Delsarte-type extremal problems and convolution roots on homogeneous spaces

TL;DR

This paper generalizes extremal problems for positive definite kernels on homogeneous spaces, establishing a correspondence with bi-invariant functions on groups, and proves the existence of extremal functions and convolution roots in this setting.

Contribution

It introduces a framework linking Delsarte-type extremal problems on homogeneous spaces to bi-invariant functions on groups, proving the existence of extremal functions and convolution roots.

Findings

Established a correspondence between kernels on homogeneous spaces and bi-invariant functions on groups.

Proved the existence of extremal functions for Delsarte problems on homogeneous spaces.

Demonstrated the existence of convolution roots for positive definite bi-invariant functions in Gelfand pairs.

Abstract

For a locally compact group $G$ and compact subgroup $K$, we consider a Delsarte-type extremal problem for $G$-invariant positive definite kernels on the homogeneous space $G/K$, generalising a certain Tur\'an problem for isotropic positive definite kernels on the unit sphere $\mathbb{S}^d$ in $\mathbb{R}^{d+1}$. We exploit a correspondence between $G$-invariant kernels on $G/K$ and $K$-bi-invariant functions on $G$ to show that the Delsarte-type problem on a homogeneous space is equivalent to a Delsarte-type problem for $K$-bi-invariant functions on its group $G$ of transformations. We use this correspondence to show the existence of an extremal function for the Delsarte problem on the homogeneous space. In the case where $(G,K)$ is a compact Gelfand pair, we show the existence of $K$-bi-invariant convolution roots for positive definite $K$-bi-invariant functions, consequently obtaining the existence of a $G$-invariant convolution root for $G$-invariant positive definite kernels.