Short courses

Title: Orthogonal Polynomials, Zeros and Applications

Professor: Dimitar K. Dimitrov (UNESP-Brazil)
I shall discuss methods from different areas of mathematics which have been used to investigate the behaviour of the zeros of orthogonal polynomials. Emphasis will be given on:
1. Discrete techniques from linear algebra, as the Theorems of Perron-Frobenius and of Hellman-Feynman;
2. Methods from Analysis. These will be divided into two main subtopics:
a) Results on zeros of solutions of ordinary differential equations, as the Sturm Comparison Theorem and its analogs and generalizations. Here we shall approach briefly the theme of electrostatic interpretation of zeros;
b)  Results from Classical Analysis on zeros of polynomials in general. This topic includes a vast number of old, beautiful and   useful but hardly remembered nowadays theorems, due to Obrechkoff, Sturm, Hermite and Biehler, Routh and Hurwitz, Marcell Riesz and Stoyanoff,  etc.

Many specific examples of the use of these methods in the study of the behaviour of zeros of orthogonal polynomials and special functions, due to Ismail, Muldoon, Elbert, Laforgia, Siafarikas, Krasikov, the author, and their collaborators, will be provided. Interesting open problems will be formulated and discussed.

I shall also report a recent result which reveals the fundamental role of the orthogonal polynomials in important mathematical problems related to zeros, as The Riemann Hypothesis and the description of the Lee-Yang measures in Statistical Mechanics.

Title: Recent trends in Orthogonal Polynomials on the Unit Circle

Professor: Francisco Marcellán (UC3M-Spain)
In this seminar we will introduce some basic background on orthogonal polynomials associated with nontrivial probability measures supported on the unit circle as well as five topics on the subject, both of a theoretical and computational flavour.
1.- Orthogonal polynomials on the unit circle. Recurrence relations. Matrix representations (GGT and CMV) of the multiplication operator.
2.- Carathéodory functions and the trigonometric moment problem. Gauss and Szegö quadrature rules. Applications in signal theory.
3.- Spectral transformations and matrix factorizations. Applications in integrable systems.
4.- Asymptotic properties of orthogonal polynomials on the unit circle. The Szegö class. The Nevai class. Applications in linear prediction.
5.- The Riemann-Hilbert approach. The main reference for a previous reading of participants in the seminar will be the monograph B. Simon, Orthogonal Polynomials on the unit circle, Amer. Math. Soc. Colloq. Publ. 54 (2 volúmenes), Amer. Math. Soc., providence, Rhode Island, 2005.
I will provide lecture notes  together with some extra material (mainly, relevant papers related to the above five topics).

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