Orthonormality of wavelet system on the Heisenberg group

Accepted Manuscript Orthonormality of wavelet system on the Heisenberg group

S. Arati, R. Radha

PII: DOI: Reference:

S0021-7824(19)30041-8 https://doi.org/10.1016/j.matpur.2019.02.004 MATPUR 3091

To appear in:

Journal de Mathématiques Pures et Appliquées

Received date:

29 May 2018

Please cite this article in press as: S. Arati, R. Radha, Orthonormality of wavelet system on the Heisenberg group, J. Math. Pures Appl. (2019), https://doi.org/10.1016/j.matpur.2019.02.004

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Orthonormality of wavelet system on the Heisenberg group S. Arati, R. Radha∗ Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Abstract On the real line, one of the interesting problems in wavelet analysis is to obtain a characterization for the orthonormality of a wavelet system. In this paper, we wish to characterize the orthonormality of the wavelet system on the Heisenberg group. We also attempt to study the completeness of this system. Further, we consider the wavelet system associated with twisted translations and dilations on L2 (C) and study its orthonormality. R´ esum´ e La caract´erisation de l’orthonormalit´e d’un syst`eme d’ondelettes d´efini sur la ligne r´eelle (R) se trouve parmi les probl`emes int´eressants dans l’analyse d’ondelettes. Ici, on caract´erise l’orthonormalit´e d’un syst`eme d’ondelettes d´efini sur le groupe de Heisenberg. On essaie ´egalement d’´etudier sa compl´etude. On consid`ere aussi l’orthonormalit´e d’un syst`eme d’ondelettes associ´e aux translations avec torsion, et aux dilations, sur L2 (C). Keywords: Heisenberg group, nonisotropic dilation, orthonormal basis, twisted translation, wavelets, Weyl transform 2010 MSC: Primary 42C40 ; Secondary 43A30, 42B10

1. Introduction Let ψ ∈ L2 (R). Define ψj,k as ψj,k (x) = 2j/2 ψ(2j x − k), for j, k ∈ Z and x ∈ R. In other words, ψj,k = D2j Tk ψ, where Tu , u ∈ R denotes the translation operator Tu f (x) = f (x − u) and Da , a ∈ R∗ denotes the dilation 5

operator Da f (x) = |a|1/2 f (ax), x ∈ R. This system {ψj,k : j, k ∈ Z} is in ∗ Corresponding

author Email addresses: [email protected] (S. Arati), [email protected] (R. Radha)

Preprint submitted to Journal de Mathematiques Pures et Appliquees

February 8, 2019

general called a wavelet system (cf [7]). The following well known result gives a characterization of the orthonormality of the wavelet system. Theorem 1.1. Let ψ ∈ L2 (R). The necessary and sufficient conditions for the orthonormality of the system {ψj,k : j, k ∈ Z} are 

ˆ + k)|2 = 1 |ψ(ξ

for a.e. ξ ∈ R

k∈Z

and



ˆ + k)ψ(2 ˆ j (ξ + k)) = 0 ψ(ξ

for a.e. ξ ∈ R, j ≥ 1.

k∈Z

Meyer, in [13], constructed a class of orthonormal wavelet bases {ψj,k : j, k ∈ Z} with ψ belonging to the Schwartz space of rapidly decreasing functions, S(R), ψ 10

compactly supported and ψ satisfying the vanishing moment property, namely,  m x ψ(x)dx = 0, ∀ m ≥ 0. In fact, there is a general approach to construct an R orthonormal wavelet basis called multiresolution analysis. This notion was first introduced by Meyer in [14] and independently studied and developed by Mallat in [11]. Further, the following theorem gives necessary and sufficient conditions

15

for the wavelet system to be an orthonormal basis for L2 (R). Theorem 1.2. Let ψ ∈ L2 (R) be such that ψ2 = 1. Then the wavelet system {ψj , k : j , k ∈ Z} is an orthonormal basis for L2 (R) if and only if 

and

j∈Z ∞ 

ˆ j ξ)|2 = 1, |ψ(2

for a.e. ξ ∈ R,

ˆ j ξ)ψ(2 ˆ j (ξ + q)) = 0, ψ(2

for a.e. ξ ∈ R, q ∈ 2Z + 1.

j=0

For the proofs of Theorem 1.1 and Theorem 1.2, we refer to [9]. In this paper, our primary objective is to obtain a characterisation for the orthonormality of the wavelet system on the Heisenberg group H. Let L be a lattice in H. In other words, L is a discrete subgroup of H such that H/L is compact. For the wavelet system on H, we need to look at the left translations by elements of L and the nonisotropic dyadic dilations on H. For the sake of computational convenience, we take the standard lattice {(2k, l, m) : k, l, m ∈ Z} in place of L. The left translations L(2k,l,m) , (k, l, m) ∈ Z3 and the nonisotropic 2

dyadic dilations δ2j , j ∈ Z are defined as follows. For ψ ∈ L2 (H), L(u,v,s) ψ(x, y, t) = ψ((u, v, s)−1 (x, y, t))   1 = ψ x − u, y − v, t − s + (y · u − x · v) , 2 (u, v, s) ∈ R3 and δa ψ(x, y, t) = |a|2 ψ(ax, ay, a2 t), a ∈ R∗ . In [12], Mayeli considered this system associated with Shannon wavelet on the Heisenberg group and proved that the resulting system forms a normalized tight frame for L2 (H). 20

Our interest is to consider a general ψ ∈ L2 (H) and find under what conditions this system will become an orthonormal system. We first give an example of an orthonormal wavelet system in L2 (H). Later, we obtain necessary and sufficient conditions for a wavelet system to be orthonormal in L2 (H). Then we illustrate with an example in which the orthonormality of the wavelet system is

25

verified using the equivalent conditions. We also wish to investigate the completeness of this orthonormal wavelet system. However, we prove that under a certain condition, the orthonormal system {δ2j L(0,0,m) ψ : m, j ∈ Z} turns out to be complete in L2 (H). But we could not succeed in our actual goal of obtaining a sufficient condition for the completeness of teh orthonormal system

30

{δ2j L(2k,l,m) ψ : k, l, m, j ∈ Z} which is naturally an interesting problem and we leave it as an open question to the interested readers. In the final part of the paper, we consider the wavelet system associated with twisted translations and dilations on L2 (C) and look into the problem of obtaining necessary and sufficient conditions for orthonormality of this twisted

35

wavelet system on C. Surprisingly, we find that the required conditions become too complicated unlike the classical result on R or the result on the Heisenberg group. Now, we shall mention a few works based on system of translates in various settings available in the literature. Characterizations of shift invariant spaces

40

in L2 (Rn ) in terms of range functions were obtained by Bownik in [2]. These results were later extended to locally compact abelian groups in [5] and [10]. 3

For nonabelian compact groups, the shift invariant spaces were explored in [17]. Characterization of the orthonormality of a system of translates on the polarised Heisenberg group was studied in [1] using the concept of bracket map. 45

The shift invariant spaces associated with the twisted translations for L2 (Cn ) were recently studied by Radha and Adhikari in [15]. In [16], characterizations of Bessel sequences, orthonormal bases, frames and Riesz bases were studied on the Heisenberg group for a shift invariant space with countably many mutually orthogonal generators. The structural properties of shift-modulation invariant

50

spaces were studied by Bownik in [3]. These results were extended to locally compact abelian groups in [6]. Recently, in [4], Bownik and Ross considered translation-invariant spaces on a locally compact abelian group G (invariant under translations by elements of closed co-compact subgroup not necessarily discrete) and characterized it in terms of range functions.

55

In the following section, we shall provide the required preliminaries.

2. Notation and background The Heisenberg group Hn is a Lie group whose underlying manifold is Rn × Rn × R which satisfies the group law   1 (x, y, t)(u, v, s) = x + u, y + v, t + s + (u · y − v · x) . 2 It is a nonabelian noncompact locally compact group. The Haar measure on Hn is the Lebesgue measure dxdydt. It follows from the well known Stone-von Neumann theorem that every infinite dimensional irreducible unitary representation on the Heisenberg group is unitarily equivalent to the representation πλ , λ ∈ R∗ , where πλ is defined by 1

πλ (x, y, t)ϕ(ξ) = e2πiλt e2πiλ(x·ξ+ 2 x·y) ϕ(ξ + y),

ϕ ∈ L2 (Rn ).

For f ∈ L1 (Hn ), the group Fourier transform fˆ is defined as follows. For λ ∈ R∗ , fˆ(λ) given by

 fˆ(λ) =

f (z, t)πλ (z, t)dzdt Cn ×R

4

is a bounded operator on L2 (Rn ). In other words, for ϕ ∈ L2 (Rn ), we have  fˆ(λ)ϕ = f (z, t)πλ (z, t)ϕdzdt, Cn ×R

where the integral is a Bochner integral taking values in L2 (Rn ). The inverse Fourier transform of f ∈ L1 (Hn ) in the t variable, denoted by f λ , is defined as  λ f (z) = f (z, t)e2πiλt dt. R

It can be seen that f λ ∈ L1 (Cn ). For f ∈ L1 (Cn ), the operator Wλ (f ) on L2 (Rn ), is defined as

 Wλ (f ) =

f (z)πλ (z, 0)dz. Cn

Clearly, there is a relation between group Fourier transform and Wλ given by fˆ(λ) = Wλ (f λ ).

(1)

Moreover, Wλ (f ) is an integral operator on L2 (Rn ) with kernel Kfλ given by  λ Kf (ξ, η) = f (x, η − ξ)eπiλx·(ξ+η) dx. (2) Rn

In particular when λ = 1, Wλ (f ) is denoted by W (f ) which is called the Weyl transform of f and the associated kernel is denoted by Kf . As in the case of the Euclidean Fourier transform, the definitions of Wλ and the group Fourier transform fˆ can be extended to functions in L2 (Cn ) and L2 (Hn ) respectively through the density argument. In fact, for f ∈ L2 (Cn ), Wλ (f ) is a Hilbert-Schmidt operator on L2 (Rn ) which satisfies Wλ (f )B2 = Kfλ L2 (Cn ) =

1 f L2 (Cn ) , |λ|n/2

(3)

where B2 = B2 (L2 (Rn )) denotes the class of Hilbert-Schmidt operators on L2 (Rn ). In other words, for f, g ∈ L2 (Cn ), Wλ (f ), Wλ (g)B2 = Kfλ , Kgλ L2 (Cn ) =

1 2 n |λ|n f, gL (C ) .

Furthermore, the group Fourier transform satisfies the Plancherel formula fˆL2 (R∗ ,B2 ;dμ) = f L2 (Hn ) , 5

(4)

where L2 (R∗ , B2 ; dμ) stands for the space of functions on R∗ taking values in B2 and square integrable with respect to the measure dμ(λ) = |λ|n dλ. Equivalently, we have

 fˆ(λ), gˆ(λ)B2 |λ|n dλ = f, gL2 (Hn ) .

fˆ, gˆL2 (R∗ ,B2 ;dμ) = R

Then, it follows from (1) and (4) that  f, gL2 (Hn ) = Kfλλ , Kgλλ L2 (Cn ) |λ|n dλ.

(5)

R

For a further study on Heisenberg group, we refer to [8] and [19]. We shall now provide some definitions and theorems that are required to

60

understand our main results. Definition 2.1. Let H be a separable Hilbert space. A sequence {fk : k ∈ Z} in H is a frame for H if there exist constants A, B > 0 such that  Af 2 ≤ |f, fk |2 ≤ Bf 2 , ∀ f ∈ H. k∈Z

The constants A and B are called frame bounds. If the right hand side inequality holds, then {fk : k ∈ Z} is said to be a Bessel sequence with bound B. If the frame bounds satisfy A = B = 1 then the frame is called a Parseval frame. A 65

sequence {fk : k ∈ Z} in H is said to be a frame sequence if it is a frame for span{fk : k ∈ Z}. Definition 2.2 ([16]). For ψ ∈ L2 (H), (k, l) ∈ Z2 , the function Gψ k,l is defined as Gψ k,l (λ) =



 + r), L(2k,l,0)  ψ(λ + r)B2 |λ + r|, λ ∈ (0, 1]. ψ(λ

r∈Z

The following theorem on orthonormality of the system of translates {L(2k,l,m) ψ : (k, l, m) ∈ Z3 } in L2 (H) is well known. Theorem 2.3 ([16]). If ψ ∈ L2 (H), then {L(2k,l,m) ψ : (k, l, m) ∈ Z3 } is an orthonormal system in L2 (H) if and only if the following conditions hold. (i)Gψ 0,0 (λ) = 1 (ii)Gψ k,l (λ) = 0

a.e. λ ∈ (0, 1]

and

a.e. λ ∈ (0, 1], for all (k, l) = (0, 0) in Z2 , 6

where Gψ k,l is as in Definition 2.2. 70

See also [1]. The following theorems on the system {L(0,0,m) ψ : m ∈ Z} can be obtained as a particular case of the analogous theorems in [16] in the context of {L(2k,l,m) ψ : (k, l, m) ∈ Z3 }. Let V 0 (ψ) denote the closed linear span of {L(0,0,m) ψ : m ∈ Z}.

75

Theorem 2.4. Let ψ ∈ L2 (H). Then {L(0,0,m) ψ : m ∈ Z} is a Parseval frame for V 0 (ψ) if and only if Gψ 0,0 (λ) = 1 a.e. λ ∈ Ωψ , where Ωψ = {μ ∈ (0, 1] : Gψ 0,0 (μ) = 0}. Definition 2.5. Let ϕ, ψ ∈ L2 (H). If the system {L(0,0,m) ϕ : m ∈ Z} is a Parseval frame for V 0 (ψ), then the function ϕ is called a Parseval frame

80

generator of V 0 (ψ). Theorem 2.6. If a subspace V of L2 (H) is invariant under translations L(0,0,m) , 2 m ∈ Z, then there exists a family of functions {ψn }∞ n=1 in L (H) such that ∞  V = V 0 (ψn ) and each ψn is a Parseval frame generator of V 0 (ψn ). Moren=1

∞

n (λ) a.e. λ ∈ R and f 2 2 rn (λ)ψ over, if f ∈ V , then f(λ) = L (H) = 85

n=1

∞ n=1

rn 2L2 (T;Gψn ) , 0,0

2

n where rn ∈ L (T; Gψ 0,0 ).

n Remark 2.7. The space L2 (T; Gψ 0,0 ) represents the space of 1-periodic com-

plex valued functions on T which are square integrable with respect to the n weight function Gψ 0,0 .

Using Theorem 2.4, we may write rn L2 (T;Gψn ) = 0,0

rn L2 (T∩Ωψn ) = rn L2 (T) , by assuming rn to be zero on T \ Ωψn .

90

3. Orthonormality of wavelet system on H Using the translation and the nonisotropic dilation defined in Section 1, we consider the wavelet system {δ2j L(2k,l,m) ψ : k, l, m, j ∈ Z} given by  δ2j L(2k,l,m) ψ(x, y, t) = 22j ψ 2j x − 2k, 2j y − l, 22j t − m + 12 2j (2ky − lx)

7

for ψ ∈ L2 (H). An example of such a wavelet system that is orthonormal in L2 (H) is as follows. H H Example 3.1. Consider ψ(x, y, t) = χH [0,1] (x)χ[0,1] (y)χ[0,1] (t), where χ[0,1] de-

notes the Haar function on [0, 1] given by ⎧ ⎪ ⎪ 1, ⎪ ⎪ ⎨ χH −1, [0,1] (x) = ⎪ ⎪ ⎪ ⎪ ⎩ 0,

0 ≤ x ≤ 12 , 1 2

< x ≤ 1,

otherwise.

Let j1 , j2 , k1 , k2 , l1 , l2 , m1 , m2 ∈ Z. Then for any ψ ∈ L2 (H), δ2j1 L(2k1 ,l1 ,m1 ) ψ, δ2j2 L(2k2 ,l2 ,m2 ) ψL2 (H) = ψ, L(−2k1 ,−l1 ,−m1 ) δ2j2 −j1 L(2k2 ,l2 ,m2 ) ψL2 (H) = ψ, δ2j2 −j1 L(2k0 ,l0 ,m0 ) ψL2 (H) ,

(6)

where k0 = k2 − 2j2 −j1 k1 , l0 = l2 − 2j2 −j1 l1 and m0 = m2 − 22(j2 −j1 ) m1 + 2j2 −j1 (k1 l2 − k2 l1 ). When j2 ≥ j1 , we note that k0 , l0 , m0 are integers. For j > 0, k, l, m ∈ Z, we have ψ, δ2j L(2k,l,m) ψL2 (H)  H H j H j = 22j χH [0,1] (x)χ[0,1] (y)χ[0,1] (t)χ[0,1] (2 x − 2k)χ[0,1] (2 y − l) R3

× χ[0,1] (22j t − m + 12 2j (2ky − lx))dxdydt    H H  (y)χ   = 22j χH (y)χ (t) χH [0,1] [0,1] [0,1] (x)χ 2k l l+1 R

R

R

2j , 2j

2k+1 2j , 2j

 (x)

× χ[0,1] (22j t − m + 12 2j (2ky − lx))dxdydt. Clearly for k ≤ −1 and k ≥ 2j−1 , we have ψ, δ2j L(2k,l,m) ψ = 0. Similarly for l ≤ −1 and l ≥ 2j , one can see that ψ, δ2j L(2k,l,m) ψ = 0. So now we consider

8

0 ≤ k ≤ 2j−1 − 1, 0 ≤ l ≤ 2j − 1 and m ∈ Z. Then ψ, δ2j L(2k,l,m) ψ    = 22j (±1)χH 2k R

= 22j

R

R

× −

m 22j

where A =



×   R

2j

R

 (±1)χH 2k

2k+1 2j , 2j

 R

 (x)χH 

l l+1 2j , 2j

 (y)

χ[0,1] (t)χ[0,1] (22j t − m + 12 2j (2ky − lx))dtdxdy  (x)χH 

l l+1 2j , 2j

χ[0,1] (t)χA,A+

1 2j+1 (2ky

when [0, 1] ∩ [A, A +

2k+1 , 2j

1 22j ]

1 22j

 (y)

 (t)dtdxdy,

− lx). It is obvious that the inner product is zero

is a set of measure zero. It can be seen that the inner

product is zero even otherwise. For A ≤ 0 < A +

1 22j

< 1, we have

ψ, δ2j L(2k,l,m) ψ =2

2j

2j

 



R

 (x)χH  (y)   (±1)χH 2k 2k+1 l l+1 , , j j j j R 2 2 2 2

R

R

 

=2 =2

2j



R

(±1)( m+1 22j −

 (±1)χH l

l+1 2j , 2j

 ×  = R

95

I1

( m+1 22j

 cj,k,l χH l

where, I1 =

2j

 2k 2j

−

2ky 2j+1

l+1 , 2j

, 2k 2j +

2ky 2j+1

1dtdxdy

0

H H lx     2j+1 )χ 2k 2k+1 (x)χ l l+1 (y)dxdy , , j j j j 2 2 2 2

 (y)

 +



lx

2j+1 )dx

 (y)dy

1 2j+1

+

1 A+ 22j

( m+1 22j

−

2ky 2j+1

1 2k 2j+1 , 2j

+

1 2j

− I2

 +

lx

2j+1 )dx

dy

= 0,

, I2 =

 2k 2j

+



and cj,k,l is a constant

dependent on the integers j, k, l. By similar calculations, it can be shown that for 0 < A < A+ 212j < 1 and for 0 < A < 1 ≤ A+ 212j , the above inner product turns out to be zero. So we have that for j > 0, ψ, δ2j L(2k,l,m) ψ = 0, ∀ k, l, m ∈ Z. By (6), δ2j1 L(2k1 ,l1 ,m1 ) ψ, δ2j2 L(2k2 ,l2 ,m2 ) ψL2 (H) = 0, for all j1 = j2 , k1 , k2 , 100

l1 , l2 , m1 , m2 ∈ Z.

9

On the other hand, for j1 = j2 , δ2j1 L(2k1 ,l1 ,m1 ) ψ, δ2j2 L(2k2 ,l2 ,m2 ) ψL2 (H) = L(2k1 ,l1 ,m1 ) ψ, L(2k2 ,l2 ,m2 ) ψL2 (H) = ψ, L(2(k2 −k1 ),l2 −l1 ,m2 −m1 +k1 l2 −k2 l1 ) ψL2 (H) . Now for k, l, m ∈ Z, ψ, L(2k,l,m) ψ =   = R

R

(7)

 R3

H H H χH [0,1] (x)χ[0,1] (y)χ[0,1] (t)χ[0,1] (x − 2k)χ[0,1] (y − l)

× χ[0,1] (t − m + 12 (2ky − lx))dxdydt  H H χH (y)χ (y)χ (t) χH [0,1] [0,1] [l,l+1] [0,1] (x)χ[2k,2k+1] (x) R

× χ[0,1] (t − m +

1 2 (2ky

− lx))dxdydt.

H For k = 0, χH [0,1] (x)χ[2k,2k+1] (x) = 0 and hence ψ, L(2k,l,m) ψ = 0. The same

can be concluded for l = 0. So we have that ψ, L(2k,l,m) ψ = 0, ∀ (k, l) = (0, 0), m ∈ Z and hence from (7), we see that for j1 = j2 , (k1 , l1 ) = (k2 , l2 ), 105

m1 , m2 ∈ Z, δ2j1 L(2k1 ,l1 ,m1 ) ψ, δ2j2 L(2k2 ,l2 ,m2 ) ψL2 (H) = 0. Further, for j1 = j2 , (k1 , l1 ) = (k2 , l2 ), m1 , m2 ∈ Z, δ2j1 L(2k1 ,l1 ,m1 ) ψ, δ2j2 L(2k2 ,l2 ,m2 ) ψL2 (H) = ψ, L(0,0,m2 −m1 ) ψL2 (H) = δm1 m2 which shows that {δ2j L(2k,l,m) ψ : k, l, m, j ∈ Z} is an orthonormal wavelet system in L2 (H). We aim at finding the necessary and sufficient conditions for the orthonor-

110

mality of the wavelet system {δ2j L(2k,l,m) ψ : k, l, m, j ∈ Z} on H. Towards this end, we provide the following definition. ψ on Definition 3.2. For ψ ∈ L2 (H) and j, k, l ∈ Z, we define the function Hj,k,l

(0, 1] as ψ (λ) = Hj,k,l



2j 2j  2j (λ + r)), (δ2j L ψ(2 (2k,l,0) ψ)(2 (λ + r))B2 |2 (λ + r)|,

r∈Z

where λ ∈ (0, 1]. 10

ψ ψ Remark 3.3. When j = 0, Hj,k,l coincides with Gψ k,l , where Gk,l is as in Defini-

tion 2.2. 115

Now, we are in a position to state our main result. Theorem 3.4. Let ψ ∈ L2 (H). The wavelet system {δ2j L(2k,l,m) ψ : k, l, m, j ∈ Z} is orthonormal in L2 (H) if and only if the following conditions hold. (i)Gψ 0,0 (λ) = 1 (ii)Gψ k,l (λ) = 0

a.e. λ ∈ (0, 1], a.e. λ ∈ (0, 1], for all (k, l) = (0, 0) in Z2 ,

ψ (λ) = 0 (iii)Hj,k,l

a.e. λ ∈ (0, 1], for j > 0, k, l in Z,

ψ where Gψ k,l and Hj,k,l are as given in Definitions 2.2 and 3.2 respectively.

In order to prove Theorem 3.4, we shall prove the following lemmas. At first, we need to determine the inverse Fourier transform of δ2j L(2k,l,m) ψ with respect to t variable. In other words, we need to determine (δ2j L(2k,l,m) ψ)λ 120

λ and also the associated kernel K(δ

2j L(2k,l,m) ψ)

λ

, in view of (5). In Lemma 3.6,

t we shall express (δ2j L(2k,l,m) ψ)λ in terms of the λ-twisted translation (T(2k,l) )λ t and dilation D2j on R2 , where (T(k,l) )λ and D2j are defined below. t )λ Definition 3.5. For (k, l) ∈ Z2 and j ∈ Z, the λ-twisted translation (T(k,l)

and the dilation D2j are defined as follows. t )λ ϕ(x, y) = eπiλ(lx−ky) ϕ(x − k, y − l) (T(k,l)

D2j ϕ(x, y) = 2j ϕ(2j x, 2j y),

and

for ϕ ∈ L2 (R2 ).

Lemma 3.6. For ψ ∈ L2 (H), (k, l, m) ∈ Z3 , j ∈ Z, the inverse Fourier transform of δ2j L(2k,l,m) ψ with respect to t variable satisfies (δ2j L(2k,l,m) ψ)λ = 2−j e2πiλ2 where ψ λ2 ψ λ2

−2j

−2j

(z) =



−2j

m

t D2j (T(2k,l) )λ2

−2j

−2j

ψ λ2

,

is the inverse Fourier transform of ψ in the t variable given by R

ψ(z, t)e2πiλ2

−2j

t

dt.

11

Proof. Consider, (δ2j L(2k,l,m) ψ)λ (x, y)   = 22j ψ 2j x − 2k, 2j y − l, 22j t − m + 12 2j (2ky − lx) e2πiλt dt. R

Now, by applying the change of variable s = 22j t − m + 12 2j (2ky − lx), we get, (δ2j L(2k,l,m) ψ)λ (x, y) −2j

= e2πiλ2

m 2j πiλ2−2j (lx−2ky)



ψ(2j x − 2k, 2j y − l, s)e2πiλ2

e

−2j

s

ds

R −2j

= e2πiλ2

m πiλ2

= 2−j e2πiλ2

e

−2j

m

−2j

j

j

(2 lx−2 2ky)

t D2j (T(2k,l) )λ2

ψ λ2

−2j

−2j

ψ λ2

(2j x − 2k, 2j y − l)

−2j

(x, y).

125

Lemma 3.7. Let ϕ ∈ L2 (R2 ). Then the kernel of Wλ (D2j ϕ) and hence that of t Wλ (D2j (T(2k,l) )λ2

−2j

λ ϕ), denoted by KD

2j ϕ

λ and KD

t λ2−2j ϕ 2j (T(2k,l) )

respectively,

satisfy the following relations. λ (ξ, η) = Kϕλ2 KD 2j ϕ λ KD

t λ2−2j ϕ 2j (T(2k,l) )

−2j

(ξ, η) = e2πiλ2

(2j ξ, 2j η),

−2j

k(2j+1 ξ+l)

(8) Kϕλ2

−2j

(2j ξ + l, 2j η).

(9)

Proof. From (2) and Definition 3.5, we have  λ (ξ, η) = 2j ϕ(2j x, 2j (η − ξ))eπiλx(ξ+η) dx. KD 2j ϕ R

Applying the change of variable u = 2j x, we get  −2j j j λ (ξ, η) = ϕ(u, 2j η − 2j ξ)eπiλ2 u(2 ξ+2 η) du. KD 2j ϕ R

The right hand side of the above equation is nothing but Kϕλ2 proving (8).

12

−2j

(2j ξ, 2j η), thus

Now, making use of (8), we see that λ KD

−2j

t λ2−2j ϕ 2j (T(2k,l) )



eπiλ2

=

−2j

λ2 (ξ, η) = K(T t

(2k,l)

(lx−2k(2j η−2j ξ))

R

= eπiλ2

−2j

2k(2j+1 ξ+l)

)λ2

−2j

ϕ

(2j ξ, 2j η)

ϕ(x − 2k, 2j η − 2j ξ − l)eπiλ2

 ϕ(u, 2j η − (2j ξ + l))eπiλ2

−2j

−2j

x(2j ξ+2j η)

u(2j ξ+2j η+l)

dx

du

R

=e

2πiλ2−2j k(2j+1 ξ+l)

Kϕλ2

−2j

(2j ξ + l, 2j η),

thereby proving (9). ψ The function Hj,k,l in Definition 3.2 can be expressed in terms of the kernel

of Wλ . In fact, using (1) and Lemma 3.6, we have ψ Hj,k,l (λ)  2j 2j = W22j (λ+r) (ψ 2 (λ+r) ), W22j (λ+r) ((δ2j L(2k,l,0) ψ)2 (λ+r) )B2 |22j (λ + r)| r∈Z

= 2−j



2j

W22j (λ+r) (ψ 2

(λ+r)

t ), W22j (λ+r) (D2j (T(2k,l) )λ+r ψ λ+r )B2 |22j (λ + r)|.

r∈Z

Now using (4) it turns out that ψ Hj,k,l (λ)  22j (λ+r) 22j (λ+r) K 22j (λ+r) , KD j (T t = 2−j ψ

2

r∈Z

= 2j

 

(2k,l)

 2 |22j (λ )λ+r ψ λ+r L (C)

+ r)|

22j (λ+r)

j j Kψ22j (λ+r) (ξ, η)Kψλ+r λ+r (2 ξ + l, 2 η)

r∈Z R R j+1

× e−2πi(λ+r)k(2

ξ+l)

dξdη|λ + r|,

(10)

by applying Lemma 3.7. 130

The following is a lemma which is crucial for the proof of Theorem 3.4. Lemma 3.8. Let ψ ∈ L2 (H). For j > 0, k, l, m ∈ Z, ψ, δ2j L(2k,l,m) ψL2 (H) = 0 ψ if and only if Hj,k,l (λ) = 0 a.e. λ ∈ (0, 1], for all j > 0, k, l in Z.

13

(11)

Proof. We shall prove this lemma by obtaining a relation between the inner ψ product in (11) and the function Hj,k,l . Using (5), we have  λ ψ, δ2j L(2k,l,m) ψL2 (H) = Kψλλ , K(δ λ L2 (R2 ) |λ|dλ. j L(2k,l,m) ψ)

(12)

2

R

Then, it follows from Lemma 3.6 that  λ K(δ (ξ, η) = (δ2j L(2k,l,m) ψ)λ (x, η − ξ)eπiλx(ξ+η) dx λ j L(2k,l,m) ψ) 2

R



−j 2πiλ2−2j m

=

2

e

R

=2

−j 2πiλ2−2j m

t (D2j (T(2k,l) )λ2

 t (D2j (T(2k,l) )λ2

e

−2j

−2j

ψ λ2 ψ λ2

−2j

)(x, η − ξ)eπiλx(ξ+η) dx

−2j

)(x, η − ξ)eπiλx(ξ+η) dx

R −2j

= 2−j e2πiλ2

m

λ KD

t λ2−2j ψ λ2−2j 2j (T(2k,l) )

(ξ, η).

Hence, λ Kψλλ , K(δ

2j L(2k,l,m) ψ) −2j

= 2−j e−2πiλ2



m

λ

L2 (R2 )

Kψλλ (ξ, η)K λ

t )λ2 D2j (T(2k,l)

R2 −2j

= 2−j e−2πiλ2



−2j

ψ λ2

−2j

(ξ, η)dξdη −2j

Kψλλ (ξ, η)Kψλ2λ2−2j (2j ξ + l, 2j η)e−2πiλ2 −2j

m

k(2j+1 ξ+l)

dξdη,

R2

using (9). Further, by applying the change of variable μ = 2−2j λ, it follows from (12) that ψ, δ2j L(2k,l,m) ψL2 (H)   2j j+1 3j −2πiμm =2 e Kψ2 22jμμ (ξ, η)Kψμμ (2j ξ + l, 2j η)e−2πiμk(2 ξ+l) dξdη|μ|dμ R

R2

Now, discretizing μ we get ψ, δ2j L(2k,l,m) ψL2 (H) 3j

1

=2

e 0

−2πiλm

 r∈Z j+1

× e−2πi(λ+r)k(2

22j (λ+r)

j j Kψ22j (λ+r) (ξ, η)Kψλ+r λ+r (2 ξ + l, 2 η)

R2

ξ+l)

dξdη|λ + r|dλ.

14

Thus, using (10), we have 2j

1

ψ, δ2j L(2k,l,m) ψL2 (H) = 2

ψ e−2πiλm Hj,k,l (λ)dλ

0

from which the lemma follows. Proof of Theorem 3.4. Since δ2j is a unitary operator, the orthonormality of the system {δ2j L(2k,l,m) ψ : k, l, m ∈ Z}, for any fixed j, boils down to the or135

thonormality of the system {L(2k,l,m) ψ : k, l, m ∈ Z}. It follows from Theorem 2.3 that this system of translates is orthonormal in L2 (H) if and only if conditions (i) and (ii) in the statement of the theorem hold true. Further, by Lemma 3.8 and (6), we have δ2j1 L(2k1 ,l1 ,m1 ) ψ, δ2j2 L(2k2 ,l2 ,m2 ) ψL2 (H) = 0 for j1 = j2 , k1 , k2 , l1 , l2 , m1 , m2 in Z if and only if condition (iii) holds.

140

We shall now see an example of an orthonormal wavelet system whose proof requires the characterization so obtained.

1 and ψ2 are the Haar Example 3.9. Let ψ(x, y, t) = ψ1 (x)ψ2 (y)ψ3 (t), where ψ

1 = ψ2 = functions on [0, 1] and ψ3 is the Shannon wavelet. In other words, ψ ⎧ ⎪ ⎪ 1, 0 ≤ x ≤ 12 , ⎪ ⎪ ⎨ H

3 = χ 1 χH and ψ [−1,− 12 ]∪[ 12 ,1] . Then for [0,1] given by χ[0,1] (x) = ⎪−1, 2 < x ≤ 1, ⎪ ⎪ ⎪ ⎩0, otherwise. ∗ j, k, l ∈ Z, λ ∈ R ,  ψ)(λ) = 22j ψ1 (2j x − 2k)ψ2 (2j y − l)πλ (x, y, 0) (δ2j L (2k,l,0) R2

 ×

3 (−2 =ψ

−2j

 λ)

 ψ3 22j t + 12 2j (2ky − lx) e2πiλt dtdxdy

R −j

ψ1 (2j x − 2k)ψ2 (2j y − l)e−πiλ2

(2ky−lx)

πλ (x, y, 0)dxdy

R2

(13) is an integral operator whose kernel can be computed directly as follows. Let

15

f ∈ L2 (R), ξ ∈ R. Consider (δ2j L (2k,l,0) ψ)(λ)f (ξ) 

3 (−2−2j λ) ψ1 (2j x − 2k)ψ2 (2j y − l)e−πiλ2−j (2ky−lx) e2πiλ(xξ+ 12 xy) =ψ R2

× f (ξ + y)dxdy 

3 (−2−2j λ) ψ1 (2j x − 2k)ψ2 (2j (η − ξ) − l)e−πiλ2−j ((η−ξ)2k−xl) =ψ R2 1

× e2πiλ(xξ+ 2 x(η−ξ)) f (η)dxdη 

3 (−2−2j λ) ψ2 (2j (η − ξ) − l)e−πiλ2−j (η−ξ)2k f (η) =ψ R

 ×

ψ1 (2j x − 2k)eπiλ2

−j

R

=

3 (−2−2j λ) 2−j ψ



1 ψ2 (2 (η − ξ) − l)ψ j

R

× e2πiλ2

−j

k(2ξ+2−j l)

xl πiλx(ξ+η)

e



dxdη

−λ2−j (ξ + η + 2−j l) 2



f (η)dη

from which it follows that the kernel of the Hilbert-Schmidt operator (δ2j L (2k,l,0) ψ)(λ) is given by λ K(δ λ (ξ, η) 2j L(2k,l,0) ψ)

=

1 2−j ψ



−λ2−j (ξ + η + 2−j l) 2

 (14)

3 (−2−2j λ)e2πiλ2−j k(2ξ+2−j l) . × ψ2 (2 (η − ξ) − l)ψ j

Now for λ ∈ (0, 1], r ∈ Z, using (14) we have  + r), L(2k,l,0)  ψ(λ + r)B2 ψ(λ      −(λ + r)(ξ + η) −(λ + r)(ξ + η + l)

= ψ1 ψ1 2 2 R2

3 (−(λ + r)))2 e−2πi(λ+r)k(2ξ+l) dξdη. × ψ2 (η − ξ)ψ2 (η − ξ − l)(ψ   ψ(λ+r)B2 = L(2k,l,0) Substituting for ψ2 , we observe that when l = 0, ψ(λ+r), 0, ∀ r ∈ Z and so Gψ k,l (λ) = 0, λ ∈ (0, 1]. But when l = 0 and k = 0, we get for

16

r∈Z  + r), L(2k,l,0)  ψ(λ + r)B2 ψ(λ  2    −(λ + r)(ξ + η)



ψ2 (η − ξ)ψ3 (−(λ + r)) e−4πi(λ+r)kξ dξdη = ψ1 2 R2

=

 

1

(ψ3 (−(λ + r)))2 2

1 ψ



−(λ + r)u 2



2 ψ2 (v)

e−4πi(λ+r)k(

u−v 2

) dudv

R2



3 (−(λ + r)))2  (ψ

1 (x))2 e4πikx dx (ψ2 (v))2 e2πi(λ+r)kv dv (ψ = |λ + r| R

=

3 (−(λ + r)))2 (ψ |λ + r|

R

1

e4πikx dx

0

1

e2πi(λ+r)kv dv = 0

0

and once again Gψ k,l (λ) = 0, λ ∈ (0, 1] thereby satisfying condition (ii) of Theorem 3.4. In the above computation, taking k = l = 0 we obtain  + r)2 = ψ(λ B2

3 (−(λ + r)))2 (ψ |λ + r|

and so Gψ 0,0 (λ) =

0  

3 (−(λ + r)))2 =

3 (−(λ + r)))2 = 1 (ψ (ψ r=−1

r∈Z

3 . This shows that ψ satisfies condition for a.e. λ ∈ (0, 1] by substituting for ψ 145

(i) of Theorem 3.4. −2j

From (13), we may write (δ2j L λ)F λ,j,k,l , where (2k,l,0) ψ)(λ) = ψ3 (−2

F λ,j,k,l is an integral operator on L2 (R). Then for j > 0, k, l ∈ Z, ψ (λ) = Hj,k,l



3 (−22j (λ + r))ψ

3 (−(λ + r)) ψ

r∈Z 2j

× F 2 =

0 

(λ+r),0,0,0

2j

,F2

(λ+r),j,k,l

B2 |22j (λ + r)|

3 (−22j (λ + r))ψ

3 (−(λ + r)) ψ

r=−1 2j

× F 2

(λ+r),0,0,0

=0

17

2j

,F2

(λ+r),j,k,l

B2 |22j (λ + r)|

3 . Thus ψ satisfies condition (iii) for a.e. λ ∈ (0, 1] using the expression for ψ of Theorem 3.4 as well and hence generates an orthonormal wavelet system in L2 (H). We shall now provide a sufficient condition for the orthonormal system 150

{δ2j L(0,0,m) ψ : m, j ∈ Z} to be complete in L2 (H). Theorem 3.10. Let ψ ∈ L2 (H) be such that {δ2j L(0,0,m) ψ : m, j ∈ Z} is orthonormal in L2 (H). If, in addition, ψ satisfies  2        ψ λ   λ  = 1 a.e. λ ∈ R,     2j 2j 2 B2 2

(15)

j∈Z

then span{δ2j L(0,0,m) ψ : m, j ∈ Z} = L2 (H). In order to prove this theorem, we require the following definition of the spectral function. Definition 3.11. Let V denote the set of all subspaces of L2 (H) that are invariant under translations L(0,0,m) , m ∈ Z. Spectral function is the mapping σ : V → L∞ (R) defined by

σV 0 (ψ) (λ) =

⎧ 2  ⎪ B2 |λ| ⎨ ψ(λ) , ψ

 λ ∈ suppψ,

⎪ ⎩0,

otherwise,

G0,0 (λ)

 where V 0 (ψ) is as in Section 2 and suppψ = {η ∈ R∗ : ψ(η) = 0}. Further, σ is 155

additive on orthogonal sums. Proof of Theorem 3.10. We make use of the idea of the proof in [18]. For a fixed j ∈ Z, let Wj0 = span{δ2j L(0,0,m) ψ : m ∈ Z}. Clearly, the subspaces Wj0 , j ∈ Z are mutually orthogonal and for j ≥ 0, Wj0 are invariant under translations L(0,0,m ) , m ∈ Z for, L(0,0,m ) δ2j L(0,0,m) ψ(x, y, t) = 22j ψ(2j x, 2j y, 22j t − (22j m + m)) = δ2j L(0,0,22j m +m) ψ(x, y, t). 18

Consider the space



Wj0 in L2 (H) and let Z 0 denote its orthogonal com-

j∈Z

plement in L2 (H). In order to prove Z 0 = {0}, we consider the space V 0 =  ⊥  0 Wj which is also invariant under L(0,0,m ) , m ∈ Z. Further, j≥0

⎞⊥

⎛



Z0 ⊂ ⎝

Wj0 ⎠ = δ2i (V 0 ),

∀i ∈ Z

j≥i

 δ2i (V 0 ). Thus, it is enough to prove that δ2i (V 0 ) = {0}. i∈Z i∈Z  δ2i (V 0 ), f = 0 and f L2 (H) = Suppose, on the contrary, there exists f ∈

and so Z 0 ⊂



i∈Z

1. Then δ2−i f ∈ V 0 , ∀ i ∈ Z. Let j ≥ 0. Then δ2j f ∈ V 0 . By Theorem 2.6, ∞  2 0 = V 0 (ψn ) and each ψn is a there exists {ψn }∞ n=1 in L (H) such that V n=1

Parseval frame generator of V 0 (ψn ). Also, there exists rj,n ∈ L2 (T) such that δ 2j f (λ) =

∞ 

n (λ) a.e. λ ∈ R rj,n (λ)ψ

n=1

and 1 = δ2j f 2L2 (H) =

∞ 

rj,n 2L2 (T)

(16)

n=1

using Remark 2.7. Now δ 2j f (λ)B2 ≤

∞ 

n (λ)B a.e. λ ∈ R. |rj,n (λ)|ψ 2

n=1

But −2j  −2j δ f (2 λ)B2 . 2j f (λ)B2 = 2

In fact, by (1), (3) and Lemmas 3.6 and 3.7, we have  2 λ 2 |K(δ δ λ (ξ, η)| dξdη 2j f (λ)B2 = 2j f ) R2  −2j −2j −2j |Kf22−2jλλ (2j ξ, 2j η)|2 dξdη = 2−4j Kf22−2jλλ 2L2 (R2 ) =2 R2

= 2−4j f(2−2j λ)2B2 . Thus, f(λ)B2 ≤ 22j

∞ 

n (22j λ)B a.e. λ ∈ R. |rj,n (22j λ)|ψ 2

n=1

19

By applying Cauchy-Schwarz inequality for the sum and integral, we obtain 4

f(λ)B2 |λ| 2 dλ 1

1

⎛ 4 ⎞ 12 ⎛ 4 ⎞ 12     ∞ ∞

n (22j λ)2 |λ|dλ⎠ |rj,n (22j λ)|2 dλ⎠ ⎝ ψ ≤ 22j ⎝ B2 1 n=1

⎛

⎜ = ⎝2−2j

1 n=1

22(j+1)   ∞

⎞ 12 ⎛ ⎟ ⎜ |rj,n (λ)|2 dλ⎠ ⎝

n=1

22j

22(j+1)   ∞

⎞ 12

n (λ)2 |λ|dλ⎟ ψ ⎠ . B2

n=1

22j

Further, by 1-periodicity of rj,n and (16), we have −2j

2

22(j+1)   ∞ 22j

2j

|rj,n (λ)| dλ = 2

=2

−2j

n=1 2j

−2j

2

∞ 3(2  )

3(2  ∞  n=1

k

2

|rj,n (λ)| dλ = 2

)

|rj,n (λ)|2 dλ

0

−2j

n=1 k=1 k−1

3(22j ) ∞  

rj,n 2L2 (T) = 3.

k=1 n=1

By using the decomposition of V 0 as V 0 =

∞ 

V 0 (ψn ) and Theorem 2.4, we

n=1

may write the spectral function of V 0 as σV 0 (λ) =

∞

n (λ)2 |λ|. So, ψ B2

n=1

4

f(λ)B2 |λ| dλ ≤

1

1 2

√

⎛ ⎞ 12  3 ⎝ χ[22j ,22(j+1) ] (λ)σV 0 (λ)dλ⎠

(17)

R

which holds for all j ≥ 0. On the other hand, by using the definition of the space V 0 , we may write σL2 (H) = σV0 +



σWj0 .

(18)

j≥0

Since {L(0,0,m) ψ : m ∈ Z} is an orthonormal system, we have σW00 (λ) =

2  ψ(λ) B2 |λ|, using Theorem 2.3. We obtain σWj0 , for j > 0 as follows. As

in the proof of Property (g) in Proposition 1.4.6 in [18], one can show that if ϕ is a Parseval frame generator of V 0 (ϕ), then {δn L(0,0,r) ϕ : r = 0, 1, 2, · · · , n2 −1} is a family of Parseval frame generators of the space δn (V 0 (ϕ)), n ∈ N using 20

the relation δn L(0,0,qn2 +r) ϕ = L(0,0,q) (δn L(0,0,r) ϕ). The spectral function of δn (V 0 (ϕ)), then turns out to be σδn (V 0 (ϕ)) (λ) =

2 n −1

2  (δn L (0,0,r) ϕ)(λ)B2 |λ|.

r=0

Once again, using (1), (3) and Lemmas 3.6 and 3.7, we obtain 2 λ 2  (δn L (0,0,r) ϕ)(λ)B2 = K(δn L(0,0,r) ϕ)λ L2 (R2 )  −2 λ 2 = |n−1 e2πiλn r KD λn−2 (ξ, η)| dξdη nϕ R2  −2 −2 2 −4 −2 2 |Kϕλn ϕ(λn  )B2 . =n λn−2 (nξ, nη)| dξdη = n R2

This shows that σδn (V

0 (ϕ))

  2  λ   (λ) =   ϕ n2 

B2

    λ   = σV 0 (ϕ) λ .  n2  n2

Using Theorem 2.6, it follows that for any subspace V of L2 (H) invariant under  L(0,0,m) , m ∈ Z, σδn (V ) (λ) = σV nλ2 , n ∈ N. Now since Wj0 = δ2j (W00 ), we have

 σWj0 (λ) = σW00

λ 22j



   2    λ  λ      . = ψ 22j B2  22j 

Further, since σL2 (H) (λ) = 1 a.e. λ ∈ R, we obtain from (18)  2        ψ λ   λ  a.e. λ ∈ R. σV0 (λ) = 1 −     2j 2j 2 B2 2 j≥0

By (15) in the hypothesis, this in turn gives σV0 (λ) =

∞      2j 2 ψ(2 λ) |22j λ| a.e. λ ∈ R. B2

j=1

Consider  R

=

|σV0 (λ)|dλ =

∞       2j 2 ψ(2 λ) |22j λ|dλ j=1

∞ 

B2

R

  ∞     2 −2j  22 ∗ 2 2−2j ψ ψ(μ) |μ|dμ = L (R ,B2 ;dμ)

j=1

= ψ2L2 (H)

R

∞ 

B2

j=1

2−2j < ∞.

j=1

21

160

Thus σV0 ∈ L1 (R). Now by applying Lebesgue dominated convergence theorem 4 1 in (17), we obtain 1 f(λ)B2 |λ| 2 dλ = 0 which then gives f(λ)B2 = 0 a.e.  λ ∈ [1, 4]. Now for any j ∈ Z, δ2j f ∈ δ2i (V 0 ) and by the above reasoning, i∈Z

 we get δ 2j f (λ)B2 = 0 a.e. λ ∈ [1, 4] which implies that f (λ)B2 = 0 a.e. λ ∈ [2−2j , 2−2j+2 ]. In other words, f(λ)B2 = 0 a.e. λ ∈ [0, ∞). Similarly,  −1 1 working with −4 f(λ)B2 |λ| 2 dλ, one can show that f(λ)B2 = 0 a.e. λ ≤ 0

165

and hence f(λ)B2 = 0 a.e. λ ∈ R. Thus, f 2L2 (H) = f2L2 (R∗ ,B2 ;dμ) = 0  δ2i (V 0 ) = {0}. which is a contradiction, thereby proving that i∈Z

We obtain the following result as a consequence of the above theorem. Corollary 3.12. Let ψ ∈ L2 (H) be such that {δ2j L(2k,l,m) ψ : k, l, m, j ∈ Z} is orthonormal in L2 (H). Then ψ will not satisfy (15). We observe from Lemma 3.6 that there is a connection between wavelet 170

system on the Heisenberg group and a wavelet system on C, namely the twisted −2j

t wavelet system {D2j (T(k,l) )2

ϕ : k, l, j ∈ Z} for ϕ ∈ L2 (C). Thus it becomes

a natural question to investigate the necessary and sufficient condition for the orthonormality of the twisted wavelet system on C. This problem is studied in the forthcoming section.

175

4. Orthonormality of twisted wavelet system on C Before stating the main result of this section, we shall prove the following lemma. Lemma 4.1. Let ϕ ∈ L2 (R2 ). Then the kernel of the Weyl transform of D2j ϕ,

22

−2j

t (T(k,l) )2

−2j

t ϕ and D2j (T(k,l) )2

−2j

(i) KD2j ϕ (ξ, η) = Kϕ2 (ii) K(T t

(k,l)

−2j

)2

ϕ respectively satisfy the following relations. j ∈ Z,

(2j ξ, 2j η),

ϕ (ξ, η)

−2j

= eπi2

kl πik(1+2−2j )ξ πik(1−2−2j )η

e

e

Ke(( 2−2j −1 )l,0)ϕ (ξ + l, η), 2

2

2

where, f or (a, b) ∈ R , the operator e(a, b) on L (R2 ) is given by (e(a, b)ϕ)(x, y) = e2πi(ax+by) ϕ(x, y), (iii )KD

−2j

t 2−2j ϕ 2j (T(k,l) )

(ξ, η) = eπi2

k(2j+1 ξ+l)

−2j

Kϕ2

(2j ξ + l, 2j η).

Proof. We note that (i) and (iii) of Lemma 4.1 immediately follow by putting λ = 1 in Lemma 3.7. In order to prove (ii), let us recall (2) and Definition 3.5. Now, consider K(T t

(k,l)

ϕ (ξ, η)

−2j

)2



−2j

eπi2

=

(lx−k(η−ξ))

ϕ(x − k, η − ξ − l)eπix(ξ+η) dx

R −2j

= eπi2

kl πik(1+2−2j )ξ πik(1−2−2j )η

e e   −2j  −1 2 l 2πiu 2 × e ϕ(u, η − (ξ + l))eπiu(ξ+l+η) du. R

The integral on the right hand side can be written as  −2j (e(( 2 2 −1 )l, 0)ϕ)(u, η − (ξ + l))eπiu(ξ+l+η) du R −2j

which turns out to be the kernel of the Weyl transform of e(( 2

2

−1

)l, 0)ϕ. Thus,

we get K(T t

(k,l)

ϕ (ξ, η)

−2j

)2

−2j

= eπi2

kl πik(1+2−2j )ξ πik(1−2−2j )η

e

e

Ke(( 2−2j −1 )l,0)ϕ (ξ + l, η), 2

180

thereby proving (ii) of the lemma. In [15], Radha and Adhikari proved the following 23

t Theorem 4.2. Let ϕ ∈ L2 (R2 ). Then {T(k,l) ϕ : k, l ∈ Z} is an orthonormal

system in L2 (R2 ) if and only if  |Kϕ (ξ + m, η)|2 dη = 1 (i)

a.e. ξ ∈ [0, 1]

and

m∈Z R

(ii)



Kϕ (ξ + m, η)Kϕ (ξ + m + l, η)dη = 0 a.e. ξ ∈ [0, 1], ∀ l = 0.

m∈Z R

We shall now state a similar result in the context of twisted wavelet system which will be proved in due course. Theorem 4.3. Let ϕ ∈ L2 (R2 ). Let ξ ∈ [0, 1]. For j1 , j2 , l1 , l2 ∈ Z, let  −2j1 Kϕ2 (2j1 +j2 (ξ + m) + l1 , 2j1 η) Pjϕ1 ,j2 ,l1 ,l2 (ξ) = m∈Z R

(19) 2−2j2

× Kϕ

(22j2 (ξ + m) + l2 , 2j2 η)dη.

For j, l, m ∈ Z, y ∈ R, let  ϕ Sj,l (ξ, m, y)

= Ke(( 2−2j −1 )l,0)ϕ 2

2y 2(ξ + m − y) + l, −2j 1+2 1 − 2−2j

 .

(20)

For j, l ∈ Z, let ϕ Rj,l (ξ) =



ϕ |Sj,l (ξ, m, y)|2 dy.

(21)

m∈Z R

For j, l1 , l2 ∈ Z, let Qϕ j,l1 ,l2 (ξ) =



ϕ ϕ Sj,l (ξ, m, y)Sj,l (ξ, m, y)dy. 1 2

(22)

m∈Z R −2j

t )2 Then the twisted wavelet system {D2j (T(k,l)

24

ϕ : k, l, j ∈ Z} is orthonormal

in L2 (R2 ) if and only if  |Kϕ (ξ + m, η)|2 dη = 1 (i)

a.e. ξ ∈ [0, 1],

m∈Z R



(ii)

Kϕ (ξ + m, η)Kϕ (ξ + m + l, η)dη = 0 a.e. ξ ∈ [0, 1], ∀ l = 0,

m∈Z R

(iii) Pjϕ1 ,j2 ,l1 ,l2 (ξ) = 0

a.e. ξ ∈ [0, 1], for j2 > j1 , l1 , l2 ∈ Z,

(iv) Qϕ j,l1 ,l2 (ξ) = 0 ϕ (v) Rj,l (ξ) =

−4j

|1−2 4

a.e. ξ ∈ [0, 1], for j ∈ Z \ {0}, l1 = l2 in Z, |

a.e. ξ ∈ [0, 1], for j ∈ Z \ {0}, l ∈ Z. −2j

t In order to establish the orthogonality of the system {D2j (T(k,l) )2 185

ϕ :

k, l, j ∈ Z}, one needs to consider the following orthogonality relations. • For j2 = j1 , k1 , k2 , l1 , l2 ∈ Z, −2j1

t D2j1 (T(k )2 1 ,l1 )

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

ϕ = 0,

• For l1 = l2 , j = 0, k1 , k2 ∈ Z, −2j

t )2 D2j (T(k 1 ,l1 )

−2j

t ϕ, D2j (T(k )2 2 ,l2 )

ϕ = 0.

As in Section 3, we shall realise each of these inner products as Fourier coefficients of certain functions. This fact is being established in the following two lemmas. Lemma 4.4. Let ϕ ∈ L2 (R2 ). For j2 > j1 , k1 , k2 , l1 , l2 ∈ Z, −2j1

t D2j1 (T(k )2 1 ,l1 )

if and only if Pjϕ1 ,j2 ,l1 ,l2 (ξ) = 0 190

Pjϕ1 ,j2 ,l1 ,l2

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

ϕ = 0

a.e. ξ ∈ [0, 1], for all j2 > j1 , l1 , l2 ∈ Z, where

is as in (19).

Proof. Let j2 > j1 , k1 , k2 , l1 , l2 ∈ Z. Using (4) we get −2j1

t )2 D2j1 (T(k 1 ,l1 )

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

ϕ

−2j1 , K −2j2  = KD j (T t t )2 ϕ D2j2 (T(k )2 ϕ (k1 ,l1 ) 2 1 2 ,l2 )  −2j1 (ξ, η)K −2j2 (ξ, η)dξdη. = KD j (T t )2 ϕ D j (T t )2 ϕ 2 1

R2

2 2

(k1 ,l1 )

25

(k2 ,l2 )

Then, using (iii) of Lemma 4.1 and applying the change of variable x = 2−j2 ξ, we get −2j1

−2j2

t D2j1 (T(k )2 1 ,l1 )

t ϕ, D2j2 (T(k )2 ϕ 2 ,l2 )  −2j1 −2j1 k1 l1 −2−2j2 k2 l2 ) Kϕ2 (2j1 ξ + l1 , 2j1 η) = eπi(2 R2 −j1

−j2

(2j2 ξ + l2 , 2j2 η)e2πi(2 k1 −2 k2 )ξ dξdη  −2j1 j2 πi(2−2j1 k1 l1 −2−2j2 k2 l2 ) =2 e Kϕ2 (2j1 +j2 x + l1 , 2j1 η) −2j2

× Kϕ2

R2 −2j2

× Kϕ2

j2 −j1

(22j2 x + l2 , 2j2 η)e2πi(2

k1 −k2 )x

dxdη.

Now, discretizing with respect to the variable x, we get −2j1

t D2j1 (T(k )2 1 ,l1 )

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

j2 πi(2−2j1 k1 l1 −2−2j2 k2 l2 )

=2 e

1  

ϕ −2j1

Kϕ2

(2j1 +j2 (ξ + m) + l1 , 2j1 η)

0 m∈Z R 2−2j2

× Kϕ

j2 −j1

(22j2 (ξ + m) + l2 , 2j2 η)e2πi(2

−2j1

= 2j2 eπi(2

k1 l1 −2−2j2 k2 l2 )

1

k1 −k2 )ξ

dηdξ

j2 −j1

Pjϕ1 ,j2 ,l1 ,l2 (ξ)e2πi(2

k1 −k2 )ξ

dξ.

0

Thus, we observe that for j2 > j1 , k1 , k2 , l1 , l2 ∈ Z, −2j1

t D2j1 (T(k )2 1 ,l1 )

1 iff

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 ) j2 −j1

Pjϕ1 ,j2 ,l1 ,l2 (ξ)e2πi(2

k1 −k2 )ξ

ϕ = 0

dξ = 0 for j2 > j1 , k1 , k2 , l1 , l2 ∈ Z

0

1 iff

Pjϕ1 ,j2 ,l1 ,l2 (ξ)e2πikξ dξ = 0 for j2 > j1 , l1 , l2 , k ∈ Z

0

iff Pjϕ1 ,j2 ,l1 ,l2 (ξ) = 0 a.e. ξ ∈ [0, 1], for j2 > j1 , l1 , l2 ∈ Z, using the uniqueness of Fourier coefficient. Lemma 4.5. Let ϕ ∈ L2 (R2 ). For j = 0 in Z, l1 = l2 in Z, k1 , k2 , ∈ Z, −2j

t D2j (T(k )2 1 ,l1 )

−2j

t ϕ, D2j (T(k )2 2 ,l2 )

26

ϕ = 0

if and only if Qϕ j,l1 ,l2 (ξ) = 0 Qϕ j,l1 ,l2

a.e. ξ ∈ [0, 1], for all j = 0, l1 = l2 in Z, where

is as in (22).

Proof. Since D2j is a unitary operator on L2 (R2 ), for j = 0 and k1 , k2 , l1 , l2 ∈ Z, we have −2j

t )2 D2j (T(k 1 ,l1 )

= K(T t

(k1 ,l1 )

−2j

= eπi2

−2j

−2j

)2

−2j

t ϕ, D2j (T(k )2 2 ,l2 ) t ϕ , K(T(k

(k1 l1 −k2 l2 )



K R2

2 ,l2 )

e((

)2

−2j

t ϕ = (T(k )2 1 ,l1 )

−2j

t ϕ, (T(k )2 2 ,l2 )

ϕ

(ξ 2−2j −1 )l1 ,0)ϕ 2

× e2πi(k1 −k2 )(

+ l1 , η)K

e((

(ξ 2−2j −1 )l2 ,0)ϕ 2

y=

+ l2 , η)

1+2−2j 1−2−2j )ξ 2πi(k1 −k2 )( )η 2 2 e dξdη,

using (ii) of Lemma 4.1. Making use of the change of variables x = 1−2−2j η 2

ϕ

1+2−2j ξ 2

and

(which is valid since j = 0), we get −2j

−2j

t )2 D2j (T(k 1 ,l1 )

t ϕ, D2j (T(k )2 ϕ 2 ,l2 )  πi2−2j (k1 l1 −k2 l2 ) K 2−2j −1 = 4e |1−2−4j | e((

2

R2

×K =

e((

4eπi2

2x ( 1+2 −2j 2−2j −1 )l ,0)ϕ 2 2

−2j (k l −k l ) 1 1 2 2 |1−2−4j |



K

e((

R2

×K

e((

( 2x−2j )l1 ,0)ϕ 1+2

+ l2 , 1−22y−2j )e2πi(k1 −k2 )(x+y) dxdy

2(ξ−y) ( 1+2 −2j 2−2j −1 )l ,0)ϕ 1 2

2(ξ−y) ( 1+2 −2j 2−2j −1 )l ,0)ϕ 2 2

+ l1 , 1−22y−2j )

+ l1 , 1−22y−2j )

+ l2 , 1−22y−2j )e2πi(k1 −k2 )ξ dξdy.

Further, discretizing with respect to the variable ξ, we have for j = 0, k1 , k2 , l1 , l2 ∈ Z, −2j

t D2j (T(k )2 1 ,l1 )

=

−2j

t ϕ, D2j (T(k )2 2 ,l2 )

ϕ

−2j (k l −k l ) 1 1 2 2 4eπi2 |1−2−4j |

×

1  

K

( 2−2j −1 )l1 ,0)ϕ e(( 2

×K

( 2−2j −1 )l2 ,0)ϕ 2

0 m∈Z R

e((

2(x+m−y) 1+2−2j

+ l1 , 1−22y−2j )

2(x+m−y) 1+2−2j

+ l2 , 1−22y−2j )e2πi(k1 −k2 )x dydx.

27

(23)

The integral on the right hand side is nothing but 1  

ϕ ϕ Sj,l (x, m, y)Sj,l (x, m, y)dy e2πi(k1 −k2 )x dx. 1 2

0 m∈Z R

Therefore, for j = 0, l1 = l2 , k1 , k2 ∈ Z, −2j

t D2j (T(k )2 1 ,l1 ) 4eπi2

=

−2j

t ϕ, D2j (T(k )2 2 ,l2 )

−2j (k l −k l ) 1 1 2 2 |1−2−4j |

1

ϕ

2πi(k1 −k2 )x Qϕ dx. j,l1 ,l2 (x)e

0

At this point, we can infer that for j = 0, l1 = l2 , k1 , k2 ∈ Z, −2j

t D2j (T(k )2 1 ,l1 )

1

−2j

t ϕ, D2j (T(k )2 2 ,l2 )

ϕ = 0

2πikx Qϕ dx = 0 for j = 0, l1 = l2 , k ∈ Z j,l1 ,l2 (x)e

iff 0

iff

Qϕ j,l1 ,l2 (x) = 0 a.e. x ∈ [0, 1], for all j = 0, l1 = l2 in Z,

which follows from the uniqueness of Fourier coefficient, thus proving our asser195

tion. Now, we shall provide the proof of the main result. Proof of Theorem 4.3. In order to prove the theorem, we shall first assume conditions (i) to (v) and establish the orthonormality of the twisted wavelet system. Suppose j1 = j2 , k1 , k2 , l1 , l2 ∈ Z. Without loss of generality, we may assume that j2 > j1 . Then by Lemma 4.4, we can conclude that −2j1

t D2j1 (T(k )2 1 ,l1 )

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

ϕ = 0.

When j1 = j2 , then either j1 = j2 = 0 or j1 = j2 = 0. In the former −2j1

t case, the inner product D2j1 (T(k )2 1 ,l1 ) t t ϕ, T(k ϕ, T(k 1 ,l1 ) 2 ,l2 )

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

ϕ reduces to

which is nothing but the inner product of system of twisted

translates on R2 . It then follows from Theorem 4.2 that −2j1

t )2 D2j1 (T(k 1 ,l1 )

−2j2

t ϕ, D2j2 (T(k )2 2 ,l2 )

28

ϕ = δk1 ,k2 δl1 ,l2 , for j1 , j2 = 0.

But when j1 = j2 = j = 0, there arises two cases, namely l1 = l2 and l1 = l2 . When l1 = l2 , the required result follows from Lemma 4.5. On the other hand when l1 = l2 = l, using (23), (20) and (21) we get for j = 0, k1 , k2 , l ∈ Z, −2j

t D2j (T(k )2 1 ,l)

−2j

t ϕ, D2j (T(k )2 2 ,l) −2j

ϕ

l(k1 −k2 ) 4eπi2 |1−2−4j |

=

1

ϕ Rj,l (x)e2πi(k1 −k2 )x dx.

(24)

0

Thus, −2j

t )2 D2j (T(k 1 ,l)

−2j

t ϕ, D2j (T(k )2 2 ,l)

ϕ = δk1 ,k2 , −2j

t using condition (v). Hence we conclude that {D2j (T(k,l) )2

ϕ : k, l, j ∈ Z} is

orthonormal. −2j

t Conversely, suppose that {D2j (T(k,l) )2

ϕ : k, l, j ∈ Z} is orthonormal.

When j1 = j2 = 0, the first two conditions follow from Theorem 4.2. Condition (iii) and condition (iv) follow from Lemma 4.4 and Lemma 4.5 respectively. By −2j

t )2 our assumption of orthonormality of {D2j (T(k,l)

ϕ : k, l, j ∈ Z}, we have for

j1 = j2 = j = 0, l1 = l2 = l, k1 , k2 ∈ Z, −2j

t )2 D2j (T(k 1 ,l)

−2j

t ϕ, D2j (T(k )2 2 ,l)

ϕ = δk1 ,k2 .

This in turn along with (24) implies that −2j

lk 4eπi2 |1−2−4j |

1

ϕ Rj,l (x)e2πikx dx = δk,0 , for j = 0, k, l ∈ Z.

0

In other words, 1

ϕ Rj,l (x)e2πikx dx =

|1−2−4j | δk,0 , 4

for j = 0, k, l ∈ Z.

0

This leads to condition (v), thus proving the theorem.

200

References [1] Barbieri, D., Hern´ andez, E., & Mayeli, A. (2014). Bracket map for the Heisenberg group and the characterization of cyclic subspaces. Appl. Comput. Harmon. Anal., 37 , 218–234. 29

[2] Bownik, M. (2000). The structure of shift-invariant subspaces of L2 (Rn ). 205

J. Funct. Anal., 176 , 282–309. [3] Bownik, M. (2007). The structure of shift-modulation invariant spaces: The rational case. J. Funct. Anal., 244 , 172–219. [4] Bownik, M., & Ross, K. A. (2015). The structure of translation-invariant spaces on locally compact abelian groups. J. Fourier Anal. Appl., 21 , 849–

210

884. [5] Cabrelli, C., & Paternostro, V. (2010). Shift-invariant spaces on LCA groups. J. Funct. Anal., 258 , 2034–2059. [6] Cabrelli, C., & Paternostro, V. (2012). Shift-modulation invariant spaces on LCA groups. Studia Math., 211 , 1–19.

215

[7] Christensen, O. (2008). Frames and bases: An introductory course. Boston: Birkh¨ auser. [8] Folland, G. B. (1989). Harmonic analysis in phase space. Princeton, New Jersey: Princeton University Press. [9] Hern´andez, E., & Weiss, G. (1996). A first course on wavelets. Boca Raton:

220

CRC Press. [10] Kamyabi Gol, R., & Raisi Tousi, R. (2008). The structure of shift invariant spaces on a locally compact abelian group. J. Math. Anal. Appl., 340 , 219–225. [11] Mallat, S. (1989). Multiresolution approximations and wavelet orthonormal

225

bases of L2 (R). Trans. Amer. Math. Soc., 315 , 69–87. [12] Mayeli, A. (2008). Shannon multiresolution analysis on the Heisenberg group. J. Math. Anal. Appl., 348(2), 671 – 684. [13] Meyer, Y. (1985-1986).

Principe d’incertitude, bases Hilbertiennes et

alg`ebres d’op´erateurs. S´eminaire Bourbaki , 662 . 30

230

[14] Meyer, Y. (December 1986). Ondelettes et fonctions splines. S´eminaire ´ EDP, Ecole Polytechnique, Paris, (pp. 1–18). [15] Radha, R., & Adhikari, S. (2016). Frames and Riesz bases of twisted shiftinvariant spaces in L2 (R2n ). J. Math. Anal. Appl., 434 , 1442–1461. [16] Radha, R., & Adhikari, S. (2017).

235

Shift-invariant spaces with count-

ably many mutually orthogonal generators on the Heisenberg group. ArXiv:1711.06902v2. [17] Radha, R., & Shravan Kumar, N. (2013). Shift-invariant subspaces on compact groups. Bull. Sci. Math., 137 (4), 485–497. [18] Rzeszotnik, Z. (2000). Characterization theorems in the theory of wavelets.

240

Ph.D. thesis Washington University. [19] Thangavelu, S. (1998). Harmonic analysis on the Heisenberg group. Boston: Birkh¨ auser.

31