Marcus Vaktnäs: Multiple orthogonal polynomials on the real line and the unit circle

Abstract

The thesis presents a series of papers in the field of orthogonal polynomials and approximation theory. The majority of research on orthogonal polynomials can be divided into two parallel groups, where the first one studies orthogonality with respect to measures supported on the real line (OPRL), while the other one studies measures supported on the unit circle (OPUC). 

The main topic of the thesis is multiple orthogonal polynomials, on the real line (MOPRL) and the unit circle (MOPUC). The former theory is well-developed, while the latter had seen very little development prior to this work. The thesis contains several fundamental results on MOPUC, which I believe will be its main contribution. 

Paper I derives recurrence relations for MOPUC that generalize the Szegő recurrence relations of OPUC, while standing in direct analogue with the nearest-neighbour recurrence relation of MOPRL. Partial difference equations for the recurrence coefficients followed from these relations, along with a Christoffel–Darboux formula. 

Paper II-IV study Christoffel transforms in MOPRL, and the more general Uvarov transforms. The papers give formulas for the transformed polynomials and recurrence coefficients, and illustrate a connection to zero location and interlacing. Applications to the zero behaviour of some families of multiple orthogonal polynomials are given. The papers also contain applications to the computation of recurrence coefficients and orthogonal polynomials of OPRL. 

Paper V-VII takes a new approach to MOPUC, working with some multiple orthogonal Laurent polynomials of integer and semi-integer degree. Analogues of the Angelesco and AT systems of MOPRL are given, together with important results on perfectness and zero location. The recurrence relations of Paper I also generalize to some of these Laurent polynomials. Szegő mapping relations are given, along with Geronimus relations for the recurrence coefficients, illustrating a connection to MOPRL that was not established for the polynomials studied in Paper I. The connection with two-point Hermite–Padé approximation is also outlined. 

The papers are complemented by an introduction with background content, followed by a summary of each paper. The first two chapters provide a friendly introduction to orthogonal and multiple orthogonal polynomials, first through a problem from the thesis, and then through motivating applications to diophantine approximation and numerical integration. The applications relate to Padé approximation and Gauss quadrature, which play a central role in the theory of orthogonal polynomials, and are used frequently in the discussions throughout the introductory chapters.