Orthogonal polynomials on the disc

Abstract

We consider the space P n of orthogonal polynomials of degree n on the unit disc for a general radially symmetric weight function. We show that there exists a single orthogonal polynomial whose rotations through the angles j π n + 1 , j = 0 , 1 , … , n forms an orthonormal basis for P n , and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Legendre polynomials on the disc. Furthermore, such a polynomial reflects the rotational symmetry of the weight in a deeper way: its rotations under other subgroups of the group of rotations forms a tight frame for P n , with a continuous version also holding. Along the way, we show that other frame decompositions with natural symmetries exist, and consider a number of structural properties of P n including the form of the monomial orthogonal polynomials, and whether or not P n contains ridge functions.