Gabor frames and asymptotic behavior of Schwartz distributions

Journal of the London Mathematical Society, 1993

Wiener Tauberian theorems are proved for integral transforms of Schwartz distributions in which the kernel of the transform belongs to a suitable testing function space.

Constructive Approximation, 2019

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions. Keywords Shift-invariant space • Sampling with derivatives • Gaussian • Totally positive function • Jensen's formula • Gabor frame • Beurling density Mathematics Subject Classification 42C15 • 42C40 • 94A20 José Luis Romero gratefully acknowledges support from the Austrian Science Fund (FWF): P 29462-N35, and from the WWTF grant INSIGHT (MA16-053).

Acta Mathematica Academiae Scientiarum Hungaricae, 1982

Functional Analysis and Its Applications, 2019

We give an Abelian type result relating the quasiasymptotic boundedness of tempered distributions to the asymptotics of their directional short-time Fourier transform (DSTFT). We also prove several Abelian-Tauberian results characterizing the quasiasymptotic behavior of distributions in S (R n) in terms of their DSTFT with fixed direction.

Journal of Mathematical Sciences, 1999

LetD′+ be the space of Schwartz distributions with support on the closed positive half-line [0, +∞). We give a generalization of the Paley-Wiener theorem to the case of the distributions inD′+.

Journal of Functional Analysis, 1997

This paper lays the foundation for a quantitative theory of Gabor expansions f (x)= k, n c k, n e 2?in:x g(x&k;). In analogy to wavelet expansions of Besov Triebel Lizorkin spaces, we show that the correct class of spaces which can be characterized by the magnitude of the coefficients c k, n is the class of modulation spaces. To analyze the behavior of the coefficients, it is necessary to invert the Gabor frame operator on these spaces. We show that the frame operator is invertible on modulation spaces if and only if it is invertible on L 2 and the atom g is in a suitable space of test functions. A similar statement for wavelet theory is false. The second part is devoted to Gabor analysis on general time frequency lattices.

Proceedings of the Edinburgh …, 2010

In this article we show that if the summability means in the Fourier inversion formula for a tempered distribution f ∈ S (R n ) converge to zero pointwise in an open set Ω, and if those means are locally bounded in L 1 (Ω) , then Ω ⊂ R n \ supp f. We prove this for several summability procedures, in particular for Abel summability, Cesàro summability, and Gauss-Weierstrass summability.

Monatshefte f�r Mathematik, 1998

The Fourier transform for Schwartz spaces on Ch6bli-Trim~che hypergroups is studied, leading to results on approximation to the identity for functions and distributions on the half-line. In particular it is shown that the heat and Poisson kernels on the half-line form approximate units in various function spaces. A characterization of the convolution of a tempered distribution and a Schwartz function is also given.

Inventiones mathematicae

We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors asĝ(ξ) = n j=1 (1 + 2πiδ j ξ) −1 e −cξ 2 for δ 1 ,. .. , δ n ∈ R, c > 0 (in which case g is called totally positive of Gaussian type). In analogy to Beurling's sampling theorem for the Paley-Wiener space of entire functions, we prove that every separated set with lower Beurling density > 1 is a sampling set for the shift-invariant space generated by such a g. In view of the known necessary density conditions, this result is optimal and validates the heuristic reasonings in the engineering literature. Using a K. G. was supported in part by the Project P26273-N25 of the Austrian Science Fund (FWF).

arXiv (Cornell University), 2024