Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. 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.

翻译：考虑三次曲线上两个变量的正弦多面体,包括椭圆曲线。对于在立方曲线上定义的适当加权函数,一个变量中两个正弦多面体的两组组成的正弦多面体构建了明确的正弦多面体基础。我们显示,这些正弦多面体可以用来以立方体和正根特性来接近函数,并展示它们用于用单式解决方案解决差异方程式的用途。