Radial Time-Frequency Analysis and Embeddings of Radial Modulation\n Spaces

2005/02/10 by Holger Rauhut, Rauhut, Holger
Mathematics · #41A46 #42C40 #46E35 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.math/0502217

openalex publication_date 2005/02/10 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

In this paper we construct frames of Gabor type for the space\nL²_rad( R^d) of radial L²-functions, and more generally, for subspaces\nof modulation spaces consisting of radial distributions. Hereby, each frame\nelement itself is a radial function. This construction is based on a\ngeneralization of the so called Feichtinger-Gr "ochenig theory -- sometimes\nalso called coorbit space theory -- which was developed in an earlier article.\nWe show that this new type of Gabor frames behaves better in linear and\nnon-linear approximation in a certain sense than usual Gabor frames when\napproximating a radial function. Moreover, we derive new embedding theorems for\ncoorbit spaces restricted to invariant vectors (functions) and apply them to\nmodulation spaces of radial distributions. As a special case this result\nimplies that the Feichtinger algebra (S₀)_rad( R^d) = M¹_rad( R^d)\nrestricted to radial functions is embedded into the Sobolev space\nH^(d-1)/2_rad( R^d). Moreover, for d\≥ 2 the embedding\n(S₀)_rad( R^d) hookrightarrow L²_rad( R^d) is compact.\n

Related

• Gabor frame characterisations of generalised modulation spaces
• Time-Frequency Shift Invariance of Gabor Spaces with an S₀-Generator
• Time-Frequency Partitions and Characterizations of Modulation Spaces with Localization Opertors
• Coorbit theory of warped time-frequency systems in ℝ^d
• The density theorem of a class of dilation-and-modulation systems on the half real line