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Delsarte-type extremal problems and convolution roots on homogeneous spaces

Mita D. Ramabulana

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 Turán 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.

Delsarte-type extremal problems and convolution roots on homogeneous spaces

Abstract

For a locally compact group and compact subgroup , we consider a Delsarte-type extremal problem for -invariant positive definite kernels on the homogeneous space , generalising a certain Turán problem for isotropic positive definite kernels on the unit sphere in . We exploit a correspondence between -invariant kernels on and -bi-invariant functions on to show that the Delsarte-type problem on a homogeneous space is equivalent to a Delsarte-type problem for -bi-invariant functions on its group 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 is a compact Gelfand pair, we show the existence of -bi-invariant convolution roots for positive definite -bi-invariant functions, consequently obtaining the existence of a -invariant convolution root for -invariant positive definite kernels.

Paper Structure

This paper contains 9 sections, 22 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. A Bijective Correspondence
  4. Delsarte-type problem on extremal problem
  5. Delsarte problem for invariant kernels
  6. Delsarte problem for $K$-bi-invariant functions on groups
  7. Existence of extremisers
  8. convolution roots
  9. Acknowledgements

Key Result

Proposition 1

Let $G$ be a locally compact group. If $f \in L^2(G)$, let $f^{\#}(x):=\overline{f(-x)}$; then $f \ast f^{\#}$ is a continuous positive definite function.

Theorems & Definitions (31)

  • Proposition 1: folland
  • Theorem 2: dijk
  • Theorem 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 21 more