Positive definite radial kernels on homogeneous trees and products

TL;DR

This paper presents new proofs and characterizations of positive definite radial kernels on homogeneous trees and their products, linking them to measures and trace-class operators, with implications for harmonic analysis.

Contribution

It introduces novel proofs and characterizations of positive definite radial kernels on homogeneous trees and their products, connecting them to measures and operator theory.

Findings

New proof of classical correspondence between kernels and measures

Characterization using positive trace-class operators

Extension to products of homogeneous trees

Abstract

We give a new proof of a classical result which provides a one-to-one correspondence between positive definite radial kernels on a homogeneous tree and finite Borel measures on the interval $[-1,1]$. Our methods allow us to find a new characterisation in terms of positive trace-class operators on $\ell_2$. Furthermore, we extend both characterisations to finite products of homogeneous trees. The proof relies on a formula for the norm of radial Schur multipliers, in the spirit of Haagerup--Steenstrup--Szwarc, and a variation of the Hamburger moment problem.