Orthogonal Polynomials and Fourier Orthogonal Series on a Cone

Orthogonal polynomials and the Fourier orthogonal series on a cone in Rd+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^{d+1}$$\end{document} are studied. It is shown that orthogonal polynomials with respect to the weight function (1-t)γ(t2-‖x‖2)μ-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-t)^{\gamma }(t^2-\Vert x\Vert ^2)^{\mu -\frac{1}{2}}$$\end{document} on the cone Vd+1={(x,t):‖x‖≤t≤1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {V}}}^{d+1} = \{(x,t): \Vert x\Vert \le t \le 1\}$$\end{document} 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 characteristic. 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.