Orthogonal structure on a wedge and on the boundary of a square
Provides a theoretical framework for orthogonal polynomials on non-standard domains, benefiting researchers in approximation theory and random matrix theory.
The paper constructs explicit bases of orthogonal polynomials on a wedge and on the boundary of a square for two large classes of weight functions, and studies convergence of Fourier expansions. It applies these results to analyze determinantal point processes and compute Stieltjes transforms.
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.