Simultaneous approximation via Laplacians on the unit ball

Abstract

We study the orthogonal structure on the unit ball $\mathbf{B}^d$ of $\mathbb{R}^d$ with respect to the Sobolev inner products $$ \left\langle f,g\right\rangle_{\Delta} =\lambda\, \mathscr{L}(f,g) + \int_{\mathbf{B}^d}{\Delta[(1-\|x\|^2) f(x)] \, \Delta[(1-\|x\|^2) g(x)]\,dx}, $$ where $\mathscr{L}(f,g) = \int_{\mathbf{S}^{d-1}}f(\xi)\,g(\xi)\,d\sigma(\xi)$ or $\mathscr{L}(f,g) = f(0) g(0)$, $\lambda >0$, $\sigma$ denotes the surface measure on the unit sphere $\mathbf{S}^{d-1}$, 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 $L^2(\mathbf{B}^d)$ space.