Mathematical Sciences: Special Functions and Orthogonal Expansions with Applications to Nonlinear Dynamics and Quantum Mechanics/Orthogonal Expansions and Potential Theory

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Work on this project will focus on problems arising in the theory of orthogonal polynomials. These are polynomials in one complex variable which are orthogonal with respect to some density (measure) defined on the complex plane. Particular emphasis will be placed on those measures which are supported on bounded sets. The setting is more general than the study of polynomials defined on a circle. It is motivated by studies of the Schrodinger equation and dynamical problems which have led to polynomials orthogonal on Julia sets and on spectral questions concerning Jacobi matrices with unbounded or periodic entries which are orthogonal on several intervals. The latter questions emerging from statistical physics. Two primary goals in the work involve the location of the roots of orthogonal polynomials and analyzing certain asymptotic relationships involving successive roots of polynomials as their degrees increase. The method to be employed involves a potential theoretic approach by which asymptotics are measured by the Green's function for the support set. Other work will involve Pade approximation and weighted polynomial approximation in Lebesgue spaces defined on plane regions.