Equation (6.3.21) implies that the column vectors in qN M × M matrix Pk0 ˜ are orthogonal. Therefore, there and the column vectors of qN M × M matrix G exists an M × M matrix Qk (ξ) such that Pk (ξ) = Qk (ξ)Hk (ξ),

ξ ∈ K, 0 ≤ k ≤ qN − 1. 114


Therefore, from equations (6.3.12) and (6.3.20), we have p ˆfk (ξ) = Pk p ξ Φ ˆ ξ N p Np p ˆ = Pk ξ Hk ξ Φ ξ N N N p ˆ k (ξ). = Qk ξ Ψ N By using the orthonormality of the system (6.3.15), we have Z Z ˆfk p−1 N ξ ˆfk p−1 N ξ dξ = Qk (ξ)Ψ ˆ k p−1 N ξ Ψ ˆ k (p−1 N ξ) Qk (ξ) dξ. K

K

Therefore, we have Z

ˆfk p−1 N ξ ˆfk p−1 N ξ dξ = q

K

Z Qk (ξ) Qk (ξ) dξ. pD

This shows that Pk (ξ) has the Fourier series expansion and let the constant M × M matrices Rλk : λ ∈ Λ, k = 0, 1, . . . , qN − 1 be its Fourier coefficients. Therefore, we have fk (x) =

X

Rλk Ψk (x − λ).

λ∈Λ

It proves the completeness of the system {Ψk (x − λ)}λ∈Λ,

k=0,1,...,qN −1

in W0 and

hence completes the proof. If Ψ0 , Ψ1 , . . . , ΨqN −1 ∈ V1 are as in Lemma 6.3.4, one can obtain from them as orthonormal basis for L2 K, CM by following the standard procedure for construction of wavelet every j ∈ Z, the n from agiven MRA. It can be easily checked that for o

collection

j

j

(qN ) 2 Ψk (p−1 N ) x − λ : λ ∈ Λ, k = 0, 1, . . . , qN − 1

is a complete

orthogonal system for Vj+1 . Therefore, it follows immediately from equation (6.3.5) that the collection o n j j −1 2 (qN ) Ψk p N x − λ : λ ∈ Λ, k = 0, 1, . . . , qN − 1 forms a complete orthonormal system for L2 K, CM .

115

6.4 Sufficient Condition for the Scaling Function

6.4

Sufficient Condition for the Scaling Function

The main goal of this section is to construct a vector-valued nonuniform multiresolution analysis on local field K of positive characteristic starting from a vector-valued refinement mask G(ξ) of the form r +χ (6.4.1) G(ξ) = ξ G2λ (ξ), N where N > 1 is an integer and r is an odd integer with 1 ≤ r ≤ qN − 1 such that G1λ (ξ)

r and N are relatively prime and G1λ (ξ) and G2λ (ξ) are M × M constant symmetric matrix sequences. In other words, we establish conditions under which the solutions of scaling equation (6.3.6) generate a VNUMRA in L2 (K) or equivalently, we find a sufficient for the orthonormality of the system Φ(x − λ) : λ ∈ Λ , where Λ = {0, r/N } + Z. Therefore, the scaling vector Φ associated with given VNUMRA should satisfy the scaling identity ˆ Φ(ξ) =G

pξ N

ˆ Φ

pξ N

.

We further assume that: qN −1 X p p G ξ + pu(s) G ξ + pu(s) = IM . N N s=0

(6.4.2)

(6.4.3)

Theorem 6.4.1. Let G(ξ) be the vector-valued refinement mask associated with the vector-valued scaling function Φ of VNUMRA and satisfies the condition (6.4.3) together with G(0) = IM and G(ξ) = G(ξ), ∀ ξ ∈ K. Then, a sufficient condition for the collection {Φ(x − λ) : λ ∈ Λ} to be orthonormal in L2 K, CM is the existence of a constant C > 0 and of a compact set E ⊂ K that contains the neighbourhood of the origin such that p k G ξ ≥ C, N

∀ ξ ∈ K, k ∈ N.

(6.4.4)

Proof. Let us assume the existence of a constant C and of the compact set E ⊂ K with properties satisfied above. For any k ∈ N, we define ( k ) k Y p j p gk (ξ) = G ξ χE ξ . N N j=1 116

6.4 Sufficient Condition for the Scaling Function ˆ pointwise as k → ∞. As the interior of the compact set E contains 0, gk → Φ Therefore, there exists a constant B > 0 such that G(ξ) − G(0) ≤ B|ξ|, for all ξ ∈ K, and thus G(ξ) ≥ 1 − B|ξ|. Since E is bounded, we can find an integer k0 ∈ N such that B|ξ| < (qN )k , for k > k0 , ξ ∈ E and hence, there exists a constant C1 > 0 such that ˆ , χE (ξ) ≤ C1 Φ(ξ)

for all ξ ∈ K.

Thus, we have gk (ξ) ≤ C1

( k ) p j p k Y ˆ ˆ . G ξ Φ ξ = C1 Φ(ξ) N N j=1

Therefore, by Lebesgue dominated convergence theorem the sequence {gk } converges ˆ in L2 -norm. We will now compute by induction the integral to Φ Z gk (ξ) gk (ξ) χ(λ−σ) (ξ) dξ, where λ, σ ∈ Λ. K

For k = 1, we have Z Z p p p ξ G ξ χE ξ χ(λ−σ) (ξ) dξ g1 (ξ) g1 (ξ) χ(λ−σ) (ξ) dξ = G N N N K K Z = (qN ) G(ξ) G(ξ) χqN (λ−σ) (ξ) dξ E

(qN −1 X

Z

−1

= (qN )p

(p/qN )D

×G

p N

G

s=0

p N

ξ + pu(s)

ξ + pu(s) χs(λ−σ) (ξ) χqN (λ−σ) (ξ) dξ.

If λ − σ ∈ Z, then the expression in the brackets in the above integral is equal to IM by (6.4.3) and thus Z g1 (ξ) g1 (ξ) χ(λ−σ) (ξ) dξ = (qN )p−1 K

Z IM χqN (λ−σ) (ξ) dξ

(p/qN )D

= p−1

Z IM χ(λ−σ) (ξ) dξ pD

= δλ,σ IM . 117


On the other hand, if λ = u(m), σ = u(n) + r/N, where m, n ∈ N0 , then the same expression will vanish and the integral becomes Z g1 (ξ) g1 (ξ) χ(λ−σ) (ξ) dξ = 0. K

When k ≥ 2, we have Z gk (ξ) gk (ξ) χ(λ−σ) (ξ) dξ K

2 k k p p p 1 p ξ G ξ ...G ξ G ξ = G N N N N K k−1 1 p p pξ ×G ξ ...G ξ χE χ(λ−σ) (ξ) dξ N N N Z k−1 −1 k−2 = (qN )k G p−1 N ξ G p N ξ . . . G(ξ) Z

E

k−1 −1 ξ χ(λ−σ) (ξ) dξ ξ ...G p N × G(ξ) G ) (k−1 ) Z (k−1 Y Y ` ` −1 −1 k G p N ξ = (qN ) G p N ξ χ(p−1 N )k (λ−σ) (ξ) dξ p−1 N

`=0

E

= (qN )k

Z (k−1 Y

`=0

G

)

(k−1 ) Y ` ` p−1 N ξ G(ξ) G(ξ) G p−1 N ξ

`=1

E

`=1

×χ(p−1 N )k (λ−σ) (ξ) dξ = (qN )k p−1

Z

(p/qN )D

k −1

Z

= (qN ) p

(p/qN )D

(k−1 Y

G

)

` p−1 N ξ G(ξ) G(ξ)

`=1

(k−1 ) Y ` × G p−1 N ξ χ(p−1 N )k (λ−σ) (ξ) dξ `=1 (k−1 ) (qN −1 Y X ` G p−1 N ξ G p−1 N ξ + pu(s) s=0

`=1

o −1 × G ξ + pu(s) G p N ξ + pu(s) G ξ + pu(s) (k−1 ) Y ` −1 × G p N ξ χ(p−1 N )k (λ−σ) (ξ) dξ `=1

118


= (qN )k p−1

Z

(p/qN )D

(k−1 Y

G

(k−1 ) ) Y ` ` p−1 N ξ A(ξ) G p−1 N ξ

`=1

`=1

×χ(p−1 N )k (λ−σ) (ξ) dξ where A(ξ) =

(qN −1 X

) G p−1 N ξ + pu(s) G ξ + pu(s) G p−1 N ξ + pu(s) G ξ + pu(s) .

s=0

Since the refinement mask G(ξ) can be expressed as r G(ξ) = G1λ (ξ) + χ ξ G2λ (ξ). N Therefore, the above relation becomes

qN −1

r 2 −1 A(ξ) = +χ p N ξ + pu(s) Gλ (ξ) G ξ + pu(s) N s=0 r o n p−1 N ξ + pu(s) G2λ (ξ) G ξ + pu(s) × G1λ (ξ) + χ N 1 2 1 2 = Gλ (ξ) Gλ (ξ) Gλ (ξ) Gλ (ξ) = G p−1 N ξ G p−1 N ξ . X

G1λ (ξ)

Thus, we have

Z

(k−1 Y

Z

gk (ξ)gk (ξ) χ(λ−σ) (ξ) dξ = (qN )k

(qN )−1 E

K

×

G

` p−1 N ξ

)

`=1

(k−1 Y

G

) ` p−1 N ξ χ(p−1 N )k (λ−σ) (ξ) dξ

G

` p N ξ

G

) ` p−1 N ξ χ(p−1 N )k (λ−σ) (ξ) dξ

`=1

= (qN )

k−1

Z (k−2 Y `=1

E

×

(k−2 Y `=1

Z =

)

−1

gk−1 (ξ) gk−1 (ξ) χ(λ−σ) (ξ) dξ. K

119


Therefore for any k ∈ N, we have Z gk (ξ) gk (ξ) χ(λ−σ) (ξ) dξ = δλ,σ ,

λ, σ ∈ Λ.

K

Passing to the limit as k → ∞ and using Plancherel’s formula, we obtain Z Z ˆ Φ(ξ) ˆ Φ(x − λ)Φ(x − σ) dx = Φ(ξ) χ(λ−σ) (ξ) dξ = δλ,σ , λ, σ ∈ Λ K

K

which proves the desired orthonormality.

120

Publications

List of Publications 1. F. A. Shah and M. Y. Bhat, Vector-valued nonuniform multiresolution analysis on local fields, Int. J. Wavelets Multiresolut. Inf. Process. 13 (4) (2015) . 2. F. A. Shah and M. Y. Bhat, Semi-orthogonal wavelet frames on local fields, Analysis, 36(3) (2016), 173-181. 3. F. A. Shah and M. Y. Bhat, Biorthogonal wavelet packets on local fields of positive characteristic, Turkish J. Math. 40 (2016) 292-309. 4. F. A. Shah and M. Y. Bhat, Nonuniform wavelet packets on local fields of positive characteristic, Filomat, accepted (2016). 5. F. A. Shah and M. Y. Bhat, Vector-valued wavelet packets on local fields of positive characteristic, NewZealand J. Math., 46 (2016) 9-20. 6. M. Y. Bhat and B. A. Khandy, On the Fourier Transform of Wavelet Packets on Local Fields of Positive Characteristic, Int. J. Inn. Res. Dev., 5(10) (2016), 394-399. 7. F. A. Shah and M. Y. Bhat, Tight framelet packets on local fields of positive characteristic, J. Class. Anal. 6 (1) (2015) 85-101. 8. F. A. Shah and M. Y. Bhat, On framelet kernels of M -band wavelet frames, Gulf J. Math. 3 (4) (2015) 59-66. 9. F. A. Shah and M. Y. Bhat, A new splitting trick for wavelet packets on local fields of positive characteristic, Poin. J. Anal. Appl. (2) (2015) 93-103.

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