Numerical study of orthogonal polynomials for fractal measures

Abstract

In recent years, potential theory has an essential effect on approximation theory and orthogonal polynomials. Basic concepts of the modern theory of general orthogonal polynomials are described in terms of Potential Theory. One of these concepts is the Widom factors which are the ratios of norms of extremal polynomials to a certain degree of capacity of a set. While there is a theory of Widom factors for finite gap case, very little is known for fractal sets, particularly for supports of continuous singular measures. The motivation of our numerical experiments is to get some ideas about how Widom factors behave on Cantor type sets. We consider weakly equilibrium Cantor sets, introduced by A.P. Goncharov in [16], which are constructed by iteration of quadratic polynomials that change from step to step depending on a sequence of parameters. Changes in these parameters provide a Cantor set with several desired properties. We give an algorithm to calculate recurrence coefficients of orthogonal polynomials for the equilibrium measure of such sets. Our numerical experiments point out stability of this algorithm. Asymptotic behaviour of the recurrence coefficients and the zeros of orthogonal polynomials for the equilibrium measure of four model Cantor sets are studied via this algorithm. Then, several conjectures about asymptotic behaviour of the recurrence coefficients, Widom factors, and zero spacings are proposed based on these numerical experiments. These results are accepted for publication [1] (jointly with G. Alpan and A.P. Goncharov).