Mathematical Sciences: Orthogonal Polynomials

点击翻译按钮获取中文摘要

Various topics in mathematical analysis associated with orthogonal polynomials will be taken up in this project. On the same interval, there are many classes of orthogonal polynomials. In each class, orthogonality is defined with respect to a measure (density) concentrated on the interval. The main theme of this work is a continuing effort to understand the relationship between the measure and the recurrence coefficients of the corresponding polynomials. Associated to each class of orthogonal polynomials is a unique algorithm which produces the higher order polynomials in terms of the lower. This recurrence is a simple linear relation, and it is the coefficients in the relation that are of considerable interest. Methods to be employed include minimizing polynomial integrals (the Christoffel function), asymptotic behavior of difference equations and eigenvalues of infinite tridiagonal matrices. The methods used for studying eigenvalues of matrices can be adapted to the study of solutions of the Schrodinger equation in a central field of force, and the Christoffel function is related to statistical prediction theory. Results of polynomial approximation, a natural by-product of this work, and prediction theory are valuable in digital processing of electronic signals.