Orthogonal polynomials on the disc

Shayne Waldron

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 ${j\pi\over n+1}$, $j=0,1,\ldots,n$ 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 $\cP_n$, 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.

Keywords: Gegenbauer (ultraspherical) polynomials, Legendre polynomials on the disc, disc polynomials, Zernike polynomials, quadrature for trigonometric polynomials, representation theory, ridge functions, tight frames

Math Review Classification: Primary 33C45, 33D50; Secondary 06B15, 42C15

Length: 15 pages

Last Updated: 10 May 2007

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