Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres

TL;DR

This paper explores conditions under which positive definite kernels on complex spheres can be represented as generalized convolution roots, advancing the understanding of kernel construction on complex manifolds.

Contribution

It introduces specific conditions for representing $L^2$-positive definite, zonal kernels on complex spheres as generalized convolution roots, extending kernel theory on manifolds.

Findings

Provides necessary conditions for kernel representation as convolution roots.

Enhances kernel construction methods on complex spheres.

Contributes to the theoretical foundation of positive definite kernels on complex manifolds.

Abstract

Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel.