# 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

## Availability: