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Orthogonal structure in and on quadratic surfaces

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Orthogonal structure in and on quadratic surfaces Text of abstract: Spherical harmonics are orthogonal polynomials on the unit sphere. They are eigenfunctions of the Laplace-Beltrami operator on the sphere and they satisfy an addition formula (a closed formula for their reproducing kernel). In this talk, we consider orthogonal polynomials on quadratic surfaces of revolution and inside the domain bounded by quadratic surfaces.  We will define orthogonal polynomials on the surface of a cone that possess both characteristics of spherical harmonics. In particular, the addition formula on the cone has a one-dimensional structure, which leads to a convolution structure on the cone useful for studying Fourier orthogonal series. Furthermore, the same narrative holds for orthogonal polynomials defined on the solid cones.

This talk is part of the Isaac Newton Institute Seminar Series series.

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