Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros

TL;DR

This paper investigates the asymptotic behavior of recurrence coefficients of orthogonal polynomials on multiple intervals, linking their accumulation points to the harmonic measure at infinity and providing a topological correspondence.

Contribution

It establishes a topological equivalence between the accumulation points of recurrence coefficients and the sequence involving harmonic measure, extending understanding of orthogonal polynomials on several intervals.

Findings

Recurrence coefficients' convergence behavior mirrors that of the harmonic measure sequence.

Explicit homeomorphism between accumulation points of coefficients and harmonic measure sequence.

Conditions under which the recurrence coefficients converge are characterized.

Abstract

Let $E = \cup_{j = 1}^l [a_{2j-1},a_{2j}],$ $a_1 < a_2 < ... < a_{2l},$ $l \geq 2$ and set ${\boldmath$\omega$}(\infty) =(\omega_1(\infty),...,\omega_{l-1}(\infty))$, where $\omega_j(\infty)$ is the harmonic measure of $[a_{2 j - 1}, a_{2 j}]$ at infinity. Let $\mu$ be a measure which is on $E$ absolutely continuous and satisfies Szeg\H{o}'s-condition and has at most a finite number of point measures outside $E$, and denote by $(P_n)$ and $({\mathcal Q}_n)$ the orthonormal polynomials and their associated Weyl solutions with respect to $d\mu$, satisfying the recurrence relation $\sqrt{\lambda_{2 + n}} y_{1 + n} = (x - \alpha_{1 + n}) y_n -\sqrt{\lambda_{1 + n}} y_{-1 + n}$. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence $(n {\boldmath$\omega$}(\infty))_{n\in \mathbb N}$ modulo 1; More precisely, putting $({\boldmath$\alpha$}^{l-1}_{1 + n}, {\boldmath$\lambda$}^{l-1}_{2 + n}) = $ $(\alpha_{[\frac{l 1}{2}]+1+n},...,$ $\alpha_{1+n},...,$ $\alpha_{-[\frac{l-2}{2}]+1+n},$ $\lambda_{[\frac{l-2}{2}]+2+n},$ $...,\lambda_{2+n},$ $...,$ $\lambda_{-[\frac{l-1}{2}]+2+n})$ we prove that $({\boldmath$\alpha$}^{l-1}_{1 + n_\nu},$ ${\boldmath$\lambda$}^{l-1}_{2 + n_\nu})_{\nu \in \mathbb N}$ converges if and only if $(n_\nu {\boldmath$\omega$}(\infty))_{\nu \in \mathbb N}$ converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of $({\boldmath$\alpha$}^{l-1}_{1 + n}, {\boldmath$\lambda$}^{l-1}_{2 + n})$ and $(n{\boldmath$\omega$}(\infty))$ modulo 1.