Discrete Decomposition of Mixed-Norm α-Modulation Spaces

We study a class of almost diagonal matrices compatible with the mixed-norm α-modulation spaces Mp,qs,α(ℝn), α [0, 1], 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 α-modulation spaces. It is shown that the "change of frame"matrix for a pair of time-frequency frames for the mixed-norm α-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 α-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm α-modulation spaces.