Discrete decomposition of mixed-norm α-modulation spaces

Abstract

We study a class of almost diagonal matrices compatible with the mixed-norm [Formula: see text]-modulation spaces [Formula: see text], [Formula: see text], introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm [Formula: see text]-modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm [Formula: see text]-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm [Formula: see text]-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm [Formula: see text]-modulation spaces.