Symplectic Analysis of Time-Frequency Spaces

We present a different symplectic point of view in the definition of weighted modulation spaces $M^{p,q}_m(\mathbb{R}^d)$ and weighted Wiener amalgam spaces $W(\mathcal{F} L^p_{m_1},L^q_{m_2})(\mathbb{R}^d)$. All of the classical time-frequency representations, such as the short-time Fourier transform (STFT), the $\tau$-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions $\mu(\mathcal{A})(f\otimes \bar{g})$, where $\mu(\mathcal{A})$ is the metaplectic operator and $\mathcal{A}$ is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [E. Cordero and L. Rodino (2022)"Characterization of Modulation Spaces by symplectic representations and applications to Schr\"odinger equations", arXiv:2204.14124], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called"shift-invertibility"condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes in to play for Wiener amalgam spaces. The shift-invertibility property is necessary: Ryhaczek and and conjugate Ryhaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-tryangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.


Introduction
The modulation and Wiener amalgam spaces were introduced by H. Feichtinger in 1983 in his pioneering work [20] and they started to become popular in the early 2000s in many different frameworks.In fact, they were successfully applied in the study of pseudodifferential and Fourier integral operators, PDE's, quantum mechanics, signal processing.Nowadays, the rich literature on these spaces witnesses their importance: see, e.g., the very partial list of works [3,5,6,8,10,36,39,41,42,43], as well as the textbooks [4,13,29,28,44].
The modulation spaces M p,q m (R d ) are classically defined in terms of the short-time Fourier transform (STFT), i.e., (1) V where g ∈ L 2 (R d )\{0} is a so-called window function and the definition is extended to (f, g) ∈ S ′ (R d ) × S(R d ) as in Section 2.3.Namely, for a tempered distribution ), for an arbitrary fixed window g.The identity V g f (x, ξ) = F (f • ḡ(• − x))(ξ) justifies the choice of the STFT as the time-frequency representation used to define modulation spaces.In fact, it states that the STFT can be used to measure the local frequency content of signals in terms of weighted mixed norm spaces (see Section 2.4 below).
Apart of its interpretation, there is no reason why the STFT shall serve as the leading time-frequency representation in the definition of modulation spaces.Actually, there are many reasons that make it unsuitable in many contexts, such as the theory of pseudodifferential operators and quantum mechanics, cf.[28,40].
In [28] M. De Gosson proved that the (cross-)Wigner distribution, defined for all f, g ∈ L 2 (R d ) as (2) W (f, g)(x, ξ) = can be used to define modulation spaces.Namely, f ∈ M p,q m (R d ) ⇔ W (f, g) ∈ L p,q m (R 2d ).Later, in [14], the above characterization was extended to (cross-)τ -Wigner distributions with τ ∈ R \ {0, 1}.The cases τ = 0, 1 correspond to the so-called (cross-) Rihacek and conjugate Rihacek distributions, respectively.Their explicit expressions W 0 (f, g)(x, ξ) = f (x)ĝ(ξ)e −2πiξ•x and W 1 (f, g)(x, ξ) = f (ξ)g(x)e 2πiξ•x , x, ξ ∈ R d , reveal that as well as The lowest common denominator of these time-frequency representations is that they can all be written as where µ(A) ∈ Mp(2d, R) is a so-called metaplectic operator and A ∈ Sp(2d, R) is the unique associated symplectic matrix, we refer to Section 2.6 for the precise definitions.In fact, Using (3) plenty of new time-frequency representations, that we call metaplectic Wigner distributions, can be defined in terms of metaplectic operators, cf.[14,15,9].The question becomes for which µ(A) ∈ Mp(2d, R) or, equivalently, for which A ∈ Sp(2d, R), the following property holds: for a moderate weight function m, 0 < p, q ≤ ∞, and a fixed non-zero g ∈ S(R d ), ( 6) . A partial answer is given in [15,9], where the authors proved that for distributions of the form 2 below for their definition), the characterization property holds.In [15] it is conjectured that the characterization (6) should hold for W A satisfying the so-called shift-invertibility property.Namely, for every W A the following equality holds (7) |W The main results of this work solve this conjecture for the Banach setting 1 ≤ p, q ≤ ∞.We summarize them is the following theorem.
Theorem 1.1.Let 1 ≤ p, q ≤ ∞, m be a v-moderate weight, g a fixed non-zero window function in S(R d ).Consider a metaplectic operator µ(A) ∈ Mp(2d, R) and let A be the unique symplectic matrix associated to µ(A).The following statements hold.(6) holds with equivalence of norms.
The core of Theorem 1.1 is that shift-invertibility alone is not sufficient to characterize modulation spaces.This is not surprising: as it is observed in [25], E A has to be upper triangular, other than invertible, for the operator f → f (E A •) to preserve the L p,q m spaces.Nevertheless, we claim that the conditions on E A stated in Theorem 1.1 are fundamental to characterize modulation spaces.To support this thesis, we stress that the Rihacek distributions are examples of non shift-invertible metaplectic Wigner distributions for which Theorem 1.1 fails.
Furthermore, we provide examples of shift-invertible metaplectic Wigner distributions which characterize modulation spaces if and only if the matrix E A is upper triangular, see Example 4.1 below (see also Remark 3.8).
A relevant contribution of this work consists of constructing explicit examples of metaplectic Wigner distributions that can be used to characterize modulation spaces, some of them extend the representations studied in [46,47] (see also references therein).Our leading idea is to substitute the time-frequency shifts in (1) and ( 2) with new time-frequency atoms.Namely, we replace the chirp e 2πiξ•t with the more general one Φ C (ξ, t) = e iπ(ξ,t) T •C(ξ,t) T , with C ∈ R 2d×2d symmetric matrix.The importance of these examples is that different atoms provide alternative ways to decompose signals into fundamental time-frequency functions, yielding to important applications in many branches of engineering, learning theory and signal analysis.In particular, discretization of time-frequency representations under this point of view could entail consequences in frame theory, phase retrival, and maybe in other aspects of signal processing, producing advances in these frameworks.
Besides the applications above, the metaplectic approach to time-frequency representations carry a high potential in many other situations were time-frequency representations play a crucial role, see e.g., [1,7,17,19,33,34].Finally, observe that a first attempt to generalize the τ -Wigner distributions is contained in the work [2], see also [16].
Outline.Section 2 contains preliminaries and notation.The main results are exposed in Section 3 whereas Section 4 exhibits the most relevant examples.In the Appendix A we extend some of the results in [25] to general invertible matrices and to the quasi-Banach setting.In the Appendix B we compute the matrices associated to tensor products of metaplectic operators.

Preliminaries
Notation.We denote t 2 = t • t, t ∈ R d , and xy = x • y (scalar product on R d ).The space S(R d ) is the Schwartz class whereas S ′ (R d ) the space of temperate distributions.The brackets f, g denote the extension to S ′ (R d ) × S(R d ) of the inner product f, g = f (t)g(t)dt on L 2 (R d ) (conjugate-linear in the second component).We write a point in the phase space (or time-frequency space) as z = (x, ξ) ∈ R 2d , and the corresponding phase-space shift (time-frequency shift) acts on a function or distribution as ( 8) ) denotes the space of smooth functions with compact support.The notation f g means that f (x) ≤ Cg(x) for all x.If g f g or, equivalently, f g f , we write f ≍ g.For two measurable functions f, g : ).Thus, for all f, g ∈ S ′ (R d ) the operator f ⊗ g ∈ S ′ (R 2d ) characterized by its action on ϕ ⊗ ψ ∈ S(R 2d ) by f ⊗ g, ϕ ⊗ ψ = f, ϕ g, ψ extends uniquely to a tempered distribution of S ′ (R 2d ).The subspace span{f ⊗ g : 2.1.Weighted mixed norm spaces.We denote by v a continuous, positive, even, submultiplicative weight function on R 2d , i.e., v(z 1 + z 2 ) ≤ v(z 1 )v(z 2 ), for all z 1 , z 2 ∈ R 2d .Observe that since v is even, positive and submultiplicative, it follows that v(z) ≥ 1 for all z ∈ R 2d .We say that w ∈ M v (R 2d ) if w is a positive, continuous, even weight function on with the obvious adjustments when min{p, q} = ∞.The space of measurable functions f having 2.2.Fourier transform.In this work, the Fourier transform of , the Fourier transform of f is defined by duality as the tempered distribution characterized by We denote with F f := f the Fourier transform operator.It is a surjective automorphism of S(R d ) and S ′ (R d ), as well as a surjective isometry of L 2 (R d ).
If 1 ≤ j ≤ d, the partial Fourier transform with respect to the jth coordinate is defined as (10) Analogously, the definition is transported on S ′ (R d ) in terms of antilinear duality pairing: for all f ∈ S ′ (R d ), Observe that holds that for all permutation σ ∈ Sym({1, . . ., d}).Finally, for all 1 ≤ j ≤ d, 2.3.Time-frequency analysis tools.The short-time Fourier transform where T L F (x, y) = F (y, y − x) and F 2 is the partial Fourier transform with respect to the second coordinate, cf.Example 2.2 below.This equality allows to extend the definition of We recall the fundamental identity of time-frequency analysis: where the symplectic matrix J is defined by (13) J = 0 d×d I d×d −I d×d 0 d×d .
Here I d×d ∈ R d×d is the identity matrix and 0 d×d is the matrix of R d×d having all zero entries.The reproducing formula for the STFT reads as follows: for all g, γ ∈ L 2 (R d ) such that g, γ = 0, ( 14) where the identity holds in L 2 (R d ) as a vector-valued integral in a weak sense (see, e.g., [13,Subsection 1.2.4]).

2.4.
Modulation spaces [4,20,21,29,26,32].Fix 0 < p, q ≤ ∞, m ∈ M v (R 2d ), and g ∈ S(R d ) \ {0}.The modulation space M p,q m (R d ) is classically defined as the space of tempered distributions f ∈ S ′ (R d ) such that defines a norm, otherwise a quasi-norm.Different windows give rise to equivalent (quasi-)norms.Modulation spaces are (quasi-)Banach spaces and the following continuous inclusions hold: ) and min{p, q} ≥ 1.We will also use the inclusion m (R d ), which coincides with the latter whenever p, q < ∞.
, where p ′ and q ′ denote the Lebesgue conjugate exponents of p and q respectively.Finally, if m 1 ≍ m 2 , then M p,q m 1 (R d ) = M p,q m 2 (R d ) for all p, q.
2.5.Wiener amalgam spaces [22,23,37].For 0 , is defined as the space of tempered distributions f ∈ S ′ (R d ) such that for some (hence, all) window g ∈ S(R d ) \ {0}, , so that where the matrix J is defined in (13).We represent A as a block matrix Let ρ be the Schrödinger representation of the Heisenberg group, that is ) defines another representation of the Heisenberg group that is equivalent to ρ, i.e. there exists a unitary operator µ(A) : This operator is not unique, but if µ ′ (A) is another unitary operator satisfying For If we add the assumptions C symmetric and invertible, then we can compute explicitly its Fourier transform, that is Example 2.2.For particular choices of A ∈ Sp(d, R), µ(A) is known.Let J, D L and V C be defined as in ( 13) and ( 20), respectively.Then, up to a sign, Other important symplectic matrices are the so-called quasi-permutation matrices [18,25].Definition 2.3.For 1 ≤ j ≤ d, the symplectic interchange matrix Π j ∈ Sp(d, R) is the matrix obtained interchanging the columns j and j + d of the 2d-by-2d identity matrix and multiplying the jth column of the resulting matrix by −1.
The corresponding metaplectic operators are the partial Fourier transforms, as we can see below.
Example 2.4.Let F j , 1 ≤ j ≤ d, be the partial Fourier transform w.r.t. the jth coordinate defined in (10).Then In fact, take any f ∈ L 1 (R d ) and compute F j ρ(x, ξ, τ )F −1 j f as follows: Observe also that j Π j = J, in line with (11).

Shift-invertibility and modulation spaces
In this section we present the features of metaplectic operators that guarantee the representations of modulation and Wiener amalgam spaces by metaplectic operators.We first need to recall the definition of metaplectic Wigner distributions and their main properties.
with equality in S ′ (R d ), the integral being meant in the weak sense.
If µ(A) ∈ Mp(2d, R) with A ∈ Sp(2d, R) having block decomposition then it was shown in [15] the equality where the matrices E A and F A are given by (31) Definition 3.5.Under the notation above, we say that W A (or, by abuse, We need the following representation formula. Lemma 3.6.Let µ(A) ∈ Mp(2d, R), γ, g ∈ S(R d ) be such that γ, g = 0 and f ∈ S ′ (R d ).Then, with equality in S ′ (R 2d ), the integral being intended in the weak sense.
Proof.Take any ϕ ∈ S(R 2d ) and use the definition of vector-valued integral in a weak sense which entails Therefore, with equality in S ′ (R 2d ).
Theorem 3.7.Let W A be shift-invertible with E A upper-triangular.Fix a non-zero window function For any γ ∈ S(R d ) such that γ, g = 0, the inversion formula for the STFT (cf.Theorem 2.3.7 in [13]) reads Multiplying both sides of the above equality by ḡ(z 2 ), for any z = (z 1 , z 2 ) ∈ R 2d , we can write ).Using Lemma 3.6, we get: with equality holding in S ′ (R 2d ).Now, if f ∈ M p,q m (R d ), the integral on the right-hand side is absolutely convergent as we shall see presently.For any z ∈ R 2d , ).Moreover, by Theorem A.2 and Theorem A.3 both applied with ). Young's convolution inequality applied to (34) entails ).Using Corollary 3.4 with g 3 = g 1 = g, g 2 = γ, for any w ∈ R 2d , Applying Theorem A.2 and Theorem A.3 with S = E A , we obtain ∞, since we considered an even submultiplicative weight v. Remark 3.8.Theorem 3.7 is sharp.Namely, if either E A is not shift-invertible or E A is not upper triangular, W A may not characterize modulation spaces.We provide two counterexamples.(a) If E A is not shift-invertible, then W A may not characterize modulation spaces.Let W 0 be the (cross-)Rihacek distribution defined in (16).Obviously, for every f ∈ L p (R d ) and g ∈ S(R d ), we obtain W 0 (f, g) L p,q = f p ĝ q .This means that the L p,q -norm of W 0 is not equivalent to the modulation norm in general.Observe that the corresponding matrix E A 0 is not shift-invertible.In fact, is not shift-invertible and does not characterize modulation spaces [14,Remark 3.7].(b) If E A is not upper-triangular, then W A may not characterize modulation spaces.Let C ∈ R 2d×2d \ {0 2d×2d } be any symmetric matrix.Then, up to a sign, where which is always invertible and lower-triangular.The metaplectic operator µ( Observe that a similar result with different methods is obtained in [25,Theorem 3.3].
As byproduct of the previous theorem we obtain new properties for shift-invertible representations W A , see ahead.
) and it is everywhere defined.
Proof.If f ∈ L 2 (R d ) and g, γ ∈ S(R d ), the inequality (34) holds pointwise (take p = q = 2, m = 1).Also, if g ∈ L 2 (R d ) the right hand-side of (34) is also well defined for all z ∈ R 2d , since ) and W A (f, g)(z) is well defined for all z ∈ R 2d .
If we limit to the case p = q, then T S : ) is bounded for all S ∈ GL(2d, R), without any further assumption on its triangularity.In this case, arguing as above, but using Theorem A.1, we obtain the following result.
Under the assumptions of Theorem 3.7, assume that is well defined by (34) (36), we obtain m (R 2d ).By (35), for all w ∈ R 2d , and Young's inequality gives ).Another consequence of Theorem 3.7 is the characterization of Wiener amalgam spaces W (F L p m 1 , L q m 2 )(R d ).Corollary 3.12.Let µ(A) ∈ Mp(2d, R) be such that W A is shift-invertible and A = π M p (µ(A)) having block decomposition in (29).Fix g ∈ S(R d ) \ {0} and define , with the analogous for max{p, q} = ∞.
Observe that ẼA = E Ã0 .Since E A is invertible and lower triangular the matrix ẼA in ( 37) is obviously invertible (and upper triangular).Hence, using the assumption , we have . The same argument also proves the case max{p, q} = ∞, simply replacing the corresponding integrals with the essential supremums.Remark 3.13.Because of (18), Corollary 3.12 is significant only for p = q.For p = q we refer to Theorem 3.10 with m = m 1 ⊗ m 2 .

Examples
We exhibit a manifold of new metaplectic Wigner distributions which may find application in time-frequency analysis, signal processing, quantum mechanics and pseudodifferential theory.which allows to rewrite the STFT V g f as composition of the metaplectic operators We may act on the window g by replacing ḡ with µ(A ′ )g, µ(A ′ ) ∈ Mp(d, R).Namely, we consider the time-frequency representation which is always shift-invertible with E A diagonal.This is not surprising, since µ(A ′ )ḡ ∈ S(R d ) for g ∈ S(R d ) and different windows in S(R d ) yield equivalent norms.
(ii) A more interesting example comes out by applying µ(A ′ ), with A ′ ∈ Sp(d, R) having block decomposition in (38), to the function f .Namely, we consider 38).This W A characterizes modulation spaces if and only if the symplectic matrix A ′ is upper triangular, since µ(A ′ ) : M p,q (R d ) → M p,q (R d ), p = q, if and only if A ′ is an upper block triangular matrix [25].

I
d×d −I d×d 0 d×d 0 d×d 0 d×d 0 d×d I d×d I d×d 0 d×d 0 d×d 0 d×d −I d×d −I d×d 0 d×d 0 d×d 0 d×d If A ∈ Sp(d, R), then det(A) = 1.The matrix A ∈ Sp(d, R) with block decomposition (19) is called free if det B = 0.For L ∈ GL(d, R) and C ∈ R d×d , C symmetric, we define (20) D L := L −1 0 d×d 0 d×d L T and V C := I d×d 0 C I d×d .J and the matrices in the form V C (C symmetric) and D L (L invertible) generate the group Sp(d, R).

( 21 )
, then µ ′ (A) = cµ(A), for some unitary constant c ∈ C, |c| = 1.The set {µ(A) : A ∈ Sp(d, R)} is a group under composition and it admits a subgroup that contains exactly two operators for each A ∈ Sp(d, R).This subgroup is called metaplectic group, denoted by Mp(d, R).It is a realization of the two-fold cover of Sp(d, R) and the projection (22) π M p : Mp(d, R) → Sp(d, R) is a group homomorphism with kernel ker(π M p ) = {−id L 2 , id L 2 }.Proposition 2.1.[24, Proposition 4.27] The operator µ(A) ∈ Mp(2d, R) maps S(R d ) isomorphically to S(R d ) and it extends to an isomorphism on S ′ (R d ).

Example 4 . 1 .
This example generalizes the STFT by applying a metaplectic operator either on the window function g or on the function f as follows.First, consider the matrix L = 0 d×d I d×d −I d×d I d×d .