Orthogonal polynomials on a class of planar algebraic curves

2022/11/13 by Fasondini, Marco, Olver, Sheehan, Xu, Yuan · 2 citations
#33D45 #65F25 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2211.06999

Abstract

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form y^m = ϕ(x) in ℝ² where m = 1, 2 and ϕ 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, …, N are computed via the Lanczos algorithm in O(Nd⁴) operations.

Cited by

• Elliptic orthogonal polynomials and OPRL
• Recurrence relations and the Christoffel-Darboux formula for elliptic orthogonal polynomials

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