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.

中文翻译：

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平面三次曲线上的正交多项式

考虑三次曲线上两个变量的正交多项式。对于关于定义在三次曲线上的适当权重函数的积分，正交多项式的显式基础是根据一个变量中的两个正交多项式族构建的。我们展示了这些正交多项式可用于逼近具有三次奇点和平方根奇点的函数，并演示它们在求解具有奇点解的微分方程时的用途。