Spherical spectral synthesis

TL;DR

This paper extends Schwartz's spectral synthesis results to higher dimensions by replacing translations with Euclidean motions and using Gelfand pairs, introducing a non-commutative spectral analysis framework.

Contribution

It introduces a new approach to spectral synthesis in multiple dimensions using Gelfand pairs and spherical functions, overcoming limitations of translation invariance.

Findings

Extended Schwartz's spectral synthesis to higher dimensions.

Developed a non-commutative spectral analysis framework.

Applied the theory to obtain new results in spectral synthesis.

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.