Orthogonal structure on a quadratic curve

TL;DR

This paper develops explicit orthogonal polynomial bases on quadratic curves such as ellipses, parabolas, and hyperbolas, enabling efficient function interpolation with singularities and analyzing convergence properties.

Contribution

It introduces explicit constructions of orthogonal polynomials on quadratic curves and studies their convergence, extending classical polynomial theory to new geometric settings.

Findings

Explicit bases for orthogonal polynomials on quadratic curves are constructed.

Fourier orthogonal expansions on these curves are shown to converge.

Bases are used for interpolating functions with singularities, achieving exponential convergence.

Abstract

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. As an application, we see that the resulting bases can be used to interpolate functions on the real line with singularities of the form $|x|$, $\sqrt{x^2+ \epsilon^2}$, or $1/x$, with exponential convergence.