The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

We characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.


Introduction
In the paper we characterise Gelfand-Shilov spaces of functions and distributions, modulation spaces and Gevrey classes in background of mapping properties of the Zak transforms. We apply these results to deduce duality properties of spaces of quasi-periodic functions and distributions and for investigating transitions of linear operators under the Zak transform.
The Zak transforms are unpredictable and exciting in several ways. They appear in natural ways when dealing with Gabor frame operators in the cases of critical sampling, where the Gabor theory cease to work properly. This ought to be the reason why the transform possess several exciting and almost magical properties, useful in Gabor theory.
For example, in critical sampling cases, the Zak transform Z , adapted to the sampling parameters, takes the Gabor frame operator S φ,ψ into the multiplication operator for some constant c which depends on the sampling parameters. (See [21,40] and Sect. 1 for notations.) We remark that this property is heavily used when showing that Gabor atoms and their canonical dual atoms often belong to the same function classes. (See [4,5,18]. ) An other example concerns the fact that if Z f is continuous, then it has zeros. This property is important when deducing various kinds of Balian-Low theorems, which are essential when finding limitations for bases and Gabor frames in Gabor analysis (see Theorem 8.4.1 and its consequences in [18]).
Before entering the Gabor theory, Zak transforms were first introduced and used in a problem in differential equation by Gelfand [16]. Subsequently, the transforms were applied in various contexts, especially in solid state physics by Zak [41] and in differential equations by Brezin [3].
In these considerations it is essential to understand various kinds of mapping properties of the Zak transforms. The transforms take suitable functions, defined on the configuration space R d into quasi-periodic functions, defined on the phase space R 2d . Hence, in similar ways as for periodic functions, Zak transformed functions are completely described by their behaviour on suitable rectangles.
For example, the (standard) Zak transform is given by when f is a suitable function or distribution (see (1.19) for the general definition of the Zak transform). By the definition it follows that if F = Z 1 f and Q d,r is the cube [0, r ] d , then F is quasi-periodic (with respect to Q d,1 × Q d,2π ). That is, It follows from these equalities that F is completely reconstructable from its data on Q d,1 × Q d,2π . It is well-known that Z 1 is bijective from L 2 (R d ) to the set of quasi-periodic elements in L 2 (Q d, 1 × Q d,2π ) and that (0.1) (Cf. e.g. [18,Theorem 8.2.3].) Consequently, L 2 (R d ) can be characterized in a convenient way by its image under the Zak transform.
An other space that can be characterized by related mapping properties concerns the Schwartz space S (R d ). In fact, it is proved in [22] by Janssen that Z 1 is continuous and bijective from S (R d ) to the set of quasi-periodic elements in C ∞ (R 2d ).
In [32,33], Heil and Tinaztepe deduce some important mapping properties for the Zak transform on modulation spaces, and apply these results to deduce Balian-Low properties in the framework of such spaces. These mapping properties on modulation spaces seems not to be (complete) characterizations, because of absence of bijectivity. In fact, apart from the spaces L 2 (R d ) and S (R d ), the whole theory seems to lack characterizations of essential function and distribution spaces via the Zak transform as remarked in Subsection 8.2 (f) in [18].
In Sect. 2 we make this part more complete and furnish the theory with various kinds of characterizations. Especially we characterize modulation and Lebesgue spaces by suitable Lebesgue estimates of short-time Fourier transforms of the Zak transforms of the involved functions. We also characterize the dual S (R d ) of S (R d ), the (standard) Gelfand-Shilov spaces and their distribution spaces by their images under the Zak transform.
For example we prove that Z 1 is continuous and bijective from S (R d ) to the set of all quasi-periodic distributions on Q d,1 × Q d,2π . (See Theorem 2.1.) In Theorems 2.2 and 2.5 we deduce similar characterizations for Gelfand-Shilov spaces and their distribution spaces. As a consequence of Theorem 2.2 we have that the Zak transform Z 1 maps the Gelfand-Shilov space S σ s (R d ) bijectively to E σ,s Z ,1 (R 2d ), the set of all quasi-periodic functions on Q d, 1 × Q d,2π in the Gevrey class E σ,s (R 2d ). In the same way it follows from Theorem 2.5 that the Zak transform Z 1 maps the Gelfand-Shilov distribution space (S σ s ) (R d ) bijectively to (E σ,s Z ,1 ) (R 2d ), the set of all quasi-periodic distributions on Q d, 1 × Q d,2π in (S σ,s s,σ ) (R 2d ). As a consequence, if s + σ < 1, then there are no non-trivial quasi-periodic functions in E σ,s (R 2d ) (cf. Corollary 2.3).
An other consequence of our results is that Z 1 maps the modulation space M p (R d ) continuously and bijectively to the set of all elements in W ∞, p (R 2d ) which are quasiperiodic on Q d,1 × Q d,2π . Furthermore, (see Theorem 2.13 and Corollary 2.14). We also use some recent results in [39] on Wiener estimates to deduce different versions of the latter characterization. For example we show that (0.2) in combination with results in [39,Section 2] give If p = 2, then an application of Parseval's formula implies that (0.3) is the same as which is a slightly weaker form of (0.1). In Sect. 3 we apply the mapping results of the Zak transform to deduce duality properties for spaces of quasi-periodic functions and distributions. For example, if p ∈ [1, ∞) and 1 p + 1 p = 1, then we prove that the dual of quasi-periodic elements in E σ,s and in W ∞, p can be identified with the set of quasi-periodic elements in the Gelfand-Shilov distribution space (S σ,s s,σ ) , respective in W ∞, p . An essential part of these investigations concerns characterizations of quasi-periodic elements in terms of estimates of their short-time Fourier transforms, given in the end of Sect. 2 and the beginning of Sect. 3.
Finally, in Sect. 4 we show how linear operators, T are transformed under conjugation of the Zak transform, It follows from our investigations that the map T → T Z is a bijection between the set of all continuous linear mappings and the set of all continuous linear mappings

Preliminaries
In this section we recall some basic facts. We start by discussing Gelfand-Shilov spaces and their properties. Thereafter we recall some properties of modulation spaces and discuss different aspects of periodic distributions.

Gelfand-Shilov Spaces and Gevrey Classes
We shall consider certain extended classes of the standard Gelfand-Shilov spaces of functions and distributions compared to [17]. Let n ∈ Z + , s j , σ j , h ∈ R + and d j ∈ Z + , j = 1, . . . , n be fixed, d = d 1 + · · · + d n , which we often identify with R d . Then is finite. Here the supremum should be taken over all α j , β j ∈ N d j and of Roumieu type (Beurling type) with parameters s 1 , . . . , s n , σ 1 , . . . , σ n is the inductive limit (projective limit) of S σ s;h (V ) with respect to h > 0. In particular, It follows that σ s (V ) is a Fréchet space with topology induced by the seminorms is complete and the topology is the largest one in order for the inclusion map i : and can be identified with the projective limit respectively inductive limit of (S σ [2,Theorem 4.16]. See also [17,24,26].) In particular, in such situations. For conveniency we set For any s j , σ j , s 0, j , σ 0, j > 0 such that s j > s 0, j , σ j > σ 0, j and s 0, j + σ 0, j ≥ 1, j = 1, . . . , n, we have with dense embeddings, where s and σ are the same as above, s 0 = (s 0,1 , . . . , s 0,n ) and σ 0 = (σ 0,1 , . . . , σ 0,n ).
Here and in what follows we use the notation A → B when the topological spaces A and B satisfy A ⊆ B with continuous embeddings. Gelfand-Shilov spaces and their distribution spaces possess convenient mapping properties under (partial) Fourier transformations. In fact, let F be the Fourier transform which takes the form and extends uniquely to homeomorphisms where t = (t 1 , . . . , t n ) and τ = (τ 1 , . . . , τ n ) are given by (See [17] and the analysis therein.) Consequently, are homeomorphisms.
Gelfand-Shilov spaces can in convenient ways be characterized in terms of estimates of the involved functions and their Fourier transforms. For example we have the following. Here f (θ ) g(θ ) means that f (θ ) ≤ cg(θ ) for some constant c > 0 which is independent of θ in the domain of f and g. We also set f (θ ) g(θ ) when f (θ ) g(θ ) and g(θ ) f (θ ).
Then the following conditions are equivalent: (2) f and f are measurable and satisfy for some r > 0 (for every r > 0); (3) f is smooth and satisfies for some h, r > 0 (for every h, r > 0).
We refer to [6,8] and their analyses for the proof of the equivalence between (1) and (2), and to [17] and their analyses for the proof of the equivalence between (1) and (3) in Proposition 1.1.
Gelfand-Shilov spaces and their distribution spaces can also be characterized by estimates of short-time Fourier transforms, (see e.g. [20,30,36]). More precisely, let φ ∈ S (R d ) be fixed. Then the short-time Fourier transform V φ f of f ∈ S (R d ) with respect to the window function φ is the Schwartz distribution on R 2d , defined by , and restricts to a continuous map from S σ Then the following is true: for some r > 0 (for every r > 0); for every r > 0 (for some r > 0).
A proof of Proposition 1.2 (1) can be found in e.g. [20] (cf. [20,Theorem 2.7]) and a proof of Proposition 1.2 (2) in the case n = 1 can be found in [36]. The general case of Proposition 1.2 (2) follows by similar arguments as in [36] and is left for the reader. See also [7] for related results.
Next we consider Gevrey classes on R d . Let σ ≥ 0. For any compact set K ⊆ R d , The Gevrey class E σ (K ) (E σ ;0 (K )) of order σ and of Roumieu type (of Beurling type) is the set of all f ∈ C ∞ (K ) such that (1.11) is finite for some (for every) h > 0. We equipp E σ (K ) (E σ ;0 (K )) by the inductive (projective) limit topology with respect to and The space B is then called a quasi-normed space. A complete quasi-normed space is called a quasi-Banach space. If B is a quasi-Banach space with quasi-norm satisfying (1.12) then by [1,29] there is an equivalent quasi-norm to · B which additionally satisfies (1.13) From now on we always assume that the quasi-norm of the quasi-Banach space B is chosen in such way that both (1.12) and (1.13) hold.
Next we recall some facts on weight functions. (1.14) If ω and v are weights on R d such that (1.14) holds, then ω is also called v-moderate. We note that (1.14) implies that ω fulfills the estimates for some r > 0. We say that v is submultiplicative if v is even and (1.14) holds with ω = v. In the sequel, v and v j for j ≥ 0, always stand for submultiplicative weights if nothing else is stated.
In the following we define a broad family of mixed quasi-normed Lebesgue spaces. Here Q E denotes the closed parallelepiped (or the E-cube) which is spanned by the ordered basis E of R d . and We is finite. Here χ is the characteristic function of the set .

Modulation and Wiener Spaces
We consider the following broad family of modulation spaces which contains the classical modulation spaces, introduced by Feichtinger [10]. where ) is finite.
The theory of modulation spaces has developed in different ways since they were introduced in [10] by Feichtinger. (Cf. e.g. [11,15,18,35].) For example, let p, q, E and ω be the same as in Definition 1.10, and let L p,q , and different choices of φ give rise to equivalent quasi-norms in Definition 1.10. We also note that [15,35].)

Similar facts hold for the space
be an ordered basis, and let Q E be the closed parallelepiped spanned by E. Also let f and F be measurable on R d respective R 2d , and let is given by [12,15,18,27,28] when E is the standard basis. In particular, is related to so-called coorbit spaces. (See [9,[12][13][14]27,28].) Remark 1.12 Let p, q, ω 0 , ω, E, f and F be the same as in Definition 1.11. Evidently, by using the fact that ω 0 is v 0 -moderate for some v 0 , it follows that

Remark 1.13
For the spaces in Definition 1.11 we set W q 0 ,r 0 = W r , when and similarly for other types of exponents and for the spaces in Definition 1.10. (See also Remark 1.9.) We also set when E 1 , E 2 are ordered bases of R d and E = E 1 × E 2 , for spaces in Definition 1.10, since these spaces are independent of E 1 .
The following result is essential when characterizing elements in modulation spaces in terms of estimates of their Zak transforms. We omit the proof since the result is a consequence of [39, Proposition 2.6].
The next result is a restatement of Proposition 1.15 in [39]. The proof is therefore omitted. Here In particular,

Classes of Periodic Elements
Let s, σ ∈ R + be such that s We note that for any where c( f , α) are the Fourier coefficients given by For any σ ≥ 0 and basis be the set of all formal expansions in (1.17) and E 0 E (R d ) be the set of all formal expansions in (1.17) such that at most finite numbers of c( f , α) are non-zero (cf. [40]). We refer to [25,40] for more characterizations of E σ E , E σ ;0 E and their duals. The following definition takes care of spaces of formal expansions (1.17) with coefficients obeying specific quasi-norm estimates.
The next result is a reformulation of Proposition 3 in [39]. The proof is therefore omitted.

Proposition 1.17 Let E be an ordered basis of
Then

Remark 1.18
Let E 0 , q, r and ω be the same as in Proposition 1.17. Also with Fourier series expansion (1.17), Then it follows from Proposition 2.7 and Remark 2.8 in [39] that .(1.18)

The Zak Transform
For any ordered basis E of R d and f ∈ S (R d ), the Zak transform is defined by Several properties for the Zak transform can be found in [18]. For example, by the definition it follows that Z E is continuous from S (R d ) to the set of all smooth functions on R 2d which are bounded together with all their derivatives. It is also clear that Z E f is quasi-periodic of order E. Here, if F is a function or an ultra-distribution, then F is called quasi-periodic of order E, when (1.20) By interpreting (1.19) as a Fourier series in the ξ variable, we regain f (x) as the zero order Fourier coefficient, which is evaluated by For conveniency we set Z 1 = Z E when E is the standard basis of R d , and recall the following important mapping properties on L 2 (R d ).

Proposition 1.19
Let E be an ordered basis of R d . Then the operator Z E is homeomorphic from L 2 (R d ) to the set of all quasi-periodic elements of order E in L 2 loc (R 2d ), and Proof Let T E be as in Remark 1.4. By straight-forward computations it follows that The assertion now follows from (0.1), (1.22) and suitable changes of variables in the involved integrals. The details are left for the reader.

Zak Transform on Gelfand-Shilov Spaces, Lebesgue Spaces and Modulation Spaces
In this section we deduce characterizations of Lebesgue spaces, modulation spaces, and Gelfand-Shilov spaces and their distribution spaces in terms of suitable estimates of the Zak transforms of the involved elements. The characterizations on modulation spaces are related to results given in [32,33].

Spaces of Quasi-Periodic Functions and Distributions
Since quasi-periodic functions depend on the phase space variable (x, ξ) ∈ R 2d , it is suitable that the Gevrey regularity with respect to x ∈ R d for such functions might be different to the Gevrey regularity with respect to ξ ∈ R d . We therefore consider two parameters analogies of E σ and E σ ;0 , where the parameter σ is replaced by the pair σ, s. More preceisely, for any compact K ⊆ R 2d and s, is finite for some h > 0 (for every h > 0). The two parameter Gevrey classes, E σ,s (R d ) and E σ,s;0 (R d ), are the projective limits of E σ,s (K j ) respective E σ,s;0 (K j ), when {K j } j≥1 is an exhaustion by compact sets of R 2d . Furthermore we let be the set of all quasi-periodic elements in respectively, with respect to the ordered basis E on R d . For conveniency we also set E σ,s;0 when E is the standard basis of R d .
Next we introduce spaces of quasi-periodic functions and distributions which correspond to Lebesgue spaces and modulation spaces. We let L p Z ,E (R 2d ) for p ∈ (0, ∞] be the set of all quasi-periodic measurable functions F on R 2d with respect to the ordered basis E such that is finite. Evidently, we may identify L p Z ,E (R 2d ) by L p (Q E×E ), and the scalar product on L 2 Z ,E (R 2d ) is given by , ω ∈ P E (R 4d ) and ∈ 1 (R 2d ) \ 0 be fixed, and let E 0 be an ordered basis in R d . Then set and . Usually we assume that ω is given by for some ω 0 ∈ P E (R 2d ).

The Zak Transform on Test Function Spaces and Their Distribution Spaces
For the classical spaces S (R d ) and its distribution space S (R d ) we have the following.
Theorem 2.1 Let E be an ordered basis of R d . Then the following is true: The assertion (1) , respectively at each occurrence.
By the previous result and the facts that S σ s (R d ) is trivially equal to {0} when s + σ < 1, and σ s (R d ) is trivial when s + σ < 1 or s = σ = 1 2 , we get the following. In similar ways we may characterize Gelfand-Shilov distributions through their Zak transforms as in the following result.

Theorem 2.5 Let s, σ > 0 be such that s + σ ≥ 1 and E be an ordered basis of R d . Then the operator Z E from
, respectively at each occurrence.
We shall first prove Theorem 2.2 before proving Theorem 2.5. We need the following lemma for the proofs.

Lemma 2.6 is a straight-forward consequence of the inequality
due to the Taylor expansion of e t . The details are left for the reader.

Proof of Theorem 2.2
Let T E be the same as in (1.22). Then the map F(x, ξ) → F(T −1 E x, T * E ξ) maps quasi-periodc elements of order E to quasi-periodic elements with respect to the standard basis. Since f → f • T maps E-periodic elements to 1-periodic functions, it follows from these observations and (1.22) that it suffices to prove the result when E is the standard basis.
We only prove the result in the Roumieu case, i.e. we prove that Z E restricts to a homeomorphism from S σ . The other case follows by similar arguments and is left for the reader.
We shall follow the proof of Theorem 8.2.5 in [18]. In fact, assume first that f ∈ S σ s (R d ), x ∈ k 0 + Q d,1 for some fixed k 0 ∈ Z d , and let F = Z 1 f . Then it follows by straight-forward applications of Proposition 1.1 that for some positive constant C which only depends on the positive constants h and r .
In particular, C is independent of x, k 0 , k and α. (See also [ for some constant h 1 > 0. By Lemma 2.6 it follows that for some constant h > 0. A combination of these estimates gives and it follows that F ∈ E σ,s (R 2d ). This shows that Z 1 is continuous from S σ Next we show that any F in E σ,s Z (R 2d ) is the Zak transform of an element in S σ s (R d ). By Theorem 8.2.5 in [18] it follows that F = Z 1 f when By applying the operator k α ∂ β x and integrating by parts we get This gives which is the same as f ∈ S σ s (R d ). For the proof of Theorem 2.5 we need the following lemma on tensor products of Gelfand-Shilov distributions.

Lemma 2.7 Let s j
The same holds true with Lemma 2.7 is essentially a restatement of Theorem 2.4 in [38]. The proof is therefore omitted.

Remark 2.8 We notice that the uniqueness assertions in Lemma 2.7 is an immediate consequence of [38, Lemma 2.3] which asserts that if
), then f = 0 (as an element in ). Proof of Theorem 2.5 By similar arguments as in the proof of Theorem 2.2 we may assume that E is the standard basis for R d .

. Then
then Assume instead that f ∈ (S σ s ) (R d ) is arbitrary. We claim that the series on the right-hand side of (2.7) converges absolutely for every as above.
In fact, let φ ∈ S σ s (R d ) and giving that By (2.6) we obtain for some r 0 > 0 and c ≥ 1 that |e r |z| Hence, if r is chosen smaller than r 0 /(2c) and letting x = y = j, we obtain The absolutely convergence of the series of the right-hand side of (2.7) now follows from (2.8).
If f ∈ (S σ s ) (R d ), then Z 1 f is defined as the element in (S σ,s s,σ ) (R 2d ), given by the right-hand side of (2.7). The previous estimates show that this definition makes sense, and that the map it also follows that the continuous extension of Z 1 to such distributions is unique.
We need to prove that any element in Then g ϕ is 2π -periodic, and it follows from Remark 2.3 and Proposition 2.5 in [40] that if φ ∈ S s σ (R d )\0, then where the series converges in (S s σ ) (R d ), and By straight-forward computations we get and it is clear that the map which takes ϕ into the right-hand side in (2.9) defines a continuous linear form on S σ s (R d ). Hence , Furthermore, by the quasi-periodicity of F we obtain A combination of these facts now gives giving that

Proposition 2.9 Let s, σ > 1 and E be an ordered basis of R d . Then the following is true:
(1) The set of all quasi-periodic elements of order E in D (R 2d ) is equal to S Z ,E (R 2d ); (2) The set of all quasi-periodic elements of order E in (D σ,s The set of all quasi-periodic elements of order E in (D σ,s;0 ) We only prove (2). The other assertions follow by similar arguments and are left for the reader. Let

If
∈ S σ,s s,σ (R 2d ), then it follows by the quasi-periodicity of F and some straightforward computations that and that T χ in (2.11) is continuous from S σ,s s,σ (R 2d ) to C ∞ 0 (R 2d ) ∩ E σ,s (R 2d ). By letting F, be defined by the right-hand side of (2.10) when F ∈ (D σ,s ) (R 2d ) and ∈ S σ,s s,σ (R 2d ), it follows that → F, T χ in (2.10) defines a linear and continuous form on S σ,s s,σ (R 2d ) which agrees with the usual distribution action, → F, when ∈ C ∞ 0 (R 2d ) ∩ E σ,s (R 2d ). The mapping properties of Gelfand-Shilov distributions also lead to some quieries concerning the inversion formula (1.21) for the Zak transform. Evidently, if F is a general quasi-periodic distribution or even Gelfand-Shilov distribution, then the integral on the right-hand side of (1.21) might not be defined. On the other hand, since , and let F = Z E f . Then F(x, · ))(ξ, y) φ(−y)e i y,ξ dy dξ. (1.21)

The Zak Transform on Lebesgue and Modulation Spaces
For completeness we begin the subsection by making a review of the Zak transform when acting on Lebesgue spaces. Here we let

Proposition 2.11
Let E be an ordered basis of R d . Then the following is true: and Proposition 2.11 (1) and (2) are evidently true for p = 2, in view of Proposition 1.19, and is presented in [32, Lemma 3.1.2], without any proof in the case p = 1. In order to be self-contained, we give a proof in Appendix A. We also observe that by choosing p = 2 in Proposition 2.11 (1) and (2) we regain Proposition 1.19.
When investigation mapping properties of the Zak transform on modulation spaces, we need to deduce various kinds of estimates on short-time Fourier transforms and partial short-time Fourier transforms of Zak transforms. Especially we search suitable estimates on V (Z E f ), and on (ZV (x, · )))(ξ, y), (2.13) which are compositions of the Zak transform and the partial short-time Fourier transforms with respect to the first and second variable, respectively. From the previous subsection it is clear that there is a one-to-one correspondence between quasi-periodic functions and distributions, and Zak transforms of functions and distributions. For a quasi-periodic function or distribution F on R 2d which satisfies (1.20), and a suitable function or distribution on R 2d , we have (2.14) which follows by straight-forward computations. We remark that functions and distributions which satisfy conditions given in (2.14) are examples of so-called echoperiodic functions and distributions, considered in [37]. First we have the following result concerning identifying Lebesgue spaces via estimates of corresponding Zak transforms.

Theorem 2.12 Let E be an ordered basis of
, and let f be a Gelfand-Shilov distribution on R d . Then In particular, Proof We only prove the result for p < ∞. The case p = ∞ follows by similar arguments and is left for the reader. The distribution ξ → Z E f (x, ξ) is E -periodic, and it follows from (1.18) that The result now follows by applying the L p (Q E ) quasi-norm with respect to the xvariable.
In the same way we may identify modulation spaces by using the Zak transform as in the next result. Here recall Definition 1.

Theorem 2.13
Let E, E 0 be an ordered bases of R d , p, r ∈ (0, ∞] 2d , ω 0 ∈ P E (R 2d ) and ω ∈ P E (R 4d ) be such that (2.5) holds. Then Z E from 1 By straight-forward computations we get  (2.19)), and the (partial) short-time Fourier transform of that distribution equals V (Z 1 f ), it follows from (1.18) that Let r 0 ≤ min( p). If we apply the L p 2 E 0 norm on (2.20) with respect to the η variable and using Hölder's inequality we get (2.21) If then the fact that r 0 ≤ min( p) and Jensen's inequality give g 1 g 0 . By applying the L r 0 E (Q E ) norm on the latter inequality, using the fact that and Jensen's inequality again we obtain .
which is the same as (2.22) in view of Proposition 1.15. In order to estimate h 0,r 0 we apply (2.14) to get By first applying the L p 1 E (R d ) norm with respect to the y variable and then the L norm with respect to the η variable we get Hence, by applying the L r 0 E (Q E ) norm on h 0,r 0 and using Hölder's and Jensen's inequalities we get A combination of (2.21)-(2.23) and Proposition 1.15 now gives In order to get the reversed estimate we again apply the L p 2 E 0 norm on (2.20) with respect to the η variable and use Hölder's inequality to get then Jensen's inequality give g 0 g 2 . By applying the L ∞ E (Q E ) norm on the latter inequality and using Jensen's inequality again we obtain That is, which is the same as (2.26) in view of Proposition 1.15. By applying the L ∞ E (Q E ) norm on h 0,∞ and using (2.23) we get , (2.27) where the last relation follows from Proposition 1.15. A combination of (2.25), (2.26) and (2.27) now gives (2.28) and the result in the case r = ∞ follows by combining (2.28) with (2.24). For general r ∈ (0, ∞] 2d , we notice that the echo-periodicity (2.14) implies that H 2,F,ω,E,E 0 , p in (2.3) is E × E periodic. The general case now follows from Proposition 1.15, the previous observation and that we have already proved the result for r = ∞. The details are left for the reader.
A consequence of the previous result is that W r, p Z ,E,E 0 ,(ω) (R 2d ) is independent of r (also in topological sense), and for this reason we set As a special case of the previous result we have the following.

Duality Properties and Some Further Characterizations of Quasi-Peridic Elements
In this section we discuss various aspects concerning duality and characterizations for quasi-periodic elements, as well as

Characterizations of Quasi-Periodic Elements via Estimates on Their Short-Time Fourier Transform
The following results are analogous to Propositions 2.7 and 2.8 in [40] concerning characterizations of periodic elements in Gelfand-Shilov distribution spaces, and to Proposition 1.2.
(2) for some r > 0 (for every r > 0), it holds (3.1) (3) for some r > 0 (for every r > 0), it holds (3) for every r > 0 (for some r > 0), it holds We only prove Proposition 3.2, and then only in the Roumieu case. The Beurling case of Proposition 3.2 as well as Proposition 3.1 follow by similar arguments and are left for the reader.

Duality Properties of Gevrey Type Quasi-Period Elements
We shall next use the previous characterizations in Propositions 3.1 and 3.2 to show that the form (2.1) can be written as Here φ ∈ S σ,s s,σ (R 2d )\0 is fixed and when E is an ordered basis of R d . We use this identity to extend the definition of (F, G) Z ,E to permit We also show that the dual of E σ,s Hence the right-hand side of (3.5) makes sense and we may evaluate the integrals with respect to x, ξ, η, y in any order. It also follows that the map (F, G) → (F, G) Z ,E defines a continuous map from · ) Z ,E be given by (3.5). Then the following is true: The same holds true with E σ,s;0 Z ,E and σ s in place of E σ,s Z ,E respective S σ s at each occurrence.
Then it follows by straight-forward computations that where X = (x, η, y) ∈ Q E × R 2d , and the Fourier coefficients c X ( f , j) and c X (g, j) are given by c X ( f , j) = (V ψ y− j f )(x − j, η)e −i j,η (3.8) and c X (g, j) = (V ψ y− j g)(x − j, η)e −i j,η . (3.9) Since the short-time Fourier transforms V φ (Z E f ) and V φ (Z E g) are smooth, it follows that ξ → F X (ξ ) and ξ → G X (ξ ) are smooth periodic functions for every X . Hence the Fourier coefficients in (3.8) and (3.9) satisfy |c X ( f , j)| j −N and |c X (g, j)| j −N (3. 10) for every N ≥ 0, when X ∈ Q E × R 2d is fixed. By integrating Hence, integrating with respect to x, η, y, using Moyal's identity and Remark 3.3, we obtain (V ψ y f )(x, η)(V ψ y g)(x, η) dxdη dy which gives (2). Here we observe that the estimates in Proposition 1.2 imply that the involved expressions in (3.11) possess suitable L 1 properties, which allow us to swap the orders of summations and integrations. This gives the result.

Transitions of Operators Under the Zak Transform
In this section we show how linear operators are transformed by the Zak transform into corresponding operators acting on quasi-periodic functions or distributions. We also present a condition on linear operators which is both necessary and sufficient in order for these operators should map quasi-periodic elements into quasi-periodic elements.

This gives
This shows that Z E • T = T Z • Z E . The continuity assertions of T Z now follows from the latter identity and Theorem 2.5.
We need the following lemma for the proof of Theorem 4.2. (1) K ( · + (z, z)) = K for every z ∈ R d ; (2) there is a unique K 0 ∈ (S σ s ) (R d ) such that K (x, y) = K 0 (x − y). The same hold true with ( σ s ) , S or D in place of (S σ s ) at each occurrence.
Lemma 4.3 is at least implicitly available in the literature, e.g. in [21]. In order to be self-contained we give a proof in Appendix A.
Proof of Theorem 4.2 Again we only prove the result when the involved spaces are given by E σ,s Z ,E or (E σ,s Z ,E ) . The other cases follow by similar arguments and are left for the reader.