Orthogonal Polynomials on Planar Cubic Curves

Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

Automated summary

This paper investigates orthogonal polynomials in two variables on cubic curves. A explicit basis of orthogonal polynomials is constructed using two families of orthogonal polynomials in one variable for integrals with respect to a suitable weight function defined on a cubic curve. It is shown that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and their usage for solving differential equations with singular solutions is demonstrated.