Abstract
The scientific aim of this project is the study of the general properties of the orthogonal polynomials, special functions and their applications to both Applied Mathematics and various areas of Pure Mathematics. This is the principal theme of the studies of our research group. These polynomials and functions have important applications to various areas of Classical Analysis and Numerical Analysis: Numerical Quadrature Formulae; Least Square Approximations, including Fourier series; Padé Approximations and the Theory of Continued Fractions; Approximations by splines; Moment Preserving Approximations; Relaxation Methods in Linear Algebra; Polynomial Inequalities and Polynomial Regression and "birth and death" processes in Statistics. While the majority of these topics can be considered to belong to Applied Mathematics, other applications to Code Theory; Potential Theory; Zeros of Polynomials and Functions are related to Algebra, Differential Equations, Complex Analysis and Classical Real Analysis. Some of the remarkable applications and connections of the orthogonal polynomials are the use of certain inequalities for sums of Jacobi polynomials in de Branges' proof of 1984 of the Bieberbach conjecture about the coefficients of univalent functions, formulated in 1916, as well as the tight relation between the distribution of the zeros of the Riemann zeta function and the eigenvalues of certain random matrices, described by the Law of Montgomery e Dyson, from one side, and these matrices and orthogonal polynomials, from the other. The members of the research group Alagacone Sri Ranga, Cleonice Fátima Bracciali, Dimitar Kolev Dimitrov and Eliana Xavier Linhares de Andrade have contributed to the theory and applications of orthogonal polynomials and special functions with more than 100 high quality research papers and supervised tens of PhD and Master students. Some of our students already do independent research; others have won national and international prizes and awards. The group has been supported by important research projects of the foundations FAPESP, CNPq and CAPES, both of national level and by international exchange programs. The purpose of the group is to continue contributing in research through new publications and transmitting the acquired knowledge to talented students in order to form a new generation of good Brazilian mathematicians. (AU)