arXiv:1703.06473v2 [math.FA] 29 Aug 2018

Noname manuscript No. (will be inserted by the editor)

A directional uncertainty principle for periodic functions

arXiv:1703.06473v1 [math.FA] 19 Mar 2017

A. Krivoshein · E. Lebedeva · J. Prestin

Abstract In this paper we introduce a notion of a directional uncertainty product for multivariate periodic functions. It measures a localization of a function along a particular direction. We study properties of the uncertainty product and give an example of well localized multivariate periodic Parseval wavelet frames. Keywords uncertainty principle · uncertainty product · wavelets Mathematics Subject Classification (2010) 42A05 · 42A63

1 Introduction A notion of uncertainty product is a sufficiently well-studied object in harmonic analysis. Initially, it was introduced for functions on the real line to measure a simultaneous localization of a function and its Fourier transform [16]. The essence of this measurement is concentrated in the fundamental Heisenberg uncertainty principle, which says that for any appropriate function the uncertainty product cannot be smaller than a positive absolute constant. Later, numerous versions of this framework were developed for different algebraic and topological structures such as abstract locally compact groups, high-dimensional spheres, etc. (see, e.g., [8], [11], [14]). For more detailed information concerning this topic, we refer the interested reader to survey [3] and the references therein. The work is supported by Volkswagen Foundation; the first and the second authors are also supported by grants from RFBR # 15-01-05796-a, St. Petersburg State University # 9.38.198.2015. A. Krivoshein St. Petersburg State University E-mail: [email protected] E. Lebedeva Peter the Great St.Petersburg Polytechnic University St. Petersburg State University E-mail: [email protected] J. Prestin University of L¨ ubeck E-mail: [email protected]

2

A. Krivoshein et al.

In this paper we focus on the case of multivariate periodic functions. For periodic functions of one variable a notion of uncertainty product was introduced in 1985 by Breitenberger in [2]. The corresponding uncertainty principle is also valid in this setup. One possible extension of this notion to the case of multivariate periodic functions was suggested by Goh and Goodman in [5] (see formula (2)). However, this approach does not take into account the main difference between periodic functions of one variable and many variables, namely the localization of a function along particular directions. The main contribution of this paper is a new approach that allows to include the directionality into the definition of the uncertainty product (see formula (3)). We compare these two approaches and show that they are not equivalent (see Lemma 3). At the same time, both definitions fit into a more general operator approach (see formula (1)). This approach was established by Folland in [4] and was extended to two normal or symmetric operators by Selig in [17] and Goh, Micchelli in [6]. For several operators this approach was generalized by Goh and Goodman in [5]. The paper is organized as follows. In Section 2 we study the properties of the directional uncertainty product for periodic functions and compare this product with one defined by Goh and Goodman. Lemma 2 gives a sequence of trigonometric polynomials such that the sequence of their directional uncertainty products tends to the optimal value. Lemma 3 illustrates a difference between these two uncertainty products. In Subsection 2.1. we study the behavior of both uncertainty products for the Dirichlet and Fej´er kernels. Lemmas 4 and 5 concern the directional case. In Lemma 6 we address the same question to the Goh and Goodman case. In Subsection 2.2. we minimize the directional angular variance for trigonometric polynomials. Theorem 4 describes the case of the directional uncertainty product, and Theorem 5 corresponds to the case of the uncertainty product defined by Goh and Goodman. In Section 3 we give an example of a multivariate periodic Parseval wavelet frame with a small directional uncertainty product (see Theorem 6).

2 Basic notations and definitions We use the standard multi-index notation. Let d ∈ N, Rd be the d-dimensional Euclidean space, {ej , 1 ≤ j ≤ d} be the standard basis in Rd , Zd be the integer lattice in Rd , Td = Rd /Zd be the d-dimensional torus. Let x = (x1 , . . . , xd )T and d y = (y1p , . . . , yd )T be column Pd vectors in R . Then hx, yi := x1 y1 + · · · + xd yd , kxk := hx, yi, kxk1 = j=1 |xj |, kxk∞ = maxj {|xj |}. We say that x ≥ y, if xj ≥ yj for all j = 1, . . . , d, and we say that x > y, if x ≥ y and x 6= y. Further, Zd+ := {α ∈ Zd : α ≥ 0}, where 0 = (0, . . . , 0) denotes the origin( in Rd . For 0, x ≤ 0, α = (α1 , . . . , αd )T ∈ Zd+ , denote |α| := α1 +· · ·+αd . For x ∈ R, x+ := x, x > 0. For a sufficiently smooth function f defined on Ω ⊂ Rd and a multi-index α ∈ |α| |α| Zd+ , Dα f denotes the derivative of f of order α and Dα f = ∂∂xαf = ∂ α1 x∂1 ...∂fαd xd . For α = ej , we also use Dej f = fj′ . The directional derivative of a sufficiently smooth function f defined on Ω along a vector L = (L1 , ..., Ld ) ∈ Rd is denoted Pd ∂f ∂f by ∂L = j=1 Lj ∂x . j

A directional uncertainty principle for periodic functions

3

R For a function f ∈ L2 (Td ) its norm is denoted by kf k2Td = Td |f (x)|2dx. The Fourier series coefficients of a function f ∈ L2 (Td ) are given by ck = ck (f ) = R fb(k) = Td f (x)e−2πihk,xi dx, k ∈ Zd . The Sobolev space H 1 (Td ) consists of functions in L2 (Td ) such that all its derivatives of the first order are also in L2 (Td ), which can be written as     X kkk2 |ck (f )|2 < ∞ . H 1 (Td ) = f ∈ L2 (Td ) :   d k∈Z

Let H be a Hilbert space with inner product h·, ·i and with norm k·k := h·, ·i1/2 . For an operator A : H → H, D(A) denotes its domain. For the operator A, the variance of a non-zero f ∈ D(A) is defined by

  2

|hAf, f i|2

A − hAf, f i f . = ∆(A, f ) = kAf k2 −

2 2 kf k kf k

For two non-commuting operators A, B on a Hilbert space H, defineTthe commutator of these operators by [A, B] := AB − BA with domain D(AB) D(BA). Theorem 1 [5, Theorem 4.1] Let A1 , . . . An , B1 , . . . Bn be symmetric or normal operators Tacting from a Hilbert space H into itself. Then for any non-zero f in D(Aj Bj ) D(Bj Aj ), j = 1, . . . , n,   2   n n n X X X 1 (1) ∆(Bj , f ) . ∆(Aj , f )  |h[Aj , Bj ]f, f i| ≤  4 j=1 j=1 j=1

If the commutator h[Aj , Bj ]f, f i is non-zero for all j = 1, . . . , n, then the uncertainty product for f is defined as −2    n n n X X X |h[Aj , Bj ]f, f i| . ∆(Bj , f )  ∆(Aj , f )  UP(f ) :=  j=1

j=1

j=1

In this terms, the uncertainty principle says that the uncertainty product UP(f ) cannot be smaller than 41 , for any appropriate function f . The well-known Heisenberg uncertainty product for functions in L2 (R) fits in this operator approach, if n = 1 and the two operators are as follows Af (x) = i df 2πxf (x), Bf (x) = 2π dx (x). The Breitenberger uncertainty product is defined for the space of periodic i df functions L2 (T). In this case, AT f (x) = e2πix f (x), BT f (x) = 2π dx (x). The comT T T mutator is [A , B ] = A . It is more convenient for the Breitenberger uncertainty product, to use the notions of the angular and frequency variance. Since kAT f k2T = kf k2T , kf k2T ∆(AT , f ) var (f ) = = |h[AT, BT ]f, f i| A

varF (f ) =



kf k2T |hATf, f i|

2

− 1,

kBT f k2Rd |hBT f, f i|2 ∆(BT , f ) = − , 2 2 kf kT kf kT kf k4T

4

A. Krivoshein et al.

UPT (f ) := varA (f )varF (f ). It is known that the lower bound for UPT does not attain on any function. But there exist sequences of functions such that UPT tends to the optimal value 14 (see, e.g., [13]). For the space L2 (Td ) of multivariate periodic functions, Goh and Goodman in [5] suggest to take the operators as follows Aj f (x) = e2πixj f (x), Td Bj f (x) = i ∂f j=1 D(Aj ) = 2π ∂xj (x), j = 1, . . . , d. Note that the domains of the operators are T L2 (Td ), dj=1 D(Bj ) = H 1 (Td ). Operators Aj are normal, Bj are self-adjoint. The commutators for f ∈ H 1 (Td ) are [Aj , Bj ]f = Aj f. The uncertainty principle for these operators is stated as follows. Theorem 2 For a function f ∈ H 1 (Td ), such that hAj f, f i 6= 0, j = 1, . . . , d, the d uncertainty product UPTGG (f ) is well-defined and 2  P  P   P 2  kf k4 d − c c k k−e kj |ck |2 kj2 |ck |2 j T k∈Zd d j=1 X  k∈Zd d 1  k∈Zd    − UPTGG (f ) = !2    ≥ 4,  2 2 kf k kf k d P P j=1 Td Td ck−ej ck j=1 k∈Zd (2) d P



where k = (k1 , . . . , kd ), ck = ck (f ) are the Fourier coefficients of f.

Defining the variances as

varA GG (f )

kf k2Td

= P n

j=1

d P

∆(Aj , f )

j=1

|h[Aj , Bj ]f, f i|

2 ,

varF GG (f ) =

d X

j=1

∆(Bj , f )/kf k2Td ,

it can be shown, that the variances attain the value ∞ if and only if hAj f, f i = 0, for all j = 1, . . . , d, or Bj f is not in L2 (Td ) for some j = 1, . . . , d. In these cases, we d can also assign to UPTGG (f ) the value ∞, except the following case varF GG (f ) = 0 A and varGG (f ) = ∞. This case happens if and only if f is a monomial, since F F varF GG (f ) = 0 if and only if f is a monomial. Indeed, since varGG (f ) = varGG (af ) d d for any appropriate f ∈ L2 (T ) and a ∈ R , a 6= 0, we can assume that kf kTd = 1. Therefore, varF GG (f ) = 0

if and only if |hBj f, f i|2 = kBj f k2Td kf k2Td

∀j = 1, . . . , d.

Due to the Cauchy-Bunyakovsky-Schwarz inequality the equality is possible only ∂f if f = αj ∂x , where αj ∈ C for all j = 1, . . . , d. Thus, f should be a monomial. j

A However, in this case, i.e., varF GG (f ) = 0 and varGG (f ) = ∞, inequality (1) takes the form 1/4 · 0 ≤ C · 0. It is trivially true. Thus, inequality (1) is valid for all non-zero functions f ∈ L2 (Td ). In fact, the above approach for the definition of the uncertainty product does not deal with a new phenomenon, that appears in the multidimensional case, namely, the localization of a function along particular directions. We suggest an approach that allows to include the directionality into the definition.

A directional uncertainty principle for periodic functions

5

The directional uncertainty product for Td along a direction L ∈ Zd is defined using the operators AL f (x) = e2πihL,xi f (x),

BL f (x) =

i ∂f (x). 2π ∂L

Note that the domains of the operators are D(AL ) = L2 (Td ), D(BL ) = H 1 (Td ) and AL is normal, BL is self-adjoint. The commutator for f ∈ D(AL ) ∩ D(BL ) is [AL , BL ]f = kLk2 AL f. Thus, the directional uncertainty product for a function f ∈ D(AL ) ∩ D(BL ) such that AL f 6= 0 is defined as d

UPTL (f ) =

kf k4Td

1 kLk42

|hAL f, f i|2

!

−1

kBL f k2Td kf k2Td

−

|hBL f, f i|2 kf k4Td

!

:=

1 F varA L (f )var L (f ), kLk4

F where varA L (f ) is the angular directional variance and varL (f ) is the frequency directional variance.

Theorem 3 For L ∈ Zd and a function f ∈ H 1 (Td ), such that hAL f, f i 6= 0, the d uncertainty product UPTL (f ) is well-defined and  !2  P P  P 2    hL, ki2 |ck |2 |ck |2 hL, ki|ck |2   d 1  k∈Zd 1   k∈Zd  k∈Zd   UPTL (f ) = P − P  2 − 1     ≥ 4,  kLk4  P |ck |2 |ck |2   k∈Zd k∈Zd c c k∈Zd k−L k (3) 

where ck = ck (f ) are the Fourier coefficients of f.

The statement easily follows from the operator approach and AL f (x) =

X

k∈Zd

ck−L e2πihk,ξi ,

BL f (x) = −

X

k∈Zd

hL, kick e2πihk,ξi .

It can be shown, that the directional variances attain the value ∞ if and only if hAL f, f i = 0 or BL f is not in L2 (Td ). In these cases, we can also assign to d A UPTL (f ) the value ∞, except the following case varF L (f ) = 0 and varL (f ) = ∞. This case happens if and only if f is a monomial, which can be shown as above. Thus analogously, inequality (1) for operators AL and BL is valid for all non-zero functions f ∈ L2 (Td ). 3 The properties of the directional uncertainty product for the periodic case First of all, we note that the standard manipulations of functions like shifts, modud lations and multiplying by numbers do not change the uncertainty product UPTL . Lemma 1 Let f ∈ H 1 (Td ). Suppose g(x) = a e2πihK,xi f (x − x0 ), where K ∈ Zd , d d a ∈ R, a 6= 0, x0 ∈ Rd , then UPTL (g) = UPTL (f ).

6

A. Krivoshein et al.

The proof can be done by straightforward computations. As for the Breitenberger uncertainty product and for the uncertainty product defined by Goh and Goodman, the optimal function for the directional uncertainty i L f,f i and b(f ) = hBkf . Since BL is product does not exist. Indeed, let a(f ) = hAkfL f,f k22 k22 self-adjoint, b(f ) is real. Due to Theorem 3.1 in [17] the equality for the uncertainty principle is attained if and only if there exist λ ∈ C such that (BL − b(f ))f = λ(AL − a(f ))f = −λ(A∗L − a(f ))f. The second identity yields   f (x) λe2πihL,xi + λe−2πihL,xi − a(f )λ − λa(f )

= 2f (x)(Re(λe2πihL,xi) − Re(a(f )λ)) ≡ 0.

This condition can be satisfied only if f = 0 or λ = 0. For the second case, we i ∂f get (BL − b(f ))f = 0 or 2π ∂L (x) = b(f )f (x), which is only possible when f is a monomial, i.e. f (x) = Ce2πihk,ξi . Recall that for monomials the directional uncertainty product is not defined. The next lemma gives a sequence of trigonometric polynomials such that the sequence of their directional uncertainty products tends to the optimal value. Lemma 2 Suppose pn (x) = (1 + cos 2πhL, xi)n for n ∈ N. Then   d 1 1 , as n → ∞. UPTL (pn ) = + O 4 n R Proof. Denote I2n := Td (1 + cos 2πhL, xi)2n dx = kpn k2Td . Since pn is even hAL pn , pn i = I2n+1 − I2n . Further, Z kBL pn k2Td = n2 kLk4 (1 + cos 2πhL, xi)2n−2 sin2 2πhL, xi dx = 2I2n−1 − I2n , Td

2

since sin 2πhL, xi = 2(1 + cos 2πhL, xi) − (1 + cos 2πhL, xi)2 . Again, since pn is even and sin 2πhL, xi is odd, we get hBL pn , pn i = 0. So, finally, I2n+1 (2I2n − I2n+1 )(2I2n−1 − I2n ) . I2n (I2n+1 − I2n )2   It remains to compute In . Since (1 + cos 2πhL, xi)n = 2n cos2n 2π hL,xi , 2 d

UPTL (pn ) = n2

In = 2n

Z

cos2n (πhL, xi)dx =

Td

2n 2d

Z

2Td

cos2n

2πhL, xi dx = 2n 2

Z

cos2n (2πhL, xi)dx.

Td

Using Euler’s formula we get ! n 1 X n 2πijhL,xi −2πi(n−j)hL,xi e e cos (2πhL, xi) = n j 2 j=0 ! n 1 X n 2πi(2j−n)hL,xi e . = n 2 j=0 j n

(4)

A directional uncertainty principle for periodic functions

7

Due to the Parseval equality for the function cosn (2πhL, xi) we obtain !2 ! Z n 1 2n 1 (2n)! 1 X n 2n = 2n . = 2n cos (2πhL, xi)dx = 2n 2 j=0 j 2 2 (n!)2 n Td d

1 Therefore, In = (2n−1)!! . Substituting this in (4), we obtain UPTL (pn ) = 14 + 8n−2 . n! ♦ Let us compare the uncertainty product defined by Goh and Goodman and the directional uncertainty product. They are not equivalent. The next lemma gives a pair of examples where the uncertainty products behave differently.

Lemma 3 Let L ∈ Zd .

(A) Suppose pen (x) = (1 + cos 2πhL, xi)n + 2 cos 2πx1 , where L is not collinear to e1 . Then d

pn ) → UPTL (e

d

UPTGG (e pn ) dkLk2 → n 4n 32

1 , 4

n → ∞.

(B) Suppose e tn (x) = (1 + cos 2πx1 )n + 2 cos 2πhL, xi, where L is not collinear to all ej and |Lj | > 1 for all j = 1, ..., d. Then d

d

UPTGG (e tn ) d−1 → , n 4

L21 UPTL (e tn ) → , n n4 32kLk4

n → ∞.

Proof. Let us prove item (A). For convenience, we will use the notation pn (x) = (1 + cos 2πhL, xi)n and some facts used in the proof of Lemma 2. Then ke pn k2Td = kpn k2Td +2 =

(4n − 1)!! +2, (2n)!

hAL pen , pen i = hAL pn , pn i = I2n+1 −I2n ,

BL pen (x) = ikLk2 n(1 + cos(2πhL, xi))n−1 sin(2πhL, xi) + 2iL1 sin 2πx1 , kBL pen k2Td = n2 kLk4 (2I2n−1 − I2n ) + 2L21 .

Since pen is even and BL pen is odd we get hBL pen , pen i = 0. Therefore,

 2  2 2 4 (4n−3)!! + 2 1    n kLk (2n)! + 2L1  Td pn ) = UPL (e   2 − 1 (4n−1)!! kLk4 +2 2n (4n−1)!! (2n)! 

(4n−1)!! (2n)!

(2n+1)!

  (2n)! 1 + 2 (2n)!(2n+1) 2 + 2 (4n−1)!! − n2 (4n−1)!! =  2 (2n + 1)(4n − 1) 2n

1 2n+1

2n+1

By the Stirling formula n! =

√ 1 )) 2πn(1+O( n 22n

√

2πn


n n e

(1 + O(1/n)), it follows that d

→ 0, n → ∞. Therefore, UPTL (e pn ) → 14 , d

 L2 1 (2n)!(4n−1) 1 + 2 4 2 kLk n (4n−1)!!   .  (2n)! 1 + 2 (4n−1)!!



n → ∞.

(2n)! (4n−1)!!

=

Now, we compute UPTGG (e pn ). Let cek = cek (e pn ) be the Fourier coefficients of pen . Then Z Z (2n − 1)!! , pn (x)dx = In = pen (x)dx = c0 = e n! Td Td

8

A. Krivoshein et al.

X

hAj pen , pen i =

k∈Zd

for j = 1, . . . , d. Further,

cek−ej e ck = δj,1 (e ce1 e c0 + e c0 e ce1 ) = 2δj,1

(2n − 1)!! , n!

Bj pen (x) = −iLj n(1 + cos(2πhL, xi))n−1 sin(2πhL, xi) − 2iδj,1 sin 2πx1 .

Therefore, kBj pen k2Td = n2 L2j (2I2n−1 − I2n ) + 2δj,1 . Since pen is even and Bj pen is odd, we get hBj pen , pen i = 0. Hence, combining all results in the definition of d UPTGG (e pn ) (2) and after some simplifications, we obtain d UPTGG (e pn )

n2 kLk2 = 4(4n − 1)

d



(4n − 1)!! n! n! +2 (2n)! (2n − 1)!! (2n − 1)!!

2

!

√ 1 2πn(1+O( n )) → 0 as n → 22n √ 1 πn(1+O( n )) (4n−1)!! n! 2n → 0 as n → ∞. Thus, (2n)! (2n−1)!! = √ (1 + 2n 2 d UPTGG (e pn ) dkLk2 → 32 as n → ∞. Finally, it follows that n4n

By the Stirling formula

(2n)! (4n−1)!!

=

2(2n)! n2 kLk2 (4n−1)!! (2n)! 1 + 2 (4n−1)!!

1+

−4

∞ and

n! (2n−1)!!

.

=

O( n1 )) as n → ∞.

Item (B) can be proved analogously. By similar arguments it can be shown that d

UPTL

and

 1  (e tn ) =  kLk4

d

tn ) = UPTGG (e

d



(4n−1)!! (2n)! (2n−1)!! n!

 2 2  (2n)! 2 4 (2n−1)! 1 + 2 (4n−1)!!  L1 /2 + 2kLk n(4n−3)!! 2n2  − 1 4 4n − 1 1 + 2 (2n)! (4n−1)!!

(2n)! 2n + 1 2n + 1 +2 2n (4n − 1)!! 2n

2

!

−1

n 2

(2n−1)! + 2kLk2 (4n−3)!! (2n)! 1 + 2 (4n−1)!!

2n . 4n − 1

The Stirling formula yields Item (B).♦

3.1 The uncertainty products for the Dirichlet and Fej´er kernels As it was noted in [13], the sequence of the Breitenberger uncertainty products of P 2πikx the Dirichlet kernels Dn (x) = n tends to infinity as n → ∞. In [12] k=−n e it was noted that the sequence of Breitenberger uncertainty products of the Fej´er n P 3 kernels Fn (x) = (1 − |k|/n)e2πinx tends to 10 as n → ∞. In the multivariate k=−n

case the analogous difference between these kernels also holds for the directional uncertainty product and the one defined by Goh and Goodman. Different methods of summation can be used for the Dirichlet kernel. Let us consider a rectangular one. P e2πihk,xi , where N ∈ Zd , N > 0, L ∈ Zd . Then Lemma 4 Let DN (x) = −N ≤k≤N d

UPTL (DN ) → ∞,

kN k → ∞.

A directional uncertainty principle for periodic functions

9

Q Proof. Let N > L, N = (N1 , . . . , Nd ). Since kDN k2Td = dj=1 (2Nj + 1) and Qd hBL DN , DN i = 0, hAL DN , DN i = j=1 (2Nj + 1 − Lj ) and kBL DN k2Td = =

X

−N ≤k≤N d Y

 

d X

(Lj kj )2 +

d X

Nj (Nj + 1) , 3

L2j

j=1

j=1

d X

j=1 n=1,n6=j

j=1

(2Nj + 1)

d X



Lj Ln kj kn 

we obtain d

Qd

j=1 (2Nj

+ 1)2

!

d X

Nj (Nj + 1) L2j 2 3 (2N + 1 − L ) j j j=1 j=1     Qd  Qd  L Lj d 1 + j=1 1 − 2N j+1 1 − j=1 1 − 2N +1 X Nj (Nj + 1) 1 j j L2j =   2 Qd Lj kLk4 3 j=1 j=1 1 − 2N +1

UPTL (DN ) =

1 kLk4

Qd

−1

j

≥ ≥

1 kLk4

1−



1 − min j

Lj 2Nj + 1

Lj d min 2d−1 kLk4 j 2Nj + 1

d X

j=1

d ! X d

L2j

j=1

L2j

Nj (Nj + 1) 3

Nj (Nj + 1) , 3 d

where the last inequality is due to the mean value theorem. Thus, UPTL (DN ) → ∞ as kN k → ∞. ♦ For the case of the multivariate Fej´er kernel Fn (x) =

X

k∈Zd ,kkk∞ 0, L ∈ Zd . Let

Fn be the Fej´er kernel, n ∈ N, L ∈ Zd . Then d

UPTGG (DN ) → ∞, as n → ∞.

kN k → ∞,

and

d

UPTGG (Fn ) →

(d + 1)2 (d + 2)2 , 6d (d + 3) (d + 4)

14

1),

A. Krivoshein et al.

Proof. Since kDN k2Td =

kBj DN k2Td =

Qd

j=1 (2Nj

Nj (Nj + 1)(2Nj + 1) 3

+ 1), hAj DN , DN i = 2Nj d Y

i=1,i6=j

Qd

(2Ni + 1) = kDN k2Td

i=1,i6=j (2Ni

+

Nj (Nj + 1) , 3

hAj DN , DN i 1 , =1− kDN k2Td 2Nj + 1

hBj DN , DN i = 0,

 2 P d d − dj=1 1 − 2N1j +1 X Nj (Nj + 1) UPTGG (DN ) =  .  2 Pd 3 j=1 d − j=1 2N1j +1 d

d

Thus, UPTGG (DN ) → ∞ as kN k → ∞. Concerning the Fej´er kernel, using the rates of growths and decays established in the previous lemma, we get kBj Fn k2Td 1 (d + 1)(d + 2) 2 n + O(n), = kFn k2Td 3 (d + 3)(d + 4)

n → ∞,

|hAj Fn , Fn i| (d + 1)(d + 2) + O(1/n3), = 1− 4dn2 kFn k2Td varA GG (Fn ) =

(d + 1)(d + 2) + O(1/n3), 2d2 n2

n → ∞,

n → ∞,

then d

UPTGG (Fn ) = varA GG (Fn )

d X kBj Fn k2Td (d + 1)2 (d + 2)2 , → kFn k2Td 6d (d + 3) (d + 4) j=1

n → ∞.♦

Also, we can place the Dirichlet and Fej´er kernels along the direction vector L. Namely, let L (x) = Dn

n X

e2πihk0 +Lm,xi ,

FnL (x) =

m=−n

  n X |m| 2πihk0 +Lm,xi 1− e , n m=−n

for some k0 ∈ Zd . d

L ) → ∞, Lemma 7 Let L ∈ Zd . Then UPTL (Dn

d

UPTL (FnL ) →

The proof can be done by straightforward computations.

3 10 ,

as n → ∞.

A directional uncertainty principle for periodic functions

15

3.2 The minimal angular variance Now we give a multivariate analogue of Rauhut’s result in [15] on minimizing the angular variance for trigonometric polynomials. For a finite subset S in Zd , denote the set of trigonometric polynomials ) ( X 2πihk,xi ΠS = ck e : ck ∈ C . k∈S

Then, one is interested in best localized polynomials, i.e., for a fixed L ∈ Zd find all trigonometric polynomials p, whose coefficient support is inside some fixed set S ⊂ Zd and its directional uncertainty product takes its minimal value, i.e. d

min {UPTL (p)}.

p∈ΠS

This problem is difficult for an arbitrary set S. Nevertheless, it is possible to minimize the angular frequency and the frequency variance separately. For the frequency variance the minimum value is equal to zero and it attains on trigonometric polynomials that have only one non-zero coefficient as it was shown above. For the angular variance the situation is not so trivial. Again, since varL A (f ) = d d varL A (af ) for any appropriate f ∈ L2 (T ) and a ∈ R , we can assume that kpkTd = 1. So, let us consider the problem min {varA L (p) : kpkTd = 1}.

p∈ΠS

(5)

Since the set {p ∈ ΠS , kpkTd = 1} is a compact set and varL A (p) is continuous (except the cases when hAL p, pi = 0), we can conclude that the minimum exists. During the proof of the theorem below, we need to split the set S into several disjoint ”threads” of points. Each ”thread” looks as follows (the order of elements is fixed) U = {k, k + L, k + 2L, . . . , k + mL},

where m ∈ N, k ∈ S such that U ⊂ S.

In other words, k is chosen such that k − L ∈ / S, and m is chosen such that k + (m + 1)L ∈ / S. These ”threads” are sorted by decreasing number of elements. Assume that the number of ”threads” is u and U0 is the longest (if there are several of them, we can take any). Therefore, S=

u−1 [ i=0

Ui =

u−1 [ i=0

{ki , ki + L, . . . , ki + mi L}.

The next Theorem states that the minimal angular variance (5) depends on the length of the longest ”thread” inside S. Theorem 4 The minimal angular variance for trigonometric polynomials with coefficient support inside S is equal to 2 min {varA L (p), kpkTd = 1} = tan

p∈ΠS

π , m0 + 2

16

A. Krivoshein et al.

where m0 +1 is the length of the longest ”thread” inside S. The minimum is attained by the trigonometric polynomial pmin whose non-zero coefficients are placed on this ”thread”. The Fourier coefficients ck = ck (pmin ) of such a polynomial are defined as follows ( if k = k0 + (j − 1)L, j = 1, . . . , m0 + 1, sin mπj 0 +2 (6) ck = 0 else. The directional uncertainty product is given by d

UPTL (pmin ) =

π 1 m0 (m0 + 4) tan2 − . 12 m0 + 2 2

−2 P − 1. Then the minimization Proof. Note that varA L (p) = ( k∈Zd ck−L ck ) 2 P P 2 problem (5) is equivalent to max ck−L ck : |ck | = 1 . Firstly, we rek∈S

k∈S iφk

duce the problem to real coefficients. Let ck = rk e , k ∈ S. Thus, we have to maximize 2 X X i(φk−L −φk ) as |rk |2 = 1. rk−L rk e , k∈S

k∈S

i(φk−L −φk )

The maximum is attained only if e = const, ∀k ∈ S or (φk−L − φk ) ≡ α mod 2π, ∀k ∈ S, for some α ∈ R. Then we can take phases as follows φk = β + α hL,ki kLk2 , where β ∈ R. Therefore, the minimization problem (5) is reduced to the following P ck−L ck k∈S P 2 → max, ck ∈ R, ck ≥ 0. (7) ck k∈S

Let us rewrite the problem using quadratic forms. We enumerate all coefficients using one index according to the order of ”threads” in S and the order of the elements inside ”threads”. Hence, (7) can be written in matrix form P ck−L ck C TM C k∈S P 2 , = C TC ck k∈S

where C = {ck }k∈S is a column vector and M is a block diagonal matrix   1 0 2 0 ... 0   1 1  2 0 2 ... 0  M0 0 . . . 0   1  0 M1 . . .   0 0 ... 0  0     2 M = . .. . . ..  , Mi =  .. .. .. . . ..  , i = 0, . . . , u − 1.    ..  . . . . . . . . 1 1   0 0 . . . Mu−1 0 0 2 0 2 0 0 0 12 0

Here Mi is a (mi + 1) × (mi + 1) tridiagonal Toeplitz matrix with zeros on the main diagonal and halves on the sub- and super-diagonal. Therefore, it remains to

A directional uncertainty principle for periodic functions

17

find the maximal eigenvalue of the matrix M and the corresponding eigenvector, T since CC TMCC ≤ λmax (M ). The eigenvalues of these matrices Mi are known (see, e.g., [10, p. 53]). They are equal to cos mπn , n = 1, . . . , mi + 1. Since the set of i +2 eigenvalues of the block-diagonal matrix M is the union of eigenvalues of its blocks, the maximum eigenvalue of M is equal to λmax (M ) = cos m0π+2 . Moreover, the corresponding eigenvector also can be found. For the matrix Mi the eigenvector v (i,n) corresponding to the eigenvalue cos mπn is given coordinate-wise as follows i +2 (i,n)

vj

= sin

πnj , mi + 2

j = 1, . . . , mi + 1,

n = 1, . . . , mi + 1.

Therefore, the eigenvectors of the block-diagonal matrix M can be easily defined. Hence, the eigenvector Cmax corresponding to the maximal eigenvalue is given by (6). The above considerations yield that min {varA L (p), kpkTd = 1} =

p∈ΠS

1 π 1 −1= . − 1 = tan2 λ2max (M ) cos2 m0π+2 m0 + 2

Now we compute the directional uncertainty product for the polynomials with the minimal angular variance. In fact, it remains to compute the frequency variance: 0 +1 X 2 mX πn m0 + 2 = , sin2 ck = kpmin k2Td = m + 2 2 0 n=1 k∈S

X

k∈S

hL, kic2k =

X

k∈S

mX 0 +1 n=1

hL, ki2 c2k = = hL, k0 i2

hL, k0 + nLi sin2

mX 0 +1 πn m0 + 2 πn = hL, k0 i + kLk2 , n sin2 m0 + 2 2 m0 + 2 n=1

mX 0 +1 n=1

hL, k0 + nLi2 sin2

πn m0 + 2

mX mX 0 +1 0 +1 m0 + 2 πn πn n sin2 n2 sin2 + 2hL, k0 ikLk2 + kLk4 . 2 m0 + 2 m0 + 2 n=1 n=1

Based on trigonometric formulas, the formulas for the Dirichlet kernel and for the conjugate Dirichlet kernel, and taking the derivatives of those kernels we compute

min )= varF L (p

P

k∈Zd

hL, ki2 |ck |2

kpmin k2Td

= kLk4



 P

 −

k∈Zd

hL, ki|ck |2

kpmin k2Td

π 1 m20 + 4m0 − cot2 12 2 m0 + 2

2  



.

Finally, d

UPTL (pmin ) =

m0 (m0 + 4) π 1 tan2 − . 12 m0 + 2 2

This finishes the proof. ♦ d Note, for m0 → ∞, we obtain UPTL (pmin ) →

π2 12

−

1 2

≈ 0.3224 > 14 .

18

A. Krivoshein et al.

Next, we establish a similar result for the uncertainty product defined by Goh Qd and Goodmand in case where the coefficients support S is a rectangle S = j=1 [−Nj , Nj ] ∩ Z , where all Nj > 0. The problem is to minimize 2   d −2 d X X X X d − ck−ej ck  ck−ej ck   varA GG (p) = 

j=1 k∈S

j=1 k∈S

the above considerations it follows that for kpk2Td = 1 the when kpk2Td = 1. From P π ck−ej ck for any j = 1, . . . , d cannot be greater than cos m+1 sum , where k∈S Qd d m is the length of the longest ”thread”. Since S = j=1 [−Nj , Nj ] ∩ Z and P L = ej , then ck−ej ck ≤ cos 2Nπj +2 , j = 1, . . . , d and for fixed j the equality k∈S P ck−ej ck = cos 2Nπj +2 attains, if k∈S

ck =

(

√

1 Nj +1

sin

πl 2Nj +2

0

if kj = −Nj − 1 + l, l = 1, . . . , 2Nj + 1,

ki = 0, for i 6= j,

else,

where k = (k1 , . . . , kd ). If it is possible to achieve for some pmin those values cos 2Nπj +2 for all j = 1, . . . , d simultaneously, then we get the minimal possible value for varA GG (p) which is equal to   −2 d d X X π π 2 A   . cos varGG (pmin ) = d − cos (8) 2Nj + 2 2Nj + 2 j=1

j=1

Theorem 5 The value min {varA GG (p), kpkTd = 1} is given by (8) and it is atp∈ΠS

tained by pmin if the Fourier coefficients ck of pmin are given by ck =

d Y

j=1

p

πlj 1 sin , 2Nj + 2 Nj + 1

where lj = kj + Nj + 1, j = 1, . . . , d, k ∈ S.

P ck−ej ck = cos 2Nπj +2 are Proof. Let us show that the maximum values k∈S

attained for all j = 1, . . . , d simultaneously. This can be checked by direct computations. Let us fix i = 1, . . . , d. Therefore, X

ck−ei ck =

k∈S

=

d Y

j=1



2N 2N d 1 +1 d +1 Y X X πlj π(lj − δij ) 1 sin ··· sin Nj + 1 2Nj + 2 2Nj + 2 j=1 j=1 d Y

d Y 1  Nj + 1

l1 =1

2Nj +1

X

j=1,j6=i lj =1

ld =1

 2N i +1 X πl πli π(li − 1) j  sin sin2 sin . 2Nj + 2 2Ni + 2 2Ni + 2 li =1

A directional uncertainty principle for periodic functions

It remains to note that

2NP j +1

sin2

lj =1

2N i +1 X li =1

sin

πlj 2Nj +2

19

= Nj + 1 and

  2Ni +1 πli π(li − 1) 1 X π(li − 1) π(li + 1) πli sin = sin + sin sin 2Ni + 2 2Ni + 2 2 l =1 2Ni + 2 2Ni + 2 2Ni + 2 i

2Ni +1

=

X

sin2

li =1

πli π π cos = (Ni + 1) cos .♦ 2Ni + 2 2Ni + 2 2Ni + 2

4 Well localized multivariate periodic Parseval wavelet frames In this section we design a family of well-localized multivariate periodic Parseval wavelet frames. This is a generalization of the wavelet family constructed in [9]. It turns out that these wavelet frames have optimal localization with respect to the dimension d of the torus T d . More precisely, we claim that d

lim UPTL (ψj ) =

j→∞

1 (d + 2)(d2 − 2d + 4) , 4 d3

d

so limd→∞ limj→∞ UPTL (ψj ) = 14 . Let A ∈ Zd×d be a dilation matrix that means that all the eigenvalues of the matrix are greater then 1. The determinant of A equal to 2. Let {0, k0 } be a full d d T d j d collection of coset representatives  of Z /AZ  , B = A , Kj = Z ∩ B [−1/2, 1/2) . 2

2

kkk Put by definition fj (k) = exp − kLk j(j−1)

, where L ∈ Zd , j ≥ 2. Let us define a

B j -periodic sequence νj (k)  k ∈ int(Kj−1 ),  fj (k)
1/2 νj (k) = k − B j−1 k0 ∈ int(Kj−1 ), 1√ − fj2 (k − B j−1 k0 )  k ∈ K j−1 \ int(Kj−1 ), 1/ 2

(9)

where int(Kj ) = Zd ∩ B j (−1/2, 1/2)d and K j = Zd ∩ B j [−1/2, 1/2]d . The B j periodicity means that νj (k +  B j p) =νj (k) for any k, p ∈ Zd . 1 1 , then on the main period Kj the sequence For instance, if d = 2, A = −1 1 νj (k) is defined as follows  k ∈ int(Kj−1 ),  fj (k)
1/2 j−2[j/2] νj (k) = 1√ − fj2 (k − B j−1 (v1 (r) v2 (r))T ) B k ∈ Qr , k ∈ Kj \ Kj−1 ,  1/ 2 k ∈ K j−1 \ int(Kj−1 ),

where Qr is the r-th quadrant of R2 , v1 (r) = − cos(πr/2), v2 (r) = − sin(πr/2), r = 1, . . . , 4, [y] = max{n ∈ N : n ≤ y}. Finally, let us define an auxiliary function ξj ∈ L2 (Td ) with the Fourier coefficients ξbj (k) :=

∞ Y

r=j+1

νr (k).

20

A. Krivoshein et al.

Later, in Theorem 6, we will prove that the infinite product converges. Then scaling masks, scaling functions, wavelet masks, and wavelet functions are defined respectively as √ µj (k) := 2νj (k), ϕ cj (k) := 2−j/2 ξbj (k), (10) −j λj (k) := e2πihk0 ,B ki µj (k + B j−1 k0 ), ψbj (k) := λj+1 (k)ϕ bj+1 (k).

Theorem 6 Suppose ϕj , ψj are the functions defined in (10) and νj is a sequence defined in (9). Then the set Ψ = {ϕj , ψj (· − (A−j k)}j∈N∪{0},k∈Lj , where Lj is a full collection of coset representatives of Zd /Aj Zd , forms a Parseval frame of L2 (Td ), and the following equalities hold true d

d

lim UPTL (ψj ) =

lim UPTL (ϕj ) = 1/4,

j→∞

j→∞

1 (d + 2)(d2 − 2d + 4) . 4 d3

(11)

The scheme of the proof repeats in the main features Theorem 4 [9]. At the same time, there are differences concerning technical details. In particular, we have to provide a new proof for an analogue of Lemma 3 [9] since the existing proof can not be rewritten for the multivariate case. We exploit Lemma 2 [9], so we cite it here for convenience. Lemma 8 (Lemma 2 [9]) Suppose α, β, γ ∈ R, m = 0, 1, . . . , and 0 < b < b1 , where b1 is an absolute constant, then X

(αk2 + βk + γ)m e−b(αk

k∈Z

∂m = (−1) ∂bm m

2

+βk+γ)

 r       π2 − ε π β2 + exp − O(1), exp −b γ − 4α bα bα

as b → 0, where ε > 0 is an arbitrary small parameter. We need also several technical lemmas. Lemma 9 Let Bjθ (0) ⊂ Rd be a ball centered at the origin with radius 21 (1 + θ)j , θ > 0. Let M ∈ Zd×d be a dilation matrix with determinant equal to 2. Then there exists θ0 > 0 and j0 ∈ N such that Bjθ0 (0) ⊂ M j Td for j ≥ j0 . Proof. Let ρ = inf j kM −j k1/j be the spectral radius of the matrix M . Since M is a dilation matrix, it follows that ρ < 1. Given 0 < ε < 1 − ρ, there exists j0 ∈ N such that (ρ + ε)j ≥ kM −j k for j ≥ j0 . Suppose x ∈ Bjθ (0), that is kxk ≤ 12 (1 + θ)j , then kM −j xk ≤ kM −j kkxk ≤

1 (1 + θ)j (ρ + ε)j for j ≥ j0 . 2

Therefore, any θ satisfying the inequality 0 < θ < (ρ + ε)−1 − 1 can be chosen as θ0 . This concludes the proof of Lemma 9. ♦

A directional uncertainty principle for periodic functions

21

Lemma 10 Suppose b = b(h) = 2h2 /(1 − h), h > 0, and

 
1 exp(−hkLk2 (kxk2 + kx − Lk2 )), F (x) := bkLk2 kxkkx − Lk 1 − bkLk2 kxk2 + kx − Lk2 4

then

X

k∈Zd

F (k) =

Z

Rd

F (x) dx + O(h2 ) as h → 0.

Proof. The Poisson summation formula X X F (k) = Fb (k) k∈Zd

k∈Zd

shows that it is sufficient to prove X Fb (k) = O(h2 ) as h → 0. k∈Zd \{0}

So, we need only to find the Fourier transform of F . To this end, we rewrite the function F as F (x) =

2 1 1 f1 (x)f1 (x − L) − f2 (x)f1 (x − L) − f1 (x)f2 (x − L), 1−h (1 − h)2 (1 − h)2

where f1 (x) = hkLkkxkexp(−hkLk2kxk2 ), f2 (x) = h3 kLk3 kxk3 exp(−hkLk2kxk2 ). Therefore, Fb can be written as

Fb (ξ) =

1 1 2 b f1 ∗ f1\ (· − L)(ξ)− fb2 ∗ f1\ (· − L)(ξ)− fb1 ∗ f2\ (· − L)(ξ). 1−h (1 − h)2 (1 − h)2

It follows from elementary properties of the Fourier transform that     ξ ξ −d 1/2−d/2 b −d 3/2−d/2 b b b f1 (ξ) = kLk h f3 , f2 (ξ) = kLk h f4 , h1/2 kLk h1/2 kLk

where

f3 (x) = kxkexp(−kxk2),

f4 (x) = kxk3 exp(−kxk2).

Since f3 is a radial function, we can exploit Theorem 3.3 chapter IV [18]. So, we get Z ∞ 2 d d r 2 +1 e−r J d−2 (2πkξkr) dr, fb3 (ξ) = 2πkξk− 2 +1 0

2

where Jn is a Bessel function of the first kind. By [1, Formula 11.4.28, p. 486] we conclude Γ (d/2 + 1/2) fb3 (ξ) = π d/2 M (d/2 + 1/2, d/2, −π 2 kξk2 ), Γ (d/2)

where M is Kummer’s (confluent hypergeometric) function. The asymptotic behavior as ξ → ∞ of this function is known and can be found, for instance, in [1, Formula 13.1.05, p. 504], therefore we obtain fb3 (ξ) = −1/2 π −d/2−3/2 Γ (d/2 + 1/2)kξk−1−d(1 + O(kξk−2 )) as ξ → ∞.

22

A. Krivoshein et al.

Analogously fb4 (ξ) = 3/4 π −d/2−7/2 Γ (d/2 + 3/2)kξk−3−d(1 + O(kξk−2)) as ξ → ∞.

Thus, we get

fb1 (ξ) = C1 (d)hkξk−1−d(1 + O(hkξk−2)) as ξ → ∞,

fb2 (ξ) = C2 (d)h3 kξk−3−d (1 + O(hkξk−2 )) as ξ → ∞,

and C1 (d) = −1/2 kLkπ −d/2−3/2 Γ (d/2+1/2), C2 (d) = 3/4 kLk3 π −d/2−7/2 Γ (d/2+ 3/2). Since, in addition, the functions fb1 and fb2 are bounded, the convolutions fb1 ∗ f1\ (· − L)(ξ), fb2 ∗ f1\ (· − L)(ξ), and fb1 ∗ f2\ (· − L)(ξ) are well-defined. Therefore, Fb (k) = O(fb1 ∗ f1\ (· − L)(k)) = O(h2 kkk−2 ) as h → 0. Thus,

X

k∈Zd \{0}

which proves the result. ♦

Fb (k) = O(h2 ) as h → 0,

Lemma 11 Let ξj0 ∈ L2 (Td ) be a function defined by its Fourier coefficients ξbj0 (k) = exp(−kLk2 kkk2 /j).

Then

d

lim UPTL (ξj0 ) =

j→∞

1 . 4

Proof. Since the coefficient ξbj0 (k) is a product of d one-dimensional corresponding it follows that it to apply Lemma 8 to the series coefficients, 2 is sufficient P P b0 2 P 2 b0 b0 b0 d ξ (k) , d hL, ki ξ (k) , and d ξ (k − L)ξ (k), and then to k∈Z

j

j

k∈Z

k∈Z

substitute the results into the definition of computations. ♦

d UPTL .

j

j

The result follows by simple

Lemma 12 Let ηj ∈ L2 (Td ) be a function defined by its Fourier coefficients ηbj (k) = e2πihk0 ,B

−j

ki

  1/2   2kLk2kkk2 kLk2 kkk2 1 − exp − . exp − j(j + 1) j+1

Then

d

lim UPTL (ηj ) =

j→∞

1 (d + 2)(d2 − 2d + 4) . 4 d3

Proof.PDenote h =P1/(j + 1), b = b(h) = 2h2 /(1 − h) = 2/(j(j + 1)). To hL, ki2 |ηbj (k)|2 one can use Lemma 8 as it was described |ηbj (k)|2 , estimate k∈Zd

k∈Zd

in Lemma 3 [9]. Namely, X

k∈Zd

|ηbj (k)|2 =

  X  2h kLk2 kkk2 exp(−2hkLk2kkk2 ) − exp − 1−h d

k∈Z

A directional uncertainty principle for periodic functions

=

d X Y

n=1 kn ∈Z

=



2 )− exp(−2hkLk2kn

π 2hkLk2

d/2

−



d X Y

n=1 kn ∈Z

π(1 − h) 2hkLk2

23

  2h 2 kLk2 kn exp − 1−h

d/2

−1

+ O(e−h ).

P Since k∈Zd kn km |ηbj (k)|2 = 0, we analogously get X 2 X k1 |ηbj (k)|2 hL, ki2 |ηbj (k)|2 = kLk2 k∈Zd

k∈Zd

= kLk2

  −1 π d/2 d/2+1 1 − (1 − h) + O(e−h ), 2(2hkLk2)d/2+1

where k = (k1 , k2 , . . . , kd )T . P However, we have to provide an alternative way to estimate ηbj (k−L)ηbj (k). k∈Zd

We write

 1/2 X  X 2 2 = η b (k − L) 1 − exp −bkLk kkk η b (k) j j k∈Zd k∈Zd

1/2   × 1 − exp −bkLk2 kk − Lk2 exp(−hkLk2(kkk2 + kk − Lk2 )).

Using the Taylor formula for the function f (b) = (1 − exp(−bkLk2kkk2 ))1/2 (1 − exp(−bkLk2kk − Lk2 ))1/2 in the neighborhood of b = 0, we get     ¯ b3 /6, f (b) = bkLk2 kkkkk − Lk 1 − bkLk2 kkk2 + kk − Lk2 /4 + f ′′′ (d)

¯ 3 = O(kkk6 h6 ). The last equality is deduced in Lemma 3 [9]. So, using and f ′′′ (d)b Lemma 8 for the remainder of the series we get   ¯ X f ′′′ (d) X 6 6 2 3 2 2 2 kkk exp(−hkkk ) = O(h3−d/2 ). b exp(−hkLk (kkk +kk −Lk )) = O h 6 d d

k∈Z

k∈Z

Therefore,    X X 1 2 2 2 2 ηbj (k − L)ηbj (k) = bkLk kkk kk − Lk 1 − bkLk kkk + kk − Lk 4 k∈Zd k∈Zd ×exp(−hkLk2 (kkk2 + kk − Lk2 )) + O(h3−d/2 ).

Next, by Lemma 10 we replace the series by the integral bkLk2

Z

Rd

 
1 kxk kx − Lk 1 − bkLk2 kxk2 + kx − Lk2 exp(−hkLk2 (kxk2 + kx + Lk2 )) dx, 4

change the variable y = x − L/2 to obtain bkLk2



     Z 2 



kLk2 2 2

y − L y + L 1 − 1 bkLk2 2kyk2 + kLk exp −hkLk dy, 2kyk +

2

2 4 2 2

Rd

24

A. Krivoshein et al.

and convert it to the polar coordinates y = rβ, where β = (sin φ1 sin φ2 . . . sin φd−1 , cos φ1 sin φ2 . . . sin φd−1 , . . . , cos φd−1 )T . So, we get   kLk4 bkLk2 exp −h 2

Z

[0, ∞)×[0, 2π)×[0, π)d−2

r2



1 + r −2

kLk2 4

2

− r −2 hL, βi2

!1/2

   kLk2 1 exp(−2hkLk2 r 2 ) × 1 − bkLk2 2r 2 + 4 2 ×r d−1 sin φ2 sin2 φ3 . . . sind−2 φd−1 dr dφ.


1/2 Next, applying the Taylor formula for (1 + r −2 kLk2 /4)2 − r −2 hL, βi2 with 1/2 respect to 1/r, changing the variable (2h) kLkr = t, integrating with respect to φ, and recalling that b = 2h2 (1 − h)−1 , we obtain   d/2 2h π 1 kLk4 exp −h 1−h 2 2hkLk2 Γ (d/2)    Z ∞ d − 2 kLk4 ht2 × td+1 exp(−t2 ) 1 + dt. h 1 − 2d t2 2(1 − h) 0

Integrating with respect to t, we finally obtain

=

X η b (k) η b (k − L) j j k∈Zd

   d/2  π d h kLk4 d−2 h d(d + 2) 4 2 exp −h + kLk h − + O(h ) . 1−h 2 2hkLk2 2 2d 1−h 8

P hL, ki|ηbj (k)|2 = 0. It remains to substitute the expresIt is easy to see that d k∈Z P P P ηbj (k−L)ηbj (k) to the definition hL, ki2 |ηbj (k)|2 , and |ηbj (k)|2 , sions for k∈Zd

of

d UPTL .

k∈Zd

k∈Zd

Lemma 12 is proved. ♦

Proof of Theorem 6. First of all, it is straightforward to see that the infinite Q product ξbj (k) := ∞ r=j+1 νr (k) converges. As usual, if an infinite product is equal to zero then it is also considered convergent. Indeed, it follows from (9) that νj (k) = fj (k) for k ∈ int(Kj−1 ), and Lemma 9 says that there exist θ0 > 0 and j0 ∈ N such that kkk ≤ (1 + θ0 )j /2 implies k ∈ int(Kj−1 ) for j ≥ j0 . Therefore, νj (k) = fj (k) for kkk ≤ (1 + θ0 )j /2. So, we get  !   j1 j1 ∞  2 2 Q Q Q   νr (k) fr (k) = νr (k) exp − kLkj1kkk , j ≥ j1  r=j1 +1 r=j+1 ξbj (k) = r=j+1   ∞ 2 2 Q  kLk kkk   fr (k) = exp − , j < j1  j r=j+1

(12) where j1 = ⌊log1+θ0 (2kkk)⌋ + 1. Therefore, ξbj (k) is well-defined and ξj ∈ L2 (Td ). Then one can check that all conditions of the unitary extension principle are

A directional uncertainty principle for periodic functions

25

fulfilled for the functions ϕj , ψj (see Theorem 2.2 [7]). Therefore, the set Ψ = {ϕj , ψj (· − (A−j k)}j∈N∪{0},k∈Lj , forms a Parseval frame of L2 (Td ). To check (11), as in the univariate case, we introduce two auxiliary functions ξj0 and ηj by the Fourier coefficients

ηbj (k) = e2πihk0 B

Now we claim that

−j

ki

ξbj0 (k) = exp(−kLk2 kkk2 /j),

1/2     kLk2 kkk2 2kLk2 kkk2 . exp − 1 − exp − j(j + 1) j+1 d X

k(ξj0 − ξj )′n kL2 (Td ) = 0,

lim kηj − 2j/2 ψj kL2 (Td ) +

d X

k(ηj − 2j/2 ψj )′n kL2 (Td ) = 0,

kξj0 − ξj k2L2 (Td ) +

d X

k(ξj0 − ξj )′n k2L2 (Td )

lim kξj0 − ξj kL2 (Td ) +

j→∞

j→∞

n=1

n=1 ′ where fn means again the partial derivative of f with respect to xn . Indeed, cj (k) for k ∈ int(Kj−1 ), and, therefore, for Since ξbj0 (k) = ξbj (k) and ηbj (k) = 2j/2 ψ j kkk ≤ (1 + θ0 ) /2 (see Lemma 9), it follows that

X

=

n=1

d 2 X b 2 ξj (k) − ξb0 (k) + 4π

X

j

kkk≥(1+θ0 )j /2

By (12), we have

n=1 kkk≥(1+θ0 )j /2

2 2 b kn ξj (k) − ξbj0 (k) .

  kLk2 kkk2 b b 0 . ξj (k) − ξj (k) ≤ 2exp − j1

Substituting this majorant to the series, we get that the series tends to zero as j → ∞ as a remainder of a convergent series. For the functions ηj and 2j/2 ψj d it can be checked analogously. functional UPTL is continuous with respect Pd The to the norm kf kL2 (Td ) + n=1 kfn′ kL2 (Td ) , which can be checked as in the oned

d

dimensional case in Lemma 1 [9]. Moreover, UPTL (ξj0 ), UPTL (ηj ) are bounded with respect to j, which follows from Lemma 11 and Lemma 12. Therefore, d

d

lim UPTL (ϕj ) = lim UPTL (ξj0 ), j→∞

j→∞

d

d

lim UPTL (2j/2 ψj ) = lim UPTL (ηj ). j→∞

j→∞

d

d

d

Finally, the functional UPTL is homogeneous that is UPTL (αf ) = UPTL (f ), α 6= 0. So d d UPTL (2j/2 ψj ) = UPTL (ψj ). Thus, d

d

lim UPTL (ϕj ) = lim UPTL (ξj0 ),

j→∞

j→∞

d

d

lim UPTL (ψj ) = lim UPTL (ηj ).

j→∞

j→∞

To conclude the proof of Theorem 6 it remains to apply Lemma 11 and Lemma 12. ♦

26

A. Krivoshein et al.

Acknowledgements The authors thank Professor O.L.Vinogradov for a valuable observation in the proof of Lemma 12.

References 1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied mathematics series. U.S. Government Printing Office, 1964. 2. E. Breitenberger. Uncertainty measures and uncertainty relations for angle observables. Foundations of Physics, 15(3):353–364, 1985. 3. G. B. Folland and A. Sitaram. The uncertainty principle: A mathematical survey. Journal of Fourier Analysis and Applications, 3(3):207–238, 1997. 4. G.B. Folland. Harmonic Analysis in Phase Space. Annals of mathematics studies. Princeton University Press, 1989. 5. S. S. Goh and T. N.T. Goodman. Uncertainty principles and asymptotic behavior. Applied and Computational Harmonic Analysis, 16(1):19 – 43, 2004. 6. S.S. Goh and C.A. Micchelli. Uncertainty principles in Hilbert spaces. Journal of Fourier Analysis and Applications, 8(4):335–374, 2002. 7. S.S. Goh and K.M. Teo. Extension principles for tight wavelet frames of periodic functions. Applied and Computational Harmonic Analysis, 25(2):168 – 186, 2008. 8. A. V. Krivoshein and E. A. Lebedeva. Uncertainty principle for the cantor dyadic group. Journal of Mathematical Analysis and Applications, 423(2):1231–1242, 2015. 9. E.A. Lebedeva and J. Prestin. Periodic wavelet frames and time-frequency localization. Applied and Computational Harmonic Analysis, 37(2):347 – 359, 2014. 10. G. Meinardus and L. L. Schumaker. Approximation of functions: theory and numerical methods. Springer Tracts in Natural Philosophy. Springer, 1 edition, 1967. 11. F. J. Narcowich and J. D. Ward. Nonstationary wavelets on the m-sphere for scattered data. Applied and Computational Harmonic Analysis, 3(4):324–336, 1996. 12. J. Prestin and E. Quak. Time frequency localization of trigonometric Hermite operators. In IN APPROXIMATION THEORY VIII, VOL. 2: WAVELETS AND MULTILEVEL APPROXIMATION, pages 343–350. World Scientific Publishing Co., Inc, 1995. 13. J. Prestin and E. Quak. Optimal functions for a periodic uncertainty principle and multiresolution analysis. Proceedings of the Edinburgh Mathematical Society (Series 2), 42:225–242, 6 1999. 14. J. F. Price and A. Sitaram. Local uncertainty inequalities for locally compact groups. Transactions of the American Mathematical Society, 308(1):105–114, 1988. 15. H. Rauhut. Best time localized trigonometric polynomials and wavelets. Advances in Computational Mathematics, 22(1):1–20, 2005. 16. E. Schr¨ odinger. About Heisenberg uncertainty relation. Proceedings of Prussian Academy of Science, pages 296–303, 1930. 17. K. Selig. Uncertainty principles revisited. ETNA. Electronic Transactions on Numerical Analysis [electronic only], 14:165–177, 2002. 18. E.M. Stein and G.L. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Mathematical Series. Princeton University Press, 1971.