Convenient descriptions of weight functions in time-frequency analysis

TL;DR

This paper characterizes submultiplicative weights satisfying the GRS-condition through exponential bounds and explores their implications for identifying relationships among weighted Lebesgue, modulation, and Gelfand-Shilov spaces.

Contribution

It provides a new characterization of the GRS-condition for weights and applies this to establish space identification properties in time-frequency analysis.

Findings

GRS-condition equivalent to exponential boundedness of weights

Identification of weighted Lebesgue and modulation spaces

Connections between modulation and Gelfand-Shilov spaces

Abstract

Let $v$ be a submultiplicative weight. Then we prove that $v$ satisfies GRS-condition, if and only if $v\cdot e^{-\ep |\cdo |}$ is bounded for every positive $\ep$. We use this equivalence to establish identification properties between weighted Lebesgue spaces, and between certain modulation spaces and Gelfand-Shilov spaces.