Excursus on modulation spaces via metaplectic operators and related time-frequency representations

Modulation spaces were originally introduced by Feichtinger in 1983. Since the 2000s there have been thousands of contributions using them as correct framework; they range from PDEs, pseudodifferential operators, quantum mechanics, signal analysis. This justifies a deep study of such spaces and the related Wiener ones. Recently, metaplectic Wigner distributions, which contain as special examples the $\tau$-Wigner distributions, the ambiguity function and the Short-time Fourier transform, have proved to characterize modulation spaces, under suitable assumptions. We investigate the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. We add a new result on this topic and conclude with an exhaustive vision of these characterizations. Similar results hold for the Wiener amalgam ones.


Introduction
This work is a detailed overview of the theory surrounding modulation spaces and their characterization using metaplectic operators and time-frequency representations.It covers the historical development of modulation spaces from their introduction by Feichtinger in 1983 to recent advancements in the field and provides new results in this framework.
Here's a breakdown of the main points discussed: Background: We introduce modulation spaces, which are fundamental in various disciplines including time-frequency analysis, PDEs, quantum mechanics, and signal analysis.Metaplectic Operators and Invariance Properties: We discuss the importance of metaplectic operators and their invariance properties in characterizing modulation spaces, presenting results regarding the conditions under which metaplectic operators extend to isomorphisms on modulation spaces.Time-Frequency Representations: We present various time-frequency representations, including the short-time Fourier transform (STFT) and the Wigner distribution, which arise as images of metaplectic operators and are called metaplectic Wigner distributions.We highlight their role in characterizing modulation spaces and provide a reproducing formula for these representations.
Characterization of Modulation Spaces: We exhibit characterizations of modulation spaces using metaplectic Wigner distributions.These characterizations involve shift-invertible properties and block decompositions of symplectic matrices.Independence of Symplectic Matrix Blocks: We showcase a new result regarding the independence of certain blocks of symplectic matrices in the characterization of modulation spaces using metaplectic Wigner distributions.
Modulation and Wiener amalgam spaces were introduced by Feichtinger in [14] (see [15,17] for the Wiener case) in the framework of time-frequency analysis and extended to the quasi-Banach setting by Galperin and Samarah [20] in 2004.Since the 2000s they have become popular in many different environments, ranging from PDEs to quantum mechanics, pseudo-differential theory and signal analysis.Hundreds of papers use these spaces as a natural framework.Since it is impossible to cite them all, we simply refer to the textbooks [3,8,24].
A key point in this study, as observed in [19], is the invariance properties of metaplectic operators, asking under which conditions a metaplectic operator Â, initially defined as a unitary operator on L 2 (R d ), extends to an isomorphism on the modulation spaces M p,q m (R d ).Recall that the metaplectic group Mp(d, R) is the two-fold cover of the symplectic group Sp(d, R), i.e., there exists a surjective Lie group homomorphism π M p : Mp(d, R) → Sp(d, R).
The translation and modulation operators are defined by Their composition yields the so-called time-frequency shift Modulation spaces M p,q m (R d ) are classically defined in terms of the short-time Fourier transform (STFT), i.e., (1) where g ∈ S(R d ) \ {0} is the so-called window function.The definition can be extended to (f, g) ∈ S ′ (R d ) × S(R d ), cf.Section 2.2 for details.
).Recently, Führ and Shafkulovska [19,Theorem 3.2] (see [5] for the quasi-Banach setting) proved the following: The following statements are equivalent: (i) Â : M p,q → M p,q is well defined; (ii) Â : M p,q → M p,q is well defined and bounded (in fact, it is an isomorphism); (iii) One out of the two conditions below holds: (iii.1) p = q, (iii.2) p = q and A is an upper triangular matrix.
This result can be lifted to weighted modulation spaces From [24,Lemma 9.4.3]we infer the action of the STFT on the metaplectic operator Â such that π M p ( Â) = A: The equality above shows the interaction between Â : M p,q (R d ) → M p,q (R d ) and D A : L p,q (R 2d ) → L p,q (R 2d ), where D A F (z) := F (A −1 z), highlighted by the diagram below. (3) Then, Theorem 1.1 is a direct consequence of the result below [19,Theorem 3.3].
Theorem 1.2.For 0 < p, q ≤ ∞, A ∈ Sp(d, R), the following statements are equivalent: ) is everywhere defined and bounded.(iii) D A : V Â−1 g (M p,q ) → L p,q (R 2d ) is everywhere defined.(iv) D A : V Â−1 g (M p,q ) → L p,q (R 2d ) is everywhere defined and bounded.(v) One out of the two conditions below holds: (v.1) p = q, (v.2) p = q and A is an upper triangular matrix.
A clarifying example, provided by [19], is the following: for p = q, consider two radially symmetric functions f ∈ L p (R d ) \ L q (R d ) and g ∈ L q (R d ) \ L p (R d ), and consider the standard symplectic matrix: where I d×d ∈ R d×d is the identity matrix, whereas 0 d×d is the matrix of R d×d having all zero entries.The tensor product (f The proof of Theorem 1.2 for p = q lies on the following issue: ) is an upper triangular matrix and 0 < p, q ≤ ∞, then Notice that the matrix S does not need to be symplectic, but only invertible.These results have inspired new characterizations of modulation spaces via timefrequency distributions built by using metaplectic operators: the so-called metaplectic Wigner distributions, defined as follows.
For Â ∈ Mp(2d, R), with π M p ( Â) = A ∈ Sp(2d, R) (notice that the number of variables is doubled!) we call the metaplectic Wigner distribution W A (f, g) the time-frequency representation Examples are the STFT, the (cross-)τ -Wigner distribution: in particular, the Wigner transform, defined as ( 7) For their related metaplectic operators Â and symplectic matrix π M p ( Â) = A we refer to the following section.
Similarly to the STFT, these time-frequency representations enjoy a reproducing formula, cf.[5, Lemma 3.6]: with equality in S ′ (R 2d ), the integral being intended in the weak sense.
From the right-hand side we infer that the key point becomes the action of W A on the time-frequency shift π(w), which can be computed explicitly.For, assume that A has got the block decomposition and consider its sub-matrix ( 10) The equality above suggests the following definition.
Definition 1.5.Under the notation above, we say that W A (or, by abuse, The representation formula (8) and the invertibility property enjoyed by E A (which plays the role of the matrix S in Theorem 1.3) are the main ingredients for the characterization of modulation spaces via metaplectic Wigner distributions, proved in Theorem 3.7 of [5]: ), with equivalence of norms.
If p = q the matrix E A does not need to be upper triangular.
For the quasi-Banach modulation spaces, the characterization above still holds true, but the proof is obtained by different methods.The main issue is to understand which symplectic matrices give a sub-matrix E A invertible, so that the corresponding distributions W A can characterize modulation spaces.
First, let us introduce the chirp function related to the symmetric matrix C ∈ R 2d×2d , defined as and, for E ∈ GL(d, R), the dilation operator T E (which is a metaplectic operator, see below) The answer to our problem is contained in the recent contribution [6] and can be summarized as follows: Theorem 1.7.Let W A be a shift-invertible metaplectic Wigner distribution with A ∈ Sp(2d, R).Then the symplectic matrix A can be split into the product of four symplectic matrices: ) is called deformation operator and can be explicitly computed, see Section 3 for details.
Roughly speaking, the result above says that W A is shift-invertible if and only if W A is a STFT up to linear change of variables and products-by-chirps.
The invariance of L p,q (R 2d ) spaces under dilations by upper-triangular matrices stated in Theorem 1.3 suggests that any dilation of this type, applied to the STFT, give equivalent modulation norms.
The characterization of Theorem 1.7 clearly depends on the blocks of A, does not clarify whether E A , M A and G A can be chosen independently of each other.The main result of this work expresses their independence: This issue is proven in Section 3.

Preliminaries
Notation.We denote by xy = x•y the scalar product on R d .The space S(R d ) is the Schwartz class whereas its dual S ′ (R d ) is the space of temperate distributions.The brackets f, g are 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).A point in the phase space (or time-frequency space) is written as z = (x, ξ) ∈ R 2d , and the corresponding phase-space shift (time-frequency shift) acts as The notation f g means that there exists C > 0 such that f (x) ≤ Cg(x) for every x.The symbol t is used to stress that C = C(t).If g f g (equivalently, f g f ), we write f ≍ g.Given two measurable functions f, g : GL(d, R) stands for the group of d × d invertible matrices, whereas Sym(d, R) = {C ∈ R d×d : C is symmetric}.

2.1.
Weighted mixed norm spaces.From now on v is a continuous, positive, even, submultiplicative weight function on R 2d , that is, v(z We write m ∈ M v (R 2d ) if m is a positive, continuous, even and v-moderate weight function on R 2d .Important examples are the polynomial weights (14) v , with the obvious adjustments when max{p, q} = ∞.The space of measurable functions f having f L p,q m < ∞ is denoted by L p,q m (R 2d ).Recall the following partial generalization of the results in [19], contained in [6, Proposition 2.1]: Define the upper triangular matrix The mapping ), S ∈ GL(2d, R) and 0 < p, q ≤ ∞, and the dilation operator 2.2.Time-frequency analysis tools.In this work, the Fourier transform of f ∈ S(R d ) is normalized as , the Fourier transform of f is defined by duality as the tempered distribution characterized by The operator F is a surjective automorphism of S(R d ) and S ′ (R d ), as well as a surjective isometry of L 2 (R d ).If f ∈ S ′ (R 2d ), we set F 2 f , the partial Fourier transform with respect to the second variables: The short-time Fourier transform of f ∈ L 2 (R d ) with respect to the window g ∈ L 2 (R d ) is defined in (1).

2.3.
Modulation spaces [3,14,15,24,20,28,30].For 0 < p, q ≤ ∞, m ∈ M v (R 2d ), and g ∈ S(R d ) \ {0}, the modulation space M p,q m (R d ) is defined as the space of tempered distributions f ∈ S ′ (R d ) such that m is a norm, otherwise a quasi-norm.Different windows give equivalent (quasi-)norms.Modulation spaces are (quasi-)Banach spaces, enjoying the inclusion properties: , where p ′ and q ′ denote the Lebesgue dual exponents of p and q, respectively.If m 1 ≍ m 2 , then M p,q m 1 (R d ) = M p,q m 2 (R d ) for all p, q.

The symplectic group Sp(d, R) and the metaplectic operators. A matrix
A T JA = J, where J is the standard symplectic matrix defined in (4).
For E ∈ GL(d, R) and C ∈ Sym(2d, R), define: The matrices J, V C (C symmetric), and D E (E invertible) generate the group Sp(d, R).
Recall the Schrödinger representation ρ of the Heisenberg group: ρ(x, ξ; τ ) = e 2πiτ e −πiξx π(x, ξ), for all x, ξ ∈ R d , τ ∈ R. We will use the property: for all f, g For every A ∈ Sp(d, R), ρ A (x, ξ; τ ) := ρ(A(x, ξ); τ ) defines another representation of the Heisenberg group that is equivalent to ρ, i.e., there exists a unitary operator Â : This operator is not unique: if Â′ is another unitary operator satisfying (20), then Â′ = c Â, for some constant c ∈ C, |c| = 1.The set { Â : A ∈ Sp(d, R)} is a group under composition and it admits the metaplectic group, denoted by Mp(d, R), as subgroup.It is a realization of the two-fold cover of Sp(d, R) and the projection: Throughout this paper, if Â ∈ Mp(d, R), the matrix A will always be the unique symplectic matrix such that π M p ( Â) = A.
Recall the following basic facts on metaplectic operators.For C ∈ R d×d , define: If C ∈ Sym(2d, R) ∩ GL(2d, R), then we can compute explicitly its Fourier transform, that is: In what follows we list the most important examples of metaplectic operators.
Example 2.3.Consider the symplectic matrices J, D L and V C defined in ( 4) and (19), respectively.Then, We will often use the following lifting-type result, proved in [5, Theorem B1]: then it is easy to show that the 4d × 4d matrix 2.5.Metaplectic Wigner distributions.Let Â ∈ Mp(2d, R).The metaplectic Wigner distribution associated to Â is defined in (5).The most famous time-frequency representations are metaplectic Wigner distributions.Namely, the STFT can be represented as where: (27) and the τ -Wigner distribution defined in ( 6) can be recast as We recall the following continuity properties.
Proposition 2.4.Let W A be a metaplectic Wigner distribution.Then, ) is bounded.The same result holds if we replace L 2 by S or S ′ .
Since metaplectic operators are unitary, for all For the 4d × 4d symplectic matrix with block decomposition (9) we define four 2d × 2d sub-matrices as follows: (30) Remark 2.5.If A ∈ Sp(2d, R), we can highlight the submatrices E A , E A , F A and F A as: where We observe that K is not symplectic.
Using the property of A symplectic matrix, one can infer (see [6, Section 2]) that: Finally, let us introduce the matrix: The relationship among the matrices above are detailed below (see [6,Lemma 2.6]).
Lemma 2.6.Consider the sub-matrices E A , F A , E A , F A defined in (30) and (31).Let L be defined as in (33).
The action of W A on the time-frequency shift π(w) can be exhibited in detail, thanks to the sub-blocks above, as explained in what follows.
Lemma 2.7.Let W A be a metaplectic Wigner distribution with A = π M p ( Â) having block decomposition (9).Then, for w ∈ R 2d , f, g ∈ L 2 (R d ), we have where M A is the symmetric matrix For the STFT we obtain: which is invertible, whereas for the τ -Wigner distributions we have: which is invertible if and only if τ = 0 and τ = 1, the cases of the Rihacek and conjugate-Rihacek distributions, which do not define modulation spaces.
We established the notation to state the characterization of shift-invertible matrices.The proof is based on the properties enjoyed by symplectic matrices and requires many computations, detailed in [6, A E A be the matrix of Lemma 2.6, with L as in (33).Then, , where Lift(G A ) is defined in (26).
In particular, W A is shift-invertible if and only if, up to a sign, where and G A is the metaplectic operator with π M p ( G A ) = G A where, assuming G A with the block decomposition in (25), (it is the matrix G A with the second diagonal multiplied by −1).

The submanifold of shift-invertible symplectic matrices
We denote with Another way to read Theorem 1.7 is that any A ∈ Sp inv (2d, R) can be written as: , where E A ∈ GL(2d, R), M A ∈ Sym(2d, R) and G A ∈ Sp(d, R).Therefore, shiftinvertible distributions are described by these three matrices, rather than the blocks of A. The possibility of defining a shift-invertible Wigner distribution with arbitrary E A and G A is fundamental for the study of metaplectic Gabor frames, cf [6].
The four submatrices E A , F A , E A and F A in (30, 31) determine the three submatrices M A , G A and E A in Lemma 2.6 and (34), and vice versa.Namely, , by the definitions of G A and M A , and by Lemma 2.6 (i), In this section, we prove rigorously that every triple E ∈ GL(2d, R), C ∈ Sym(2d, R) and S ∈ Sp(d, R) determines uniquely a symplectic matrix in Sp inv (2d, R).
To simplify the notation, let and denote: , where L is defined as in (33).Proof.We first observe that A = α(E, C, S) ∈ Sp inv (2d, R) for every (E, C, S) ∈ CG(2d, R) with E A = E. Indeed, a simple computation shows that: where K is defined as in Remark 2.5.This highlights that E A = E and G A = S.In particular, A ∈ Sp inv (2d, R).
Proof.By (35), if W A is shift-invertible, then (39) holds for the triple E A , M A + L and δ A .It remains to check that (39) defines a shift-invertible metaplectic Wigner distribution.Let W A be as in (39) and A be the related symplectic matrix.Then, L Lift(J δ) where δ is the projection of the metaplectic operator: δg = δḡ.Thus, the shift-invertibility of W A follows by Theorem 1.7.
In view of the issues above, the characterization of modulation spaces can be easily rephrased as follows.
(i) If E is upper-triangular, then f M p,q m ≍ W A (f, g) L p,q m .
(ii) If p = q, the upper triangularity assumption in (i) can be dropped.
It is not difficult to show similar results for Wiener amalgam spaces [17].We leave the details to the interested reader.

Declarations
Conflict of interest.All authors declare that they have no conflicts of interest.

F 2 I
is the Fourier transform with respect to the second variables, then π M p (F 2 ) = A F T 2 , where A F T 2 ∈ Sp(2d, R) is the 4d × 4d matrix with block decomposition d×d 0 d×d 0 d×d 0 d×d 0 d×d 0 d×d 0 d×d I d×d 0 d×d 0 d×d I d×d 0 d×d 0 d×d −I d×d 0 d×d 0 d×d
Theorem 4.2 and Corollary 4.3].Theorem 2.8.Let W A be a shift-invertible metaplectic Wigner distribution and