  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.
