Abstract
We provide a characterization of the Gelfand–Shilov-type spaces of test functions and their dual spaces of tempered ultradistributions by means of Wilson bases of exponential decay. We offer two different proofs and extend known results to the Roumieu case.
Highlights
IntroductionWe give a description of Gelfand–Shilov spaces and their dual spaces of tempered ultradistributions in terms of Wilson bases
Since both proofs are essentially based on the exponential decay of elements of Wilson bases and asymptotic behavior of the short-time Fourier transform (STFT), the techniques from the present paper can be modified to include other time–frequency representations and more general spaces of test functions and their distribution spaces
From the general theory of coorbit spaces it follows that Wilson bases of exponential decay are unconditional bases for CoYh,s(Rd) and FCoYh,s(Rd), [2,14]
Summary
We give a description of Gelfand–Shilov spaces and their dual spaces of tempered ultradistributions in terms of Wilson bases This extends some results from [14] given for Beurling type Gelfand–Shilov spaces. Both Wilson bases and Hermite functions are orthonormal bases for L2(Rd) consisting of functions that are well localized in phase– space (time–frequency plane). If that relation is taken for 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 more general (for example anisotropic) spaces of test functions and their distribution spaces. If φ ∈ Σ1(Rd), Mξ Txφ ∈ Σ1(Rd), so by (1) it follows that the STFT can be extended to Σ1(Rd), and restricted to Σ1(Rd)
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