Mathematical Sciences: Orthogonal Polynomials

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Work on this project will continue investigations into the nature of orthogonal polynomials defined on subsets of the real and complex numbers. The orthogonality is defined with regard to some measure (density) and, depending on the nature of that measure, one encounters radically different sets of polynomials. The polynomials may arise in different contexts - as solutions of differential equations or as the result of recurrence relations. But the measure always exists in the background. Particular emphasis will be placed on understanding polynomials orthogonal to measures supported on several disjoint intervals. Asymptotic formulas for polynomials have been developed in terms of integral representations when measures are supported on a single interval. Efforts will be made to carry over this work to the case of support on several intervals.Another line of investigation will consider polynomials orthogonal with respect to measures supported on Julia sets in the complex plane. Two goals are set forth. The first is to show that in general the Jacobi matrix with these polynomials is limitperiodic when the Julia set is a Cantor set. A second goal is to determine the moments of the orthogonality measure via the recurrence formulas. A new line of investigation uses ergodic theory to study properties of orthogonal polynomials on the unit circle. When the recurrence relation is generated by a stochastic process, there is a Lyaponov exponent. When the exponent is zero, there is evidence that the measure has an absolutely continuous part. Efforts will be made to verify this.