Wilson bases and ultradistributions

We give a characterization of Gelfand-Shilov type spaces of test functions and their dual spaces of tempered ultradistributions by the means of Wilson bases of exponential decay. We offer two different proofs, and extend known results to the Roumieu case.


Introduction
Wilson bases were constructed by I. Daubechies, S. Jaffard, and J. Journé in [8] to overcome constraints arising from the Balian-Low theorem, and soonafter shown to be unconditional bases for modulation spaces, see [12]. By combining the Wilson bases and tools from timefrequency analysis, approximate diagonalization of different classes of pseudodifferential operators is obtained in [23,24]. Wilson bases of Meyer type were used in the study of gravitational waves, cf. [3,20]. We refer to [2] for recent construction of orthonormal Wilson bases in multidimensional case which overcomes a deficiency of the tensor product construction used in e.g. [24,25].
Gelfand-Shilov spaces were initially introduced for the analysis of solutions of certain parabolic initial-value problems [13], and thereafter applied in different contexts when precise estimates of global decay and regularity are needed, see [14] for an overview. Recently, Hermite expansions of Fourier transform invariant Gelfand-Shilov spaces, and more generally Pilipović spaces, are considered in [31], see also [19,21].
In this paper we give a description of Gelfand-Shilov spaces and their dual spaces of tempered ultradistirbutions in terms of Wilson bases. This extends some results from [22] given for Beurling type Gelfand-Shilov spaces. Both Wilson bases and Hermite functions are orthonormal bases for L 2 (R d ) consisting of functions which are well localized in phase-space (time-frequency plane). From such perspective our results are expected. However, due to the specific structure of Wilson bases, the proofs are based on entirely different arguments then those related to the Hermite basis which utilize recursive relation between Hermite functions and the fact that they are eigenfunctions of the harmonic oscillator.
Instead, we apply the powerful general theory of coorbit spaces, [10,11]. The key auxiliary result is the fact that Wilson bases are unconditional bases for coorbit spaces, [12]. We modify and simplify the approach from [22] related to Beurling case, and provide detailed proofs since the more involved Roumieu case contains nontrivial modifications of arguments given there. As a consequence of our results, we recover the well known relation between Gelfand-Shilov spaces and modulation spaces. Furthermore, if that relation is taken as granted, we give an alternative proof of our main results without an explicit reference to coorbit spaces.
Since both proofs are essentially based on the exponential decay of elements of Wilson bases and asymptotic behavior of the STFT, the techniques from the present paper can be modified to include other time-frequency representations and also more general (for example anisotropic) spaces of test functions and their distribution spaces. Such investigations are out of scope of the present paper and will be the subject of our future work.
We end the introduction by recalling basic notation which will be used in the sequel.
Notation. Operators of translations and modulations of a given function f are respectively given by T x f (·) = f (· − x) and M y f (·) = e 2πiy· f (·), x, y ∈ R d . The notation A ֒→ B means that the topological spaces A and B satisfy A ⊆ B with continuous embeddings. We write A(θ) B(θ), θ ∈ Ω, if there is a constant c > 0 such that A(θ) ≤ cB(θ) for all θ ∈ Ω.
The scalar product in L 2 (R d ) is given by and · 2 = ·, · . The Fourier transform of an integrable function f is given bŷ It extends uniquely to a unitary operator on L 2 (R d ).
Let φ ∈ L 2 (R d ) be fixed. Then the short-time Fourier transform The following fundamental identity of timefrequency analysis ( [6,15]) is often used: By Σ 1 (R d ) we denote the Gelfand-Shilov space of smooth functions given by: , so by (1.1) it follows that the STFT can be extended to Σ ′ 1 (R d ), and restricted to Σ 1 (R d ).

Preliminaries
In this section we recall the Wilson bases, weight functions, coorbit spaces and Gelfand-Shilov type spaces. We also prove some auxiliary results (Lemmas 2.2 and 2.3 and Theorem 2.5) which will be used in Sections 3 and 4.
2.1. Wilson bases. Following the idea of K. Wilson [33], Daubechies, Jaffard and Journe constructed a real-valued function ψ such that for some constants a, b, C > 0, and obtain an orthonormal basis (ONB) {ψ l,n } l∈N 0 ,n∈Z , of L 2 (R), where see [8]. From (2.1) it follows that 3) for some constants a, b, C > 0 depending on l and n, and {ψ l,n } is therefore called the Wilson basis of exponential decay.
3 Moreover, following Gröchenig [15], we may rewrite (2.2) as ψ 0,n (x) = T n ψ(x), and To obtain an orthonormal basis of L 2 (R d ), Tachizawa in [24] considered d−dimensional Wilson basis given by the tensor product 4) for some constants a, b, C > 0, depending on l and n.
The tensor product Wilson bases are 2 d −modular, i.e. their elements have 2 d peaks in frequency, which may have undesirable consequences in applications, see [2] for details. That motivated, Bownik et al. [2] to construct a family of orthonormal Wilson bases with 2 k −modular covering of the frequency domain with k = 1, . . . , d. The tensor product Wilson bases turned out to be the special case of their construction.

Weight functions.
for some constant C ≥ 1. If ω and v are weights on R d such that (2.5) holds, then ω is also called v-moderate. If v can be chosen as polynomial, then ω is called a weight of polynomial type. The set of all moderate weights on R d is denoted by P E (R d ).
The weight v on R d is called submultiplicative, if it is even and (2.5) holds for ω = v. From now on, v always denotes a submultiplicative weight if nothing else is stated. In particular, if (2.5) holds and v is submultiplicative, then it follows by straight-forward computations that If ω is a moderate weight on R d , then there is a submultiplicative weight v on R d such that (2.5) and (2.6) hold, see [16,30]. Moreover if v is submultiplicative on R d , then for some constant r > 0 (cf. [16]). In particular, if ω is moderate, then for some r > 0. We will consider only weight functions w satisfying Beurling-Domar's non-quasianalyticity condition The most important examples of weight functions which satisfy (2.9) For the purpose of this paper we focus our attention to subexponential weights of the form ω h,s (·) = e h|·| 1/s , s > 1, h ≥ 0.
. This terminology (and notation) is justified by the general theory of coorbit spaces developed in [10,11], see also [7] for a more recent survey.
From the results given there, it follows that CoY h,s (R d ) is a Banach space invariant under translations, modulations, and complex conjuga- We will use the following simple results.
3) and the change of variables we have: Thusf ∈ CoY h,s (R d ) if and only if (2.12) holds true.
We write Proof. We again choose φ(x) = e −πx 2 in the definition of CoY h,s (R d ) and follow the idea of the proof of [15, Proposition 11.3.1]. By (1.1) and the Plancherel theorem we formally have Since e 2h|x| 1/s is a moderate weight, it follows that 6 and a) follows. Part b) follows from a) and the arguments given in the proof of Lemma 2.2.
From the general theory of coorbit spaces it follows that Wilson bases of exponential decay are unconditional bases for CoY h,s (R d ) and F CoY h,s (R d ), [12,22]. The precise statement is the following. Proof. The proof is omitted since it follows by the arguments given in the proof of Theorem 4 in [12], where polynomial type weights are considered instead. The subexponential type weights considered here are treated in [22], see Theorem 4.4 and Remark 4.5 given there.
Theorem 2.4 is the main auxiliary result which will be used to prove representation theorem for Gelfand-Shilov spaces, Theorem 3.1 a).

2.4.
Gelfand-Shilov spaces. Gelfand and Shilov introduced the spaces of type S, for the analysis of solutions of certain parabolic initialvalue problems. A comprehensive study of those spaces which are afterwards called Gelfand-Shilov spaces is given in [13]. We focus our attention to the case when regularity and decay are controlled by the so called Gevrey sequences M p = p! s , when s > 0, and refer to e.g. [29] for an overview of a more general situation.
Let 0 < s be fixed. Then the (Fourier invariant) Gelfand-Shilov space is finite for some h > 0 (for every h > 0). The semi-norms · S s,h induce inductive limit topology for the space S s (R d ), and projective limit topology for Σ s (R d ). Thus the former space becomes an LS space, while the latter space is an FS space (Fréchet-Schwartz space) under these topologies.
, if and only if s ≥ 1 2 (s > 1 2 ). The Gelfand-Shilov distribution spaces S ′ s (R d ) and Σ ′ s (R d ) (also known as spaces of tempered ultradistributions) are the dual spaces of S s (R d ) and Σ s (R d ), respectively.
We have (2.18) The Fourier transform F extends uniquely to homeomorphisms on S ′ s (R d ) and on Σ ′ s (R d ). Furthermore, F restricts to homeomorphisms on S s (R d ) and on Σ s (R d ). Similar facts hold true when the Fourier transform is replaced by a partial Fourier transform.
Fourier transform invariance of S s (R d ) and Σ s (R d ) follows from the following result which also gives a characterization of Gelfand-Shilov spaces in terms of coorbit spaces. c) there exists h > 0 (for every h > 0) such that Proof. a) ⇔ b) is well known, and holds for all s > 0, [4,17,18]. b) ⇔ c Note that, since the equivalence between a) and b) in Theorem 2.5 holds even if s = 1, if ψ satisfies (2.1), then the Wilson basis elements ψ l,n , l ∈ N 0 , n ∈ Z, given by (2.2) belong to S 1 (R d ), cf. (2.3).
The restriction s > 1 when proving b) ⇔ c) in Theorem 2.5 comes from Definition 2.1. In fact, in the general theory of coorbit spaces, as presented in [10,11], an important role is played by BUPUs (bounded uniform partitions of unity) consisting of compactly supported smooth functions. In such setting, coorbit spaces consist of non-quasianalytic functions.
From definitions of CoY h,s (R d ), F CoY h,s (R d ) and Theorem 2.5 c) it follows that Gelfand-Shilov spaces are essentially characterized by the decay estimates of the short-time Fourier transform. Note that the estimates given in Proposition 2.6 below employ the sup-norm (L ∞norm) whereas in Theorem 2.5 c) the L 2 -norm related to CoY h,s (R d ) and F CoY h,s (R d ) is considered instead. (In fact, any L p −norm (1 ≤ p ≤ ∞) can be used, see [21].) ) and let f be a Gelfand-Shilov distribution on R d . Then the following is true:
We omit the proof since the first part follows from [ , and restricts to a continuous map from S s (R d ) × S s (R d ) to S s (R 2d ). The same conclusion holds with Σ s in place of S s , at each place. Therefore, Definition 2.1 can be appropriately modified to include ultradistributions f ∈ S ′ s (R d ) (or f ∈ Σ ′ s (R d ), and we will use such extension from now on.

Main results
In this section we discuss Wilson bases expansions in the context of Gelfand-Shilov spaces and their dual spaces of tempered ultradistributions.
Theorem 3.1. Let s > 1 and let there be given a Wilson basis of exponential decay {ψ l,n } l∈N 0 ,n∈Z .

2)
for some (for all) k ≥ 0, then there exists a function f ∈ S s (R d ) (f ∈ Σ s (R d )) such that (3.1) holds with c l,n = f, ψ l,n , l ∈ N 0 , n ∈ Z.
Proof. a) We prove the Roumieu case, since the Beurling case is given in [22,   To prove b), we note that (3.2) obviously implies l∈N 0 ,n∈Z |c l,n | 2 e 2k|n/2| 1/s < ∞, and l∈N 0 ,n∈Z |c l,n | 2 e 2k|l| 1/s < ∞, and since the Wilson basis is an ONB we have that c l,n = f, ψ l,n , l ∈ N 0 , n ∈ Z. Now, by Theorem 2.5 we conclude that f ∈ S s (R d ), and the proof is finished.
For the proof of Theorem 3.3 we need a simple lemma on divergent series. We note that a similar argument is used in the proof of [34, Theorem 9.6-1]. To be self-contained we provide the proof in Appendix.
Lemma 3.2. Let (a n ) n∈N 0 be a zero convergent sequence of non-negative numbers such that n∈N 0 a n = +∞.
Then there exists an increasing sequence of integers m l , l ∈ N, such that 1 < m l −1 n=m l−1 a n < 3. a

3)
for every (for some) h ≥ 0, where c l,n = f, ψ l,n , l ∈ N 0 , n ∈ Z. b) Conversely, if (3.3) holds for some sequence (c l,n ) l∈N 0 ,n∈Z and for every (for some) h ≥ 0, then there exists Proof. The Beurling case can be proved by making appropriate changes if the proof of [34, Theorem 9.6-1], cf. [26]. However, since the proof for the Roumieu case contains nontrivial modifications of Zemanian's proof, we provide it here. b) Let (3.3) holds for some h ≥ 0, and let f = l∈N 0 ,n∈Z c l,n ψ l,n .
If φ ∈ S s (R d ) then we have Since φ ∈ S s (R d ), by Theorem 3.1 a) it follows that we can choose h ≥ 0 such that and for such choice of h ≥ 0, by (3.3) it follows that l∈N 0 ,n∈Z |c l,n | 2 e −2h(| n 2 |+|l|) 1/s < ∞, and we conclude that | f, φ | < ∞, so that f ∈ S ′ s (R d ), and b) is proved.
Next we prove that the sequence (e −k(| n 2 |+|l|) 1/s c l,n ) l∈N 0 ,n∈Z is bounded for every k > 0.
We give the proof by contradiction: suppose that there exists k 0 > 0 such that the sequence (e −k 0 (| n 2 |+|l|) 1/s c l,n ) l∈N0,n∈Z is unbounded. Then there exists a sequence of increasing (by components) indices (l m , n m ) m∈N such that e −k 0 (|lm|+| nm 2 |) 1/s |c lm,nm | ≥ m m ∈ N. Next we consider the sequence (a l,n ) with the following properties (1) a l,n c l,n = |a l,n c l,n |, This gives l∈N 0 ,n∈Z By Theorem 3.1 b) it follows that φ = l∈N 0 ,n∈Z a l,n ψ l,n ∈ S s (R d ), so that l∈N 0 ,n∈Z a l,n c l,n < ∞. On the other hand, which gives the contradiction. Thus, we conclude that the sequence (e −k(| n 2 |+|l|) 1/s c l,n ) l∈N 0 ,n∈Z is bounded for every k > 0.
Finally, we prove that (3.3) holds for every h > 0. Again we give the proof by contradiction. Suppose that there exists h 0 > 0 such that Since (e −h 0 (| n 2 |+|l|) 1/s c l,n ) l∈N 0 ,n∈Z is bounded, it follows that (b l,n ) is a zero convergent sequence. By Lemma 3.2 it follows that there is an increasing sequence of indices (l m , n m ) m∈N such that for every m ∈ N, where we used (3.5). Thus, By Theorem 3.1 b) it follows that l∈N 0 ,n∈Z a l,n ψ l,n ∈ S s (R d ), and therefore l∈N 0 ,n∈Z a l,n c l,n < ∞. This is a contradiction with (3.6). We conclude that the assumption (3.4) can not hold. Therefore l∈N 0 ,n∈Z |c l,n | 2 e −2h(| n 2 |+|l|) 1/s < ∞ for every h > 0 which completes the proof. 14

Alternative proof via modulation spaces
Modulation spaces, originally introduced by Feichtinger in [9], are recognized as appropriate family of spaces when dealing with problems of time-frequency analysis, see [1,6,9,15], to mention just a few references. A broader family of modulation spaces, including quasi-Banach spaces when the Lebesgue parameters p, q belong to (0, 1) is studied in e.g. [32].
The next theorem is analogous to Theorem 2.4. It follows from [15, Chapter 12.3] so we omit the proof.
Theorem 4.1. Let p, q ∈ [1, ∞], ω ∈ P E (R 2d ), and let there be given a Wilson basis of exponential decay {ψ l,n } l∈N 0 ,n∈Z . Then the Banach spaces M p,q ω (R d ) and l p,q ω are isomorphic. An explicit isomorphism is provided by the coefficient operator C ψ : M p,q ω (R d ) → l p,q ω given by C ψ f = ( f, ψ l,n ) (l,n)∈N 0 ×Z . with the unconditional convergence in M p,q ω (R d ) if 1 ≤ p, q < ∞, and weak * convergence in M ∞ 1/v (R d ) otherwise.