Dynamics of Orthonormal Bases Associated to Basins of Attraction

TL;DR

This paper investigates how orthonormal bases in reproducing kernel Hilbert spaces evolve as basins of attraction change, focusing on a specific polynomial family and their associated dynamics.

Contribution

It introduces a study of the variation of orthonormal bases linked to basins of attraction for a family of polynomials, extending previous kernel construction techniques.

Findings

Explicit descriptions of basis changes for the polynomial family

Insights into the dynamics of orthonormal bases in complex basins

Extension of kernel construction methods to varying basins

Abstract

In the paper "Infinite Product Represenations for Kernels and Iterations of Functions", a technique was developed which allows for the construction of a reproducing kernel Hilbert space on basins of attraction containing $0$. When the right conditions are met, an explicit orthonormal basis can be constructed using a particular class of operators. It is natural then to consider how the orthonormal basis changes as we let the basin of attraction vary. We will consider this question for the basins of attraction containing $0$ of the family of polynomials $\mathcal{F} = \{az^{2^{n+2}}-2az^{2^{n+1}}:a\neq0\}$, where $n\in\mathbb{N}$.