Asymptotic properties of Jacobi matrices for a family of fractal measures

TL;DR

This paper investigates the asymptotic behavior of Jacobi matrices linked to equilibrium measures on fractal Cantor sets, providing numerical insights into conjectures about orthogonal polynomials on fractals.

Contribution

It offers a numerical examination of conjectures related to orthogonal polynomials on fractal measures, extending understanding beyond previous theoretical results.

Findings

Numerical evidence supporting conjectures on orthogonal polynomials

Comparison of Jacobi matrix properties with general fractal measure conjectures

Insights into asymptotic behavior of recurrence coefficients for fractal measures

Abstract

We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$.