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October 5, 2025Open Access

A novel approach through spherical functions in the characterization of invariant polynomials

Key Points

This proof shows that all smooth K-invariant functions can be derived from invariant polynomials.
Key evidence lies in the innovative application of Gelfand theory to establish the relationship.
The approach utilizes spherical functions to simplify the understanding of K-invariant functions.
This work emphasizes the importance of spherical functions in the domain of invariant theory.

Abstract

Given a compact subgroup K of the orthogonal group acting on the Euclidean space Rn, Gerald Schwarz proved that every smooth K-invariant function on Rn can be expressed as a smooth function of a generating set of K-invariant polynomials on n variables. The goal of this work is to provide an alternative and more straightforward proof of this result, based on Gelfand theory, with a particular focus on spherical functions.

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Cite This Study

Marín et al. (Wed,) studied this question.

synapsesocial.com/papers/68e2537cd6d66a53c24743d4 https://doi.org/https://doi.org/10.48550/arxiv.2510.01043

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