Orthogonal Polynomials on Planar Cubic Curves

Article Mathematics, Applied

Orthogonal structure on a quadratic curve

Sheehan Olver et al.

Summary: The study focuses on orthogonal polynomials on quadratic curves in the plane, such as ellipses, parabolas, hyperbolas, and two lines. An explicit basis of orthogonal polynomials is constructed for integrals with respect to weight functions on these curves using two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is analyzed in each case, with applications ranging from Fourier extension problems to interpolation of functions with singularities.

IMA JOURNAL OF NUMERICAL ANALYSIS (2021)

添加到收藏夹

Article Mathematics

ORTHOGONAL STRUCTURE AND ORTHOGONAL SERIES IN AND ON A DOUBLE CONE OR A HYPERBOLOID

Yuan Xu

Summary: In this paper, orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution are studied, showing similarities to spherical harmonics. These polynomials are eigenfunctions of a second order linear differential operator and satisfy an addition formula. The addition formula leads to a convolution structure, providing a powerful tool for studying Fourier orthogonal series on these domains.

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY (2021)

添加到收藏夹

Article Mathematics, Applied

Non-homogeneous wave equation on a cone

Sheehan Olver et al.

Summary: The wave equation on the cone has a unique solution under certain conditions, which can be explicitly given in the Fourier series of orthogonal polynomials. This provides a particular solution for solving the boundary value problems of the non-homogeneous wave equation, which can be combined with a solution to the homogeneous wave equation for obtaining the full solution.

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS (2021)

添加到收藏夹

Article Mathematics, Applied