Asymptotics of Carleman polynomials for level curves of the inverse of a shifted Zhukovsky transformation

TL;DR

This paper investigates the asymptotic behavior of Carleman orthogonal polynomials on a specific level curve derived from a shifted Zhukovsky transformation, revealing complex convergence properties influenced by geometric singularities.

Contribution

It provides a detailed analysis of the asymptotics of orthogonal polynomials on a novel curve, highlighting the influence of geometric singularities on their convergence and limiting behavior.

Findings

Subsequences of polynomials converge based on logarithmic conditions.

Limiting points form a one-parameter family of functions.

Behavior is driven by geometric singularities, not weight functions.

Abstract

This paper complements the recent investigation of \cite{DM} on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve $L$. We study the specific case of $L=\{z= w-1 +(w-1)^{-1},\ |w|=R\}$, for some $R>2$, providing an example that exhibits the new features discovered in \cite{DM}, and for which the asymptotic behavior of the orthogonal polynomials is established over the entire domain of orthogonality. Surprisingly, this variation of the classical example of the ellipse turns out to be quite sophisticated. After properly normalizing the corresponding orthonormal polynomials $p_n$, $n=0,1,...$, and on certain critical subregion of the orthogonality domain, a subsequence $\{p_{n_k}\}$ converges if and only if $\log_{\mu^4}(n_k)$ converges modulo 1 ($\mu$ being an important quantity associated to $L$). As a consequence, the limiting points of the sequence $\{p_n\}$ form a one parameter family of functions, the parameter's range being the interval $[0,1)$. The polynomials $p_n$ are much influenced by a certain integrand function, the explained behavior being the result of this integrand having a nonisolated singularity that is a cluster point of poles. The nature of this singularity sparks purely from geometric considerations, as opposed to the more common situation where the critical singularities come from the orthogonality weight.