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Irregular sampling for hyperbolic secant type functions

Anton Baranov, Yurii Belov

Abstract

We study Gabor frames in the case when the window function is of hyperbolic secant type, i.e., $g(x) = (e^{ax}+e^{-bx})^{-1}$, ${\rm Re}\,a, {\rm Re}\,b>0$. A criterion for half-irregular sampling is obtained: for a separated $Λ\subset\mathbb{R}$ the Gabor system $\mathcal{G}(g, Λ\times α\Z)$ is a frame in $L^2(\R)$ if and only if $D^-(Λ) >α$ where $D^-(Λ)$ is the usual (Beurling) lower density of $Λ$. This extends a result by Gröchenig, Romero, and Stöckler which applies to the case of a standard hyperbolic secant. Also, a full description of complete interpolating sequences for the shift-invariant space generated by $g$ is given.

Irregular sampling for hyperbolic secant type functions

Abstract

We study Gabor frames in the case when the window function is of hyperbolic secant type, i.e., , . A criterion for half-irregular sampling is obtained: for a separated the Gabor system is a frame in if and only if where is the usual (Beurling) lower density of . This extends a result by Gröchenig, Romero, and Stöckler which applies to the case of a standard hyperbolic secant. Also, a full description of complete interpolating sequences for the shift-invariant space generated by is given.
Paper Structure (8 sections, 13 theorems, 51 equations)

This paper contains 8 sections, 13 theorems, 51 equations.

Table of Contents

  1. Introduction and main results
  2. Relation between $V^2(g)$ and a space of Cauchy transforms
  3. Small Fock type spaces
  4. Estimate of the norm of a reproducing kernel
  5. Complete interpolating sequences in $\mathcal{F}_{\beta, 0}$
  6. Gaussian invariant spaces and Fock spaces
  7. The space $\mathcal{H}_{a,b}$ coincide with $\mathcal{F}_{\beta, \gamma}$
  8. Proof of Theorem \ref{['main1']}

Key Result

Theorem A

Let $S \subset \mathbb{R}^2$ be a separated set and let $g(x) = e^{-\pi x^2}$. Then $\mathcal{G}(g, S)$ is a frame for $L^2(\mathbb{R})$ if and only if

Theorems & Definitions (22)

  • Theorem A
  • Theorem B
  • Theorem 1.1
  • Theorem C
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Theorem D
  • proof
  • ...and 12 more