Mathematical Sciences: Studies in Approximation Theory and Orthogonal Polynomials

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940577 Nevai This project is concerned with mathematical research in approximation theory, orthogonal polynomials and related areas of analysis. The work involves extremal problems, ordinary and generalized polynomial inequalities, difference and differential equations, spectral theory of real, complex and matrix-valued tridiag gonal (Jacobi) and banded matrices, Toeplitz and Hankel and Hilbert space operators. Work will also continue on the development of software directed at analyzing the numerical aspects of orthogonal polynomials. Specifics of the research include investigations into conditions on assuring boundedness for solutions of homogeneous and nonhomogeneous perturbations oflinear difference equations in terms of their unperturbed counterparts. The ratio and comparative asymptotics of extremal problems involving weighted best approximation of monomials by lower degree polynomials will be taken up. Efforts to understand the connection between a Jacobi matrix and its spectral measure and its analogues on the unit circle will also be made. Work will be done extending recent results on the asymptotic behavior of Christoffel functions to measures in the complex plane and on generalized polynomials and their suitability as means of approximation. Approximation theory and especially the theory of orthogonal polynomials has fruitful connections with many branches of mathematics such as differential equations, operator theory, numerical integration, continued fractions and control theory, to name a few. These subjects have received strong impulses from the theory of orthogonal polynomials in the past and this process continues with increasing vigor. ***