Orthogonal polynomials on the disc

We consider the space Pn of orthogonal polynomials of degree n on the unit disc for a general radially symmetricweight function.We show that there exists a single orthogonal polynomialwhose rotations through the angles j n+1 , j = 0, 1, . . . , n forms an orthonormal basis for Pn, 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 theweight in a deeperway: its rotations under other subgroups of the group of rotations forms a tight frame for Pn, with a continuous version also holding.Along theway,we show that other framedecompositionswith natural symmetries exist, and consider a number of structural properties of Pn including the form of the monomial orthogonal polynomials, and whether or not Pn contains ridge functions. © 2007 Elsevier Inc. All rights reserved. MSC: primary 33C4533D50; secondary 06B1542C15