Mathematical Sciences: Orthogonal Polynomials

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The research planned in this project centers on classes of special functions known as orthogonal polynomials. Orthogonality can only be considered in the presence of a measure; in this work the measures will be concentrated on the real line or on a circle. The work focuses on the connection between a measure and the recurrence coefficients of the corresponding orthogonal polynomials. Work will also be done investigating the summability and convergence of orthogonal polynomial expansions. The methods involve minimizing polynomial integrals (the Christoffel function), asymptotic behavior of difference equations and eigenvalues of infinite tridiagonal matrices. These same techniques can be used for studying solutions of the Schrodinger equation in a central field of force and the Christoffel function is related to statistical prediction theory. Orthogonal polynomials form a subclass of special functions developed over centuries for the solution or approximate solution of fundamental equations modeling the physical world. They are generally simple to express, relatively easy to compute and often occur as fundamental solutions of certain differential equations.