Orthogonal structure on a wedge and on the boundary of a square

TL;DR

This paper constructs explicit orthogonal polynomial bases on a wedge and square boundary, studies their convergence, and applies these results to analyze related determinantal point processes and Stieltjes transforms.

Contribution

It introduces explicit bases for orthogonal polynomials on a wedge and square boundary, and investigates their convergence and applications in point process analysis.

Findings

Explicit orthogonal polynomial bases are constructed for wedge and square boundary.

Convergence of Fourier orthogonal expansions is established.

Applications include analysis of determinantal point processes and Stieltjes transforms.

Abstract

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.