Discrete Decomposition of Mixed-Norm $α$-modulation spaces

We study a class of almost diagonal matrices compatible with the mixed-norm $\alpha$-modulation spaces $M_{\vec{p},q}^{s,\alpha}(\mathbb{R}^n)$, $\alpha\in [0,1]$, introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. 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 $\alpha$-modulation spaces. We show that the "change of frame" matrix for a pair of time-frequency frames for the mixed-norm $\alpha$-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 $\alpha$-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm $\alpha$-modulation spaces.