Discrete Decomposition of Mixed-Norm α-modulation spaces

2023/03/28 by Morten Nielsen, Nielsen, Morten
Mathematics · #41A17 #42B35 #42C15 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2303.15791

openalex publication_date 2023/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a class of almost diagonal matrices compatible with the mixed-norm α-modulation spaces M_p,q^s,α(ℝⁿ), α∈ [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 α-modulation spaces. We show 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.

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