Functional matrix multipliers for Parseval Gabor multi-frame generators (Q2422321)

The authors characterize bounded linear operator multipliers on $L^2(\mathbb R)$ that map Gabor frame generators to Gabor frame generators. They prove that a functional matrix $M(t) = (f_{ij} (t))_{ m\times m}$, with $f_{ij} \in L^\infty (\mathbb R), \ i, j = 1, \dots, m$, is a multiplier for Parseval Gabor multi-frame generators with parameters $a, b > 0$ if and only if $M(t)$ is unitary, $M^* (t) M(t + \frac{1}{b} )$ is $a$-periodic, and $M^* (t) M(t + \frac{1}{b} ) = \lambda(t) I$ for some unimodular function $\lambda(t)$. Finally, the authors obtain a characterization of functional multipliers for Parseval Gabor frames with single function generators as a special case.