Advances in Pure Mathematics, 2014, 4, 601-609 Published Online November 2014 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2014.411069
Some Results on Wavelet Frame Packets Sana Khan, Mohammad Kalimuddin Ahmad Department of Mathematics, Aligarh Muslim University, Aligarh, India Email:
[email protected],
[email protected] Received 14 September 2014; revised 18 October 2014; accepted 29 October 2014 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding frame bounds.
Keywords Wavelet, Wavelet Packets, Frame Packets
1. Introduction Let us consider an orthonormal wavelet of L2 ( ) . The orthonormal wavelet bases {ψ j , k : j , k ∈ } have a frequency localization which is proportional to 2 j at the resolution level j . If we consider a bandlimited wavelet ψ (i.e. ψˆ is compactly supported), the measure of supp (ψˆ j , k ) is 2 j times the measure of supp (ψˆ ) , since = (ψˆ j ,k ) (ω ) 2
ψˆ ( 2 ω ) e
−j 2
−j
ω −i 2j
k
,
j , k ∈ ,
where = ψ j , k 2 j 2ψ ( 2 j x − k ) . The wavelet bases have poor frequency localization when j is large. For some applications, it is more convenient to have orthonormal bases with better frequency localization. This will be provided by the wavelet packets. The wavelet packets introduced by Coifman, Meyer and Wickerhauser [1] [2] played an important role in the applications of wavelet analysis. But the theory itself is worthy for further study. Some developments in the wavelet packet theory should be mentioned, for instance Shen [3] generalized the notion of univariate orthogonal wavelet packets to the case of multivariate wavelet packets. Chui and Li [4] generalized the concept of orthogonal wavelet packets to the case of nonorthogonal wavelet packets. Yang [5] constructed a-scale orthogonal multiwavelet packets which were more flexible in applications. In [6], Chen and Cheng studied compactly supported orthogonal vector-valued wavelets and wavelet packets. Other notable generalizations are biorthogonal
How to cite this paper: Khan, S. and Ahmad, M.K. (2014) Some Results on Wavelet Frame Packets. Advances in Pure Mathematics, 4, 601-609. http://dx.doi.org/10.4236/apm.2014.411069
S. Khan, M. K. Ahmad
wavelet packets [7] and non-orthogonal wavelet packets with r-scaling functions [8]. For a nice exposition of wavelet packets of L2 ( ) , see [9]. The main tool used in the construction of wavelet packets is the splitting trick [10]. Let {V j : j ∈ } be an MRA of L2 ( ) with the corresponding scaling function φ and the wavelet ψ . Let W j be the corresponding wavelet subspaces W j = span {ψ j , k : k ∈ } . In the construction of a wavelet from an MRA, the space V1 is split into two orthogonal components V0 and W0 , where V1 is the closure of the linear span of the functions {21 2 φ ( 2 x − k ) : k ∈ } and V0 and W0 are the closure of the span of {φ ( x − k ) : k ∈ } and
k k ) φ 2 x − , we see that the above procedure splits the respectively. Since φ ( 2 x −= 2 half integer translates of a function into the integer translates of two functions. k We can also choose to split W j which is the span of ψ 2 j x − k : k= ∈ ψ 2 j x − j : k ∈ . We 2 − ( j −1) then have two functions whose 2 k translates will span the same space W j . Repeating the splitting procedure j times, we get 2 j functions whose integer translates alone span the space W j . If we apply this to each W j , then the resulting basis of L2 ( ) will give us a better frequency localization. This basis is called “wavelet packet basis”. There are many orthonormal bases in the wavelet packets. Efficient algorithms for finding the best possible basis do exist; however for certain wavelet applications in signal analysis, frames are more suitable than orthonormal bases, due to the redundancy in frames. Therefore, it is worthwhile to generalize the construction of wavelet packets to wavelet frame packets in which there are many frames. The wavelet frame packets on was studied in [11], and the frame packets on d were studied by Long and Chen in [12] [13]. Also, multiwavelet packets and frame packets of L2 ( d ) were discussed in [14]. Throughout the paper, the space of all square integrable functions on the real line will be denoted by L2 ( ) and the inner product and Fourier transform of functions in L2 ( ) is given by
{ψ ( x − k ) : k ∈ }
{ (
f,g =
)
}
∞
∫ f ( x ) g ( x ) dx,
−∞
and ∞ fˆ (ω ) := ∫−∞ f ( x ) e − iω x dx
respectively. Also the norm of any f in L2 ( ) will be denoted by
and the relationship bef = f, f ˆ tween functions and their Fourier transform is defined by 2π f , g = f , gˆ . For f ∈ L1 ( ) ∩ L2 ( ) , the Fourier transform fˆ of f is in L2 ( ) and satisfies the Parseval identity
12
fˆ
2 2
= 2π f
2 2
. Also, let L∞ ( )
be the collection of almost everywhere (a.e.) bounded functions, i.e., functions bounded everywhere except on sets of (Lebesgue) measure zero and equipped with the norm
f
∞
= ess sup f ( x ) . −∞< x 0,
∑
δ ( µ ) δ ( − µ ) < ∞.
µ ∈2 +1
and + = +
µ ∈2 +1
Then {n; j , k } constitutes wavelet frame packets with frame bounds and . Proof. Let be the class of all those functions f ∈ L2 ( ) such that fˆ ∈ L∞ ( ) and fˆ is compactly supported in \ 0 . By using the Parseval identity, we have = f , n; j , k
(
)
Since, ˆn; j , k (ω ) = 2− j 2 ˆn 2− j ω e
1 ˆ ˆ 1 ∞ ˆ = f , n; j , k f (ω ) ˆn; j , k (ω ) dω. 2π 2π ∫−∞
ω −i 2j
f , n; j , k
k
, we have
(
)
ω
i k 1 −j 2 ∞ ˆ j = 2 ∫−∞ f (ω ) ˆn 2− j ω e 2 dω 2π 1 −j 2 ∞ j ˆ j = 2 ∫−∞ 2 f 2 ω ˆn (ω ) eikω dω 2π 1 j2 ∞ ˆ j = 2 ∫−∞ f 2 ω ˆn (ω ) eikω dω. 2π
(
(
Hence,
604
)
)
∑∑
n j , k ∈
(
f , n; j , k
2
=
1
( 2π )
2 j ∫ fˆ ( 2 j ω ) ˆn (ω ) eikω dω 2 ∑∑ ∑ n j∈ k ∈
S. Khan, M. K. Ahmad 2
(12)
.
)
Let F j (ω ) = fˆ 2 j ω ˆn (ω ) for j , n ∈ . Each F j is compactly supported in \ 0 and belongs to L ( ) . If F is such a function, 2
∑F (ω + 2kπ ) ,
k ∈
1 ˆ F ( k ) , k ∈ , then by Poisson sum formula we 2π
which is 2π -periodic and whose Fourier coefficients are have,
1
∑F (ω + 2kπ ) = ∑Fˆ ( k ) eikω . 2π
k ∈
k ∈
Hence,
1 F (ω ) ∑ = Fˆ ( k ) eikω dω 2π ∫ k ∈
F (ω + 2kπ ) dω. ∫ F (ω )k∑ ∈
(13)
But the left side of (13) equals 2 1 1 Fˆ ( k ) ∫ F (ω ) e −ikω dω = Fˆ ( k ) . ∑ ∑ 2π k∈ 2π k∈
(14)
It follows that 2 1 = Fˆ ( k ) ∑ 2π k∈
F (ω + 2kπ ) dω. ∫ F (ω )k∑ ∈
(15)
Applying (15) when F = F j in (12) we obtain
(
)
(
)
1 2 j ∫ fˆ 2 j ω ˆn (ω ) ∑ fˆ 2 j (ω + 2kπ ) ˆn (ω + 2kπ ) dω ∑∑ 2π n j∈ k ∈
= ∑ ∑ f , n; j ,k 2
n j , k ∈
1 ∑∑ 2 j fˆ 2 j ω ˆn (ω ) fˆ 2 j ω ˆn (ω ) + ∑ fˆ 2 j (ω + 2kπ ) ˆn (ω + 2kπ ) dω 2π n j∈ ∫ k ≠0
(
= =
(
)
(
2 1 fˆ (ω ) ∑ ∑ ˆn 2− j ω ∫ 2π n j∈
)
2
(
)
)
dω + I ,
where,
I =
(
)
(
)
1 ∑∑ 2 j fˆ 2 j ω ˆn (ω ) ∑ fˆ 2 j (ω + 2kπ ) ˆn (ω + 2kπ ) dω. 2π n j∈ ∫ k ≠0
In the expression for I , the parameter k is a non-zero integer. For each such k there is a unique nonnegative integer l and a unique odd integer q such that k = 2l q . Therefore, we have
∑∑ ∫ fˆ (ω )ˆn ( 2− j ω ) ∑ fˆ (ω + 2 j 2kπ ) ˆn ( 2− j ω + 2kπ ) dω
2πI =
n j∈
k ≠0
∑∑ ∫ fˆ (ω )ˆn ( 2− j ω ) ∑ ∑ fˆ (ω + 2 j +l 2qπ ) ˆn ( 2l ( 2− j −l ω + 2qπ ) ) dω ∞
=
n j∈
= ∫ fˆ (ω )
=∫
fˆ (ω )
l =0 q∈2 +1
∑ ∑∑∑ fˆ (ω + 2 p 2qπ ) ˆn ( 2l 2− p ω ) ˆn ( 2l ( 2− p ω + 2qπ ) ) dω ∞
q∈2 +1 l = 0 p∈ n
∑ ∑ fˆ (ω + 2 p 2qπ ) α q ( 2− p ω ) dω.
q∈2 +1 p∈
Thus,
605
S. Khan, M. K. Ahmad
∑∑
n j , k ∈
(
)
(
)
2 1 fˆ (ω ) ∑∑ ˆn 2− j ω ∫ 2π n j∈
2
f= , n; j , k
2 1 fˆ (ω ) ∑∑ ˆn 2− j ω ∫ 2π n j∈
=
2
2
(
) (
)
dω +
1 fˆ (ω ) ∑ ∑ fˆ ω + 2 p 2qπ α q 2− p ω dω 2π ∫ q∈2 +1 p∈
dω +
1 ( f ), 2π
(16) for all f ∈ . By using Schwarz’s inequality we have
( f ) ≤
∑ ∑ ∫ fˆ (η ) q∈2 +1 p∈
2
12
12
α q ( 2− p η ) dη ⋅ ∫ fˆ (η + 2 p 2qπ ) α q ( 2− p η ) dη .
2
By changing variables in the second integral and using the fact that α q (ω − 2qπ ) = α − q (ω ) , and applying Schwarz’s inequality for series we have ( f ) ≤
12
12
2 2 ∑ ∑ ∫ fˆ (η ) α q 2− pη dη ⋅ ∑ ∫ fˆ (η ) α − q 2− pη dη q∈2 +1 p∈ p∈
(
∑
≤
q∈2 +1
)
δ ( q ) δ ( −q )
(
)
2
12
fˆ . 2
Hence, −
∑
δ ( q ) δ ( −q )
12
q∈2 +1
fˆ
≤ ( f )≤
2 2
∑
q∈2 +1
δ ( q ) δ ( −q )
12
2
fˆ . 2
These inequalities together with (16) give us f
2 2
≤∑
∑
n j , k ∈
f , n; j , k
2
≤ f
2 2
,
. ∀ f ∈
Since is dense in L2 ( ) , the above inequality holds for all f ∈ L2 ( ) . Theorem 2. The system {n; j , k } , n, j ∈ + , k ∈ is orthonormal if and only if kπ ) ∑∑ ˆn (ω + 2= 2
n k ∈
1 for a.e. ω ∈
(17)
and + 2kπ ) ∑∑ˆn ( 2 j (ω + 2kπ ) ) ˆn (ω= n k ∈
0 for a.e. ω ∈ ,
j ≥ 1.
(18)
Proof. By using the Plancherel theorem we have
∑∫n ( x ) n ( x − k ) dx
= δ k ,0
n
2 1 ˆn (ω ) eikω dω ∑ ∫ 2π n 2 1 ∞ 2( l +1) π ˆn (ω ) eikω dω = ∑ ∑ ∫ l 2 π 2π l = −∞ n 2 1 ∞ 2π ˆn ( µ + 2lπ ) eik µ dµ = ∑ ∑ ∫ 0 2π l = −∞ n
=
= Thus,
{ } n ;0, k
2 1 2π ˆ ( µ + 2lπ ) eik µ dµ . ∑∑ n ∫ 2π 0 n l∈
is orthonormal if and only if
Performing a change of variables, we see that
∑∑ ˆn ( µ + 2lπ )
2
= 1 a.e. The converse is immediate.
n l∈
n; j , k , n; j ,l = n;0, k , n;0,l ; this tells us that the system
606
{ }
S. Khan, M. K. Ahmad
is orthonormal for each fixed j when (17) is satisfied. The proof of condition (18) is similar.
n; j , k
Lemma 1. If
{ } n; j , k
is an orthonormal system, then ∞
∑ˆn ( 2m ω )= ∑∑∑ˆn ( 2m (ω + 2kπ ) ) ˆn ( 2 j (ω + 2kπ ) ) ˆn ( 2 j ω )
a.e.
(19)
n =j 1 k ∈
n
for all m ≥ 1 . Proof. Let Am (ω ) be the R.H.S of (19). We have to show that Am (ω ) = ∑ˆn 2m ω
(
)
for a.e. ω ∈ . We
n
first show that Am (ω ) = Am −1 ( 2ω ) and then that A1 (ω ) = ∑ˆn ( 2ω ) ; this will clearly give us (19). Using (18), n
with j replaced by m , we have
(
Am (ω ) = ∑∑ˆn 2m (ω + 2kπ ) n k ∈
∞
) ∑ˆ ( 2 (ω + 2kπ ) ) ˆ ( 2 ω ) j n n 1 j=
j
∞
∑∑ˆn ( 2m (ω + 2kπ ) ) ˆn (ω + 2kπ )ˆn (ω ) + ∑∑ˆn ( 2m (ω + 2kπ ) ) ∑ˆn ( 2 j (ω + 2kπ ) ) ˆn ( 2 j ω )
=
n k ∈
n k ∈
(
= ∑∑ˆn 2m (ω + 2kπ ) n k ∈
∞
1 j=
) ∑ˆ ( 2 (ω + 2kπ ) ) ˆ ( 2 ω ) . j n n 0 j=
j
Replacing k by 2l , we have
(
Am (ω ) = ∑∑ˆn 2m (ω + 4lπ ) n l∈
∞
) ∑ˆ ( 2 (ω + 4lπ ) )ˆ ( 2 ω ) j n n 0 j=
j
ω ω ∞ ω = ∑∑ˆn 2m +1 2 + 2lπ ∑ˆn 2 j +1 2 + 2lπ ˆn 2 j +1 2 j= 0 n l∈ ω ω ∞ ω =∑∑ˆn 2m +1 + 2lπ ∑ˆn 2 j + 2lπ ˆn 2 j 2 2 j= 2 1 n l∈ ω = Am +1 . 2 This shows that Am (ω ) = Am −1 ( 2ω ) a.e. Now, we calculate A1 (ω ) and show that A1 (ω ) = ∑ˆn ( 2ω ) . n Changing variables in the sum over j , we have ∞
(
)
(
)
(
)
(
)
(
) (
)
A1 (ω ) = ∑∑ˆn ( 2 (ω + 2kπ ) ) ∑ˆn 2 j (ω + 2kπ ) ˆn 2 j ω ∞
n k ∈
j= 1
= ∑∑ˆn ( 2ω + 4kπ ) ∑ˆn 2 j ( 2ω + 4kπ ) ˆn 2 j 2ω n k ∈
j= 0 ∞
= ∑∑ˆn ( 2ω + 2kπ ) ∑ˆn 2 j ( 2ω + 2kπ ) ˆn 2 j 2ω n k ∈
j= 0
∞ =∑∑ˆn ( 2ω + 2kπ ) ˆn ( 2ω + 2kπ )ˆn ( 2ω ) + ∑ˆn 2 j ( 2ω + 2kπ ) ˆn 2 j 2ω . n k ∈ j= 1
(
) (
)
By using (17) and (18), we have
A1 (ω ) = ∑ˆn ( 2ω ) . n
Theorem 3. Let
{n }
be a sequence of wavelet frame packets with bounds and . Define
= γm
∞
∑λmn n , n =1
607
m ∈ .
{γ m }
by (20)
S. Khan, M. K. Ahmad
If the numbers
{λmn }m,n∈
satisfy the two conditions ∞
= l : sup ∑
∞
∑λmn λmp
< ∞,
(21)
n = p 1= m 1
∞ ∞ 2 k :=inf ∑ λmn − ∑ ∑λmn λmp > 0, n = = p≠n m 1 m 1
then
{γ m }m =1 ∞
(22)
defined by (20) is a wavelet frame packet with bounds k and l .
Proof. Let f ∈ L2 ( ) . Then ∞
∞
= ∑ γm, f
2
∞
∞
= ∑ ∑λmn n , f
2
= m 1
∑
= m 1 = n 1 ∞
∞
∑∑ λmn
=
2
m= 1 n= 1
∑λmn n , f
2
= m 1= n 1 ∞
∞
+ ∑∑∑λmn λmp n ,f
2
n , f
∞
m= 1 n= 1 p ≠ n
f , p
(23)
= R1 ( f ) + R2 ( f ) . By Cauchy-Schwarz inequality, we get ∞
R2 ( f ) ≤ ∑∑ n , f
∞
∑λmn λmp
f , p
n= 1 p≠n
∞ ≤ ∑∑ n , f 1 p≠n n=
2
m= 1 12
∞
1 p≠n n=
∞
∑λmn λmp ⋅ ∑∑ f , p
m= 1
12
∞
∑λmn λmp .
2
m= 1
On solving the second term in the last product, we have ∞
∞
∞
∞
∞
= f , p ∑λmn λmp ∑∑ = f , p ∑λmn λmp ∑∑ ∑∑ 2
1 p≠n n=
2
1 1n ≠ p m= p=
n , f
2
1 1 p≠n m= n=
∞
∑λmn λmp
1 m=
Thus, ∞
R2 ( f ) ≤ ∑∑ n , f
2
1 p≠n n=
∞
∑λmn λmp .
1 m=
By (23), we have ∞
∑
γm, f
∞
∞
≥ ∑∑ λmn
2
2
n , f
1 1n = 1 m= m=
=
∞
∑
n 1 =
2
∞
− ∑∑ n , f
1 p≠n n=
∞
2
∑λmn λmp
1 m=
2 ∑ λmn − ∑ ∑λmn λmp p≠n m 1 = = m 1
n , f
∞
∞
2
≥ k ∑ n , f
2
k f
2
∞
.
n =1
Thus, ∞
≤ ∑ γm, f
2
.
m =1
Similarly, one can prove the upper frame condition.
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[2]
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Hernandez, E. and Weiss, G. (1996) A First Course on Wavelets. CRC Press, Boca Raton. http://dx.doi.org/10.1201/9781420049985
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