Orthogonal Polynomials with Respect to Self-Similar Measures

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 $x$.