Infinite product representations for kernels and iterations of functions

TL;DR

This paper explores infinite product representations of kernels, their applications in dynamics, harmonic analysis, and stochastic processes, introduces new Cuntz relation representations, and links Julia sets with Hilbert spaces through complex analysis.

Contribution

It introduces novel infinite product kernel representations, constructs new Cuntz relation representations, and connects Julia sets with Hilbert spaces via complex dynamics.

Findings

New family of Cuntz relation representations

Association of Julia sets with Hilbert spaces

Applications in dynamics and harmonic analysis

Abstract

We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping $R$ in one complex variable, and its iterations.