Mathematical Sciences: Studies in Orthogonal Polynomials and Approximation Theory

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Nevai Abstract Nevai plans to continue his research in approximation theory, orthogonal polynomials, and related areas of analysis involving various extremal problems, ordinary and generalized polynomial inequalities, difference and differential equations, spectral theory of real, complex, and matrix-valued Jacobi, Hessenberg, and banded matrices, Toeplitz and Hankel forms, and Hilbert space operators. In addition, he will continue working on his "Orthogonal Polynomials" software project in Mathematica and on numerical aspects of orthogonal polynomials. The primary focus of his research will be concentrated on three areas: orthogonal polynomials on the unit circle and on arcs of the unit circle, generalized polynomials and polynomial inequalities, and linear difference equations and growth of their solutions. Approximation theory and orthogonal polynomials form an essential part of mathematical analysis in the sense that (i) they provide theoretical foundations for real life applications of various results in "pure" mathematics, and that (ii) they yield a natural bridge between theory and practice. The extraordinary usefulness of orthogonal polynomials stems from the facts that among others (i) they are easily computable by a stable three term recursion formula, (ii) they are a natural medium for expanding "general" functions into well behaved series, and that (iii) their zeros are especially suitable for interplation and quadrature processes. Quadrature processes enable one to evaluate very complicated expressions involving integrals with high degree of precision. The primary subject of this proposal, that is, extensions of Szego's theory of orthogonal polynomials, is especially useful for theses purposes. The proposer hopes to find efficient methods with solid theoretical foundations.