Orthogonal polynomials on a class of planar algebraic curves

11/13/2022

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by   Marco Fasondini, et al.

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We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form y^m = ϕ(x) in ℝ^2 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(Nd^4) operations.