Behavior of Gabor frame operators on Wiener amalgam spaces

TL;DR

This paper proves that Gabor frame operators converge to the identity in both operator norm and weak* sense on Wiener amalgam spaces as sampling density increases, extending known results and validating key representations.

Contribution

It establishes the convergence of Gabor expansions in operator norm on Wiener amalgam spaces and confirms Janssen's and Wexler-Raz conditions in this context.

Findings

Convergence of Gabor expansions in operator norm on Wiener amalgam spaces.

Validation of Janssen's representation for Gabor frame operators.

Confirmation of Wexler-Raz biorthogonality condition on these spaces.

Abstract

It is well known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on $W(L^p, \ell^q)$ as the sampling density tends to infinity. Also we show the validity of the Janssen's representation and the Wexler-Raz biorthogonality condition for Gabor frame operator on $W(L^p, \ell^q)$.