Square-integrability of multivariate metaplectic wave-packet representations

This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space . The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group , which guarantees the square-integrability of the associated metaplectic wave-packet representation on .

. The abstract notions of symplectic wave-packet groups and metaplectic wavepacket representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group ( ) R Sp d , which guarantees the square-integrability of the associated metaplectic wave-packet representation on ( ) R L d 2 .
The abstract theory of covariant/coherent state transforms is the mathematical basis of modern high frequency approximation techniques and time-frequency (resp. time-scale) analysis [37,44,48,49]. Over the last decades, abstract and computational aspects of covariant/ coherent state transforms have achieved significant popularity in mathematical and theoretical physics, see [3,5,37,47] and references therein. Coherent state transforms are classically obtained by a given coherent function systems. Then admissibility conditions on the coherent system imply analyzing of functions with respect to the system by the inner product evaluation [22,23,35]. From harmonic and functional analysis aspects such coherent structures are classically originated from squar-integrable representations of locally compact groups, see [33,46,50,59] and references therein. Commonly used coherent states transforms in theoretical physics, computational science and engineering are wavelet transform [49], Gabor transform [37], wave-packet transform [27][28][29][30]32].
The mathematical theory of Gabor analysis is based on the coherent state generated by modulations and translations of a given window function [4,6,31,34]. Wavelet analysis is a time-scale analysis which is based on the continuous affine group as the group of dilations and translations [9]. Abstract harmonic analysis extensions of wavelet analysis are studied in [7,49]. The theory of wave packet transform over the real line has been extended for higher dimensions by several authors, see [11]. The mathematical theory of classical wave-packet analysis on the real line is originated from classical dilations, translations, and modulations of a given window function. The mathematical theory of wave-packet analysis as a coherent state analysis has been recently abstracted in the setting of locally compact Abelian groups in [28]. In a nutshell, wave-packet analysis which is also well-known as Gabor-wavelet analysis is a shrewd extensions of the two most prominent coherent states analysis, namely Gabor and wavelet analysis.
The following paper consists of abstract aspects of nature of metaplectic wave-packet transforms over ( ) R L d 2 . This paper aims to introduce the notion of metaplectic wave-packet transform over the Hilbert function space ( ) R L d 2 . We shall address analytic aspects of metaplectic wave-packet transforms over ( ) R L d 2 using tools from representation theory of locally compact groups and abstract harmonic analysis.
This article contains 6 sections. Section 2 is devoted to fix notations and a summary of classical Fourier analysis on R d and classical harmonic analysis on projective representations and square-integrable representations over locally compact groups. In section 3 we present a brief study of harmonic analysis over the real symplectic group ( ) R Sp d . We introduce the abstract notion of symplectic wave-packet groups associated to closed subgroups of ( ) R Sp d . We shall also show that the group structure of symplectic wave-packet groups canonically determines an irreducible projective (unitary) group representation of the group, which is called as metaplectic wave-packet representation. We then present an admissibility criterion on closed subgroups of ( ) R Sp d to guarantee the square-integrability of the associated metaplectic wave-packet representation on ( ) R L d 2 .
As an application of our results we study analytic aspects of metaplectic wave-packet transforms associated to closed subgroups of the real symplectic goup ( ) R Sp d . It is also shown that, if H is a compact subgroup of ( ) R Sp d , for all non-zero window functions we can continuously reconstruct any L 2 -function from metaplectic wave-packet coefficients. Finally, we will illustrate application of these techniques in the case of well-known compact subgroups of the real symplectic group ( ) R Sp d .

Preliminaries and notations
Let G be a locally compact group and H be a Hilbert space. Let ( ) U H be the multiplicative group of all unitary operators on H. A projective group representation of G on H is a mapping for some right Haar measure n G of G.
Since R d is an LCA (locally compact Abelian) group, according to the Schur's lemma, all irreducible representations of R d are one-dimensional. Thus any irreducible unitary repre- and hence there exists a continuous homomorphism ω of R d into the circle group T, such that for each x z. Such homomorphisms are called characters of R d and the set of all such characters of R d is denoted by R d . If R d equipped with the topology of compact convergence on R d which coincides with the w * -topology that R d inherits as a subset of ( ) ∞ R L d , then R d with respect to the product of characters is an LCA group which is called the dual (character) group of R d . The character group R d , that is the multiplicative group of all continuous additive homomorphisms of R d into the circle group T, can be parametrizes by R d via the following is called the Fourier transform on R d . It is a norm-decreasing * -homomorphism from ( ) and hence it can be extended uniquely to a unitary isomorphism from ( ) , see [24]. Then each is unitary as well. The modulation and translation operators are connected via the Fourier transform by , ω ∈ R d , and ∈ R k d , see [24,38,52]. From now on and in this article, for a fixed Haar (Lebesgue) measure R m d on is defined by ( ) π λ = ω M T x . Then, it is well-known as the Moyal's formula, that , see [37] and classical references therein.

Harmonic analysis over symplectic groups
Throughout this section, we briefly present basics of classical harmonic analysis over the real symplectic group ( ) R Sp d , for a complete picture of this matrix group we referee the readers to [18][19][20][44][45][46] and the comprehensive list of classical references therein. For be the linear map given by ( ) ( ) . The group consists of all symplectic matrices is called the (real) symplectic group which is denoted by ( ) R Sp d . It is a simple noncompact finite-dimensional real Lie group. In block-matrix notation, the symplectic group The real symplectic group ( ) R Sp d satisfies the following decomposition, namely Iwasawa (Gram-Schmidt) decomposition, ( where [55,56] { } ( ) which satisfies the following intertwining identity In coordinate terms, a metaplectic operator on ( ) which satisfies the following intertwining identity , , In this case, the operator U is called as the metaplectic operator on ( ) R L d 2 associated to the symplectic matrix S.
is given by The following proposition [43] shows that the Fourier transform, dilations, and chrip multiplications can be considered as metaplectic operators.
and satisfies the following intertwining identity is a metaplectic operator on and satisfies the following intertwining Then the following [43] result gives us a unified and also explicit construction of metaplectic operators on ( ) R L d 2 by splitting them into simple operators given in proposition 3.1. .
is the metaplectic operator associated to the symplectic matrix S.

Multivariate metaplectic wave packet representations
In this section we present the abstract structure of multivariate symplectic wave-packet groups associated to closed subgroups of the real symplectice group ( ) R Sp d . Then we introduce the associated multivariate metaplectic wave-packet representation. We shall also study classical properties of these representations.
For a closed subgroup H of the real symplectic group ( ) R Sp d , the underlying manifold is a group with the identity element ( ) 1, 0, 0 . We call this group as symplectic wave-packet group associated to the subgroup H over R d . For simplicity, we may use ( ) G H instead of ( ) G H d, , at times. The groups H and × R R d d can be considered as closed subgroups of ( ) G H . Then we present the following theorem concerning basic properties of the symplectic wave-packet group ( ) G H in the framework of harmonic analysis.

and a right Haar measure given by
Proof. It can readily be checked that the mapping is continuous. This automatically implies that the symplectic wave-packet group Similarly, using (3.4), Fubini's theorem and also since the Lebesgue measure µ Next we deduce the following consequences.
. In particular, the symplectic wave-packet group ( ) G H is unimodular if and only if H is unimodular. Proof.
. Then we can write 3) The following theorem shows that ( ) Γ H g g given by ( Proof. Plainly, we have ( ) . Invoking definition of ( ) λ Γ H S, , it is evident to check that ( ) λ Γ H S, is a unitary operator, because it is the composition of two unitary operators, namely be a second degree character such that the intertwining identity (3.5) holds for ′ S . Hence, we get   [28,42,57,58] and the comprehensive list of references therein.

Square-integrability of multivariate metaplectic wave-packet representations
Throughout this section, we study the square-integrability of multivariate metaplectic wavepacket representations. We still assume that H is a closed subgroup of the symplectic group ( ) R Sp d .
It should be mentioned that in the framework of classical voice/coherent state transforms [59], the problem of admissibility conditions for subgroups of the symplectic group studied from an algebraic perspective in [1,2,12,13,17,21].
be a window function. The metaplectic wave-packet transform of ( ) ∈ R f L d 2 with respect to the window function ψ is given by the voice transform associated to the metaplectic wave-packet representation, that is Then, using Fubini's theorem and also the Moyal's formula (2.4), we get

Then, each non-zero function
which implies the square-integrability of the metaplectic wave-packet representation Γ H over the symplectic wave-packet group ( ) G H . □ As a consequence of theorem 5.2, we deduce the following orthogonality relation concerning the metaplectic wave-packet transforms.
Proof. The same argument used in theorem 5.2 implies that

Then (5.4) and also twice applying the Polarization identity guarantees (5.3). □
Next result is an inversion (reconstruction) formula for the metaplectic wave-packet transform defined by (5.1). , from metaplectic wave-packet coefficients generated by ψ, via the following resolution of the identity formula; in the weak sense of the Hilbert function space ( ) , which equivalently implies the reconstruction formula (5.5) in the weak sens of the Hilbert function space ( ) □ Then we can present the following reproducing property for the metaplectic wave-packet representations. . Then (2) ψ H is the unique reproducing kernel Hilbert space (RKHS) over ( ) G H associated to the positive definite kernel given by Next corollary summarizes our recent results in terms of continuous frame theory [8,53]. In this case [26], the real symplectic group ( ) R Sp is precisely the special linear group ( ) R SL 2, , that is the the multiplicative matrix group, consists of all real × 2 2 matrices with determinant one. That is, SL 2,  : : , , , a nd 1 .

Corollary 5.6. Let H be a compact subgroup of the real symplectic group
It is a simple real 3-dimensional Lie group. The special linear group ( ) R SL 2, satisfies the following decomposition, namely Iwasawa (Gram-Schmidt) decomposition, ( ) = R KAN SL 2, where ( ) = K SO 2 is the special orthogonal group consists of all × 2 2-orthogonal matrices with real entries and the subgroups A N , are given by The group ( ) R SL 2, is non-compact but unimodular. A Haar measure of ( ) R SL 2, is given by

The subgroup
( ) SO 2 is isomorphic, as a real Lie group, to the circle group, also known as ( ) = T U 1 , via the canonical Lie group isomorphism which sends the complex number θ e i of absolute value 1, to the special orthogonal matrix ( ) θ H . From now on, we may call ( ) SO 2 as the circle group, at times. It can be readily checked that, any closed subgroup of ( ) R SL 2, conjugated to ( ) SO 2 is also compact in ( ) R SL 2, . In addition, the circle group ( ) SO 2 is a maximal compact subgroup of the multiplicative matrix Lie group ( ) R SL 2, , which means that ( ) SO 2 is a compact subgroup and it is maximal among such subgroups as well. Thus, any continuous (non-discrete) and compact subgroup is one-dimensional. Then by proposition 3.2 of [45], it is conjugated to the compact subgroup ( ) SO 2 .
(i) The circle group. By the above argument and theoretical motivation, first we shall focus on analytic and constructive analysis of metaplectic wave-packet representations over the compact subgroup ( ) SO 2 . The normalized Haar measure ( ) σ SO 2 of the circle group ( ) SO 2 is given by The following theorem characterizes analytic aspects of the metaplectic wave-packet representation associated to the compact subgroup ( ) SO 2 . Proof.

Proposition 6.2.
( ( )) G SO 2 is a non-Abelian, non-compact, and unimodular group with a Haar measure given by be a non-zero window function. The metaplectic wave-packet transform can be regarded as given by , , 0, 2 . The Plancherel formula for (6.2) reads as follows; Then (6.3) guarantees the following reconstruction formula; ). Since every subgroup of the circle group is either dense or finite, we deduce that any closed proper subgroup of the circle group is finite.
Let ∈ N N be a positive integer and Then T N is a finite subgroup of T of order N. One can also check that, , is a finite subgroup of ( ) SO 2 of order N. Also, it is easy to check that any finite subgroup of ( ) (i) Finite circle groups Let ∈ N N be a positive integer. The normalized Haar measure of ( ) SO 2 N is given by is a non-Abelian, non-compact, and unimodular group with a Haar measure given by be a non-zero window function. The metaplectic wave-packet transform can be regarded as given by The Plancherel formula for (6.5) reads as follows; Then (6.6) guarantees the following reconstruction formula; It is well-known that K d is the maximal compact subgroup of the real symplectic group ( ) R Sp d , see [18][19][20]45] and the classical list of references therein. Also, it can readily be check that The following theorem presents an explicit construction for metaplectic operators associated to the maximal compact subgroup K d .
is the metaplectic operator associated to the symplectic matrix S.
Next we can also present the following characterizations.
Proof. Let d > 1 and Then, Λ = 0 and hence H = A and Q = B. Thus, using theorem 6.4, we deduce that -dimensional real Lie group and it is non-connected. The probability (normalized Haar) measure over ( ) d O is given by ω Concluding Remarks. The main purpose of this article was dedicated to presenting a constructive admissibility criterion on closed subgroups of the real symplectic group ( ) R Sp d which guarantees square integrability of the associated multivariate metaplectic wave-packet representations and hence a valid resolution of the identity operator in the sense of the Hilbert function space ( ) R L d 2 .
Invoking topological and geometric structure of the real Lie group ( ) R Sp d , there is a high degree of freedom in selecting an admissible subgroup H of ( ) R Sp d . Among all closed subgroups of ( ) R Sp d , just compact ones are admissible and hence they guarantee a squareintegrable multivariate metaplectic wave-packet representation and valid reconstruction formula.