Orthogonal Polynomials with Respect to Self-Similar Measures

TL;DR

This paper explores the properties of orthogonal polynomials with respect to self-similar measures, especially on Cantor sets, introducing visualization techniques and a dynamical systems perspective.

Contribution

It presents new experimental insights into orthogonal polynomials on fractal supports and proposes a novel dynamical systems approach for analyzing these functions.

Findings

Distinct behaviors of polynomials on Cantor sets and gaps

A new visualization method for functions on fractals

A dynamical systems framework for polynomial families

Abstract

We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theory of dynamical systems, for studying families $P_{n}(x)$ of orthogonal functions as functions of $n$ for fixed values of $x$.