Orthogonal polynomials on a class of planar algebraic curves

Abstract

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form $y^m=\varphi (x)$ in ${\mathbb{R}}^2$ where $m=1,2$ and $\varphi$ is a polynomial of arbitrary degree $d$, in terms of univariate semiclassical OPs. We compute connection coeffeicients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree $0,\dots ,N$ are computed via the Lanczos algorithm in $O(N{d}^4)$ operations.