Simultaneous Approximation via Laplacians on the Unit Ball

Abstract

We study the orthogonal structure on the unit ball textbf{B}^d of mathbb {R}^d with respect to the Sobolev inner products f,gΔ=λL(f,g)+∫BdΔ[(1-‖x‖2)f(x)]Δ[(1-‖x‖2)g(x)]dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\langle f,g\\right\\rangle _{\\Delta } =\\lambda \\, \\mathscr {L}(f,g) + \\int _{\ extbf{B}^d}{\\Delta [(1-\\Vert x\\Vert ^2) f(x)] \\, \\Delta [(1-\\Vert x\\Vert ^2) g(x)]\\,\ extrm{d}x}, \\end{aligned}$$\\end{document}where mathscr {L}(f,g) = int _{textbf{S}^{d-1}}f(xi ),g(xi ),textrm{d}sigma (xi ) or mathscr {L}(f,g) = f(0) g(0), lambda >0, sigma denotes the surface measure on the unit sphere textbf{S}^{d-1}, and Delta is the usual Laplacian operator. Our main contribution consists in the study of orthogonal polynomials associated with langle cdot , cdot rangle _{Delta }, giving their explicit expression in terms of the classical orthogonal polynomials on the unit ball, and proving that they satisfy a fourth-order partial differential equation, extending the well-known property for ball polynomials, since they satisfy a second-order PDE. We also study the approximation properties of the Fourier sums with respect to these orthogonal polynomials and, in particular, we estimate the error of simultaneous approximation of a function, its partial derivatives, and its Laplacian in the L^2(textbf{B}^d) space.