Functional Matrix Multipliers for Parseval Gabor Multi-frame Generators

Abstract

We consider the problem of characterizing the bounded linear operator multipliers on $L^{2}(\mathbb{R})$ that map Gabor frame generators to Gabor frame generators. We prove that a functional matrix $M(t)=[f_{ij}(t)]_{m \times m}$ (where $f_{ij}\in L^{\infty}(\mathbb{R})$) is a multiplier for Parseval Gabor multi-frame generators with parameters $a, b >0$ if and only if $M(t)$ is unitary and $M^{*}(t)M(t+\frac{1}{b})= \lambda(t)I$ for some unimodular $a$-periodic function $\lambda(t)$. As a special case ($m =1$) this recovers the characterization of functional multipliers for Parseval Gabor frames with single function generators.