(The organization thanks Prof. Yamilet Quintana for the contents of this presentation)
The development of the Theory of Orthogonal Polynomials (TOP) has been focused in two closely related directions: Algebraic and analytical aspects.
The first direction is linked to the theory of special functions, combinatorics and linear algebra, and it is mainly dedicated to the study of specific orthogonal systems or its hierarchies, such as Jacobi, Hahn and Askey-Wilson polynomials, among others. It also deals with discrete and q-analog orthogonal polynomials, as well as many of the recent contributions on the study of orthogonal polynomials in several variables.
The second direction is characterized by the use of methods of more general mathematical analysis. The general properties of the orthogonal polynomials are just a small part of these analytical aspects, whereas the bigger part includes two extremely rich areas: the TOP on the real line, and the TOP on the unit circle. Historically, the appearance of the classical orthogonal polynomials on the real line, presented in form of continuous fractions, goes back to the XVIII century, but were widely developed during the XIX and (beginning of) XX centuries. The theory of orthogonal polynomials on the unit circle, on the other hand, is more recent, appearing in the works of Szegö and Geronimus on the first half of the XX century. The two-volume monographies of Barry Simon present exhaustively the development of the theory since then.
The connection of the TOP with other areas of Mathematics from both theoretical and applications perspectives must be emphasized. Among others, there are close relations with continuous fractions theory, operator (Jacobi and Toeplitz) theory, moment problem, analytic functions, interpolation, rational and polynomial approximation (in particular, Padé approximation), gaussian quadratures and its extensions, spectral methods for the numerical analysis of differential equations and boundary value problems, electrostatic potential theory, quantum mechanics, information entropies, special functions, number theory, graph theory, combinatorics, random matrices, stochastic processes, and control theory.
In this context, the Iberoamerican Workshop in Orthogonal Polynomials and Applications (EIBPOA) is an international mathematical event, whose main objective is to establish a forum that facilitates the interaction and exchange of ideas between researchers, academics and students working in orthogonal polynomials or related areas. The first edition of EIBPOA was held at Universidad Nacional de Colombia (Bogotá) on June 1-3, 2011.