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. We discuss applications to the Fourier extension problem, interpolation of functions with singularities or near singularities and the solution of Schrödinger’s equation with nondifferentiable or nearly nondifferentiable potentials.

中文翻译：

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二次曲线上的正交结构

研究平面二次曲线上的正交多项式。这些包括椭圆，抛物线，双曲线和两条线的正交多项式。对于关于在任意二次曲线上定义的适当权重函数的积分，根据一个变量中的两个正交多项式族，构造了正交多项式的显式基础。还在每种情况下研究傅立叶正交展开的收敛性。我们讨论了傅里叶扩展问题的应用，具有奇异性或近奇异性的函数插值以及具有不可微或几乎不可微势的薛定ding方程的解。