Orthogonal polynomials on a class of planar algebraic curves

TL;DR

This paper develops a method to construct and compute bivariate orthogonal polynomials on certain algebraic curves using univariate semiclassical OPs, with efficient algorithms for connection coefficients.

Contribution

It introduces a novel construction of bivariate orthogonal polynomials on algebraic curves and provides an efficient Lanczos-based algorithm for their computation.

Findings

Connection coefficients form a banded matrix.

Lanczos algorithm computes coefficients in O(Nd^4) operations.

Bivariate OPs are expressed in terms of univariate semiclassical OPs.

Abstract

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form $y^m = \phi(x)$ in $\mathbb{R}^2$ where $m = 1, 2$ and $\phi$ 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 $0, \ldots, N$ are computed via the Lanczos algorithm in $O(Nd^4)$ operations.