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.

2000 AMS Subject Classification:

• Primary 33C45
• Secondary 28A80

Keywords:

• orthogonal polynomials
• self-similar measures
• dynamical systems

Acknowledgments

We are grateful to Giorgio Mantica for allowing us to use his codes, and to the anonymous reviewer for a very helpful report.

Steven M. Heilman’s research was supported by the National Science Foundation through the Research Experiences for Undergraduates Program at Cornell. Philip Owrutsky was supported in his work as a Cornell Presidential Research Scholar. Robert S. Strichartz’s research was supported in part by the National Science Foundation, grant DMS-0652440.

Notes

¹By a generic point we mean a point chosen, with respect to a uniform distribution, among a suitable set of (rational) values in floating-point arithmetic. In all cases, we treat only x representable in double-precision floating-point arithmetic.