Orthogonal polynomials and Fourier orthogonal series on a cone

2019/05/18 by Yuan Xu, Xu, Yuan · 3 citations
Mathematics · #33C50 #42C05 #42C10 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1905.07587

openalex publication_date 2019/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in ℝ^d+1 are studied. It is shown that orthogonal polynomials with respect to the weight function (1-t)^γ(t²-‖x‖²)^μ-\frac12 on the cone \mathbbV^d+1 = \(x,t): ‖x‖ ≤ t ≤ 1\ are eigenfunctions of a second order differential operator, with eigenvalues depending only on the degree of the polynomials, and the reproducing kernels of these polynomials satisfy a closed formula that has a one-dimensional chacteristic. The latter leads to a convolution structure on the cone, which is then utilized to study the Fourier orthogonal series. This narrative also holds, in part, for more general classes of weight functions. Furthermore, analogous results are also established for orthogonal structure on the surface of the cone.

Citations

• Differential-difference operators associated to reflection groups

Cited by

• Approximation and orthogonality on fully symmetric domains
• Fourier transforms of orthogonal structures on the paraboloid
• Best approximation by polynomials on the conic domains

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• Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid
• Fourier transforms of orthogonal polynomials on the cone
• Sobolev orthogonal polynomials on the conic surface
• Monomial and Rodrigues orthogonal polynomials on the cone
• Fourier orthogonal series on a paraboloid