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

TL;DR

The paper addresses how to characterize smooth $K$-invariant functions on $\mathbb{R}^n$ through invariant polynomials, offering an elementary proof via Gelfand theory and spherical functions. It proves a version of Schwarz's theorem for integrable $K$-invariant functions with compactly supported Fourier transforms, showing the representing function $h$ can be chosen real-analytic on $\mathbb{R}^{\ell}$. The approach reframes Schwarz's extension problem as an extension of the Gelfand transform in abelian (Euclidean) Gelfand pairs and provides explicit constructions using spherical functions and their Taylor expansions. The results are illustrated with examples, including the $\mathrm{SO}(n)$-invariant radial setting and a $\mathbb{Z}_2$ action on $\mathbb{R}^2$, highlighting how non-independence among generators can be managed and how coefficients relate to differential operators evaluated at the origin. Overall, the work yields a conceptually streamlined, analytic pathway from $K$-invariant data to a single-variable generating function $h$ with strong regularity properties.

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.

Theorems & Definitions (19)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• Remark 3.1
• Definition 3.2
• Lemma 3.3
• proof
• proof : Proof of Proposition \ref{['prop: aux sph f']}