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Foundations of Computational Mathematics
Published in: Foundations of Computational Mathematics 6/2016

01-12-2016

Do Orthogonal Polynomials Dream of Symmetric Curves?

Authors: A. Martínez-Finkelshtein, E. A. Rakhmanov

Published in: Foundations of Computational Mathematics | Issue 6/2016

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Abstract

The complex or non-Hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not clear. Nevertheless, numerical experiments, such as those presented in this paper, show that the zeros not simply cluster somewhere on the plane, but persistently choose to align on certain curves, and in a very regular fashion. The problem of the limit zero distribution for the non-Hermitian orthogonal polynomials is one of the central aspects of their theory. Several important results in this direction have been obtained, especially in the last 30 years, and describing them is one of the goals of the first parts of this paper. However, the general theory is far from being complete, and many natural questions remain unanswered or have only a partial explanation. Thus, the second motivation of this paper is to discuss some “mysterious” configurations of zeros of polynomials, defined by an orthogonality condition with respect to a sum of exponential functions on the plane, that appeared as a results of our numerical experiments. In this apparently simple situation the zeros of these orthogonal polynomials may exhibit different behaviors: for some of them we state the rigorous results, while others are presented as conjectures (apparently, within a reach of modern techniques). Finally, there are cases for which it is not yet clear how to explain our numerical results, and where we cannot go beyond an empirical discussion.
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Footnotes
1
The procedure used to compute these zeros numerically is briefly explained in Sect. 6.2.
 
2
As a referee pointed out, the configuration of Fig. 7, top right, could be formally considered as a limit case of what we describe in Theorem 6.1 below, where the S-property does play a role. Nevertheless, the situation is completely different here due to the existence of a string of zeros of the weight (6.6) along the imaginary axis. In fact, zeros of \(Q_n\), at least visually, are equally spaced on \(i{\mathbb {R}}\), following the distribution of zeros of \(w_n\) and not an equilibrium measure on an interval of \(i{\mathbb {R}}\).
 
3
Strictly speaking, even the situation when the components of the support of \(\vec {\lambda }\) are simple arcs with common endpoints is already interesting.
 
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Metadata
Title
Do Orthogonal Polynomials Dream of Symmetric Curves?
Authors
A. Martínez-Finkelshtein
E. A. Rakhmanov
Publication date
01-12-2016
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 6/2016
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-016-9313-0

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