Spherical spectral synthesis

2016/07/24 by László Székelyhidi, Székelyhidi, László · 1 citation
Mathematics · Physics and Astronomy · #22D15 #43A45 #43A90 #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Relativity and Gravitational Theory #math.FA #msc:22D15 #msc:43A45 #msc:43A90

paper · pdf · doi:10.48550/arxiv.1607.07079

22 pages

arxiv created 2016/07/24 · openalex publication_date 2016/07/24 · arxiv updated 2016/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we make an attempt to extend L. Schwartz's classical result on spectral synthesis to several dimensions. Due to counterexamples of D. I. Gurevich this is impossible for translation invariant varieties. Our idea is to replace translations by proper euclidean motions in higher dimensions. For this purpose we introduce the basic concepts of spectral analysis and synthesis in the non-commutative setting based on Gelfand pairs, where "translation invariance" will be replaced by invariance with respect to a compact group of automorphisms. The role of exponential functions will be played by spherical functions. As an application we obtain the extension of L. Schwartz's fundamental result.

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• The Pompeiu problem and spherical spectral analysis

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