Orthogonal polynomials for the weakly equilibrium Cantor sets

TL;DR

This paper investigates orthogonal polynomials on weakly equilibrium Cantor sets, revealing their connection to Chebyshev polynomials and providing methods to compute all degrees and Jacobi parameters, with bounded Widom factors.

Contribution

It establishes that monic orthogonal polynomials on these sets are Chebyshev polynomials and offers procedures to determine all orthogonal polynomials and Jacobi parameters.

Findings

Monic orthogonal polynomials $Q_{2^s}$ coincide with Chebyshev polynomials.

Procedures to find $Q_{n}$ and Jacobi parameters are proposed.

Widom factors are shown to be bounded below.

Abstract

Let $K(\gamma)$ be the weakly equilibrium Cantor type set introduced in [10]. It is proven that the monic orthogonal polynomials $Q_{2^s}$ with respect to the equilibrium measure of $K(\gamma)$ coincide with the Chebyshev polynomials of the set. Procedures are suggested to find $Q_{n}$ of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below.