Two problems in Computational Mathematics: Multiple Orthogonal Polynomials and Greedy Energy Points

Abstract

In this thesis we investigate the asymptotic behavior of greedy energy sequences on locally compact metric spaces and Euclidean spaces, and the asymptotic behavior of multiple orthogonal polynomials associated with measures supported on star-like sets in the complex plane.

Greedy energy sequences are constructed through a greedy (iterative) algorithm, in which the Nth point of the sequence is selected optimally (from the energy point of view) at the Nth step. In the context of Euclidean spaces, we assume that the interaction between points is governed by the Riesz potential V=1/r^s, where s>0 and r denotes Euclidean distance. We show that for s>1, greedy energy sequences on Jordan arcs or curves are not asymptotically s-energy minimizing (i.e. the energy of the first N points of any such sequence has a limiting behavior that differs from the limiting behavior satisfied by optimal N-point configurations). In fact, we show that for s>1 no infinite sequence of points on Jordan arcs or curves can be asymptotically s-energy minimizing.

Corresponding to s=infinity, we disprove a conjecture attributed to L. Bos on the asymptotic distribution of greedy best-packing configurations. Several other topics are investigated, such as second-order asymptotics on the unit circle, and greedy energy sequences with external fields.

In this thesis we also study the ratio and nth root asymptotics of multiple orthogonal polynomials associated with a Nikishin-type system consisting of two measures supported on a star-like set. These polynomials satisfy a three-term recurrence relation of third order with positive coefficients. We prove the existence of different periodic limits for the sequence of ratios of consecutive polynomials and the sequence of recurrence coefficients (this situation is analyzed here for the first time in the context of polynomials generated by three-term recurrences of higher order). The ratio asymptotic limits are expressed in terms of the branches of a three-sheeted compact Riemann surface of genus zero, and the nth root asymptotics is described in terms of the solution to a vector equilibrium problem for logarithmic potentials.