Best polynomial approximation on the unit ball

Abstract

Let $E_n(f{)}_{\mu }$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2({\varpi }_{\mu },{\mathbb{B}}^d)$, where ${\mathbb{B}}^d$ is the unit ball in ${\mathbb{R}}^d$ and ${\varpi }_{\mu }(x)=(1-\Vert x{\Vert }^2{)}^{\mu }$ for $\mu >-1$. Our main result shows that, for $s\in \mathbb{N}$, $$E_n(f{)}_{\mu }\le c{n}^{-2s}[E_{n-2s}({\Delta }^{s}f{)}_{\mu +2s}+E_n({\Delta }_0^{s}f{)}_{\mu }],$$ where $\Delta$ and ${\Delta }_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.

Final form in IMA J. Numer. Anal

References in corpus (3)

• A Spectral Method for Elliptic Equations: The Neumann Problem
• A spectral method for elliptic equations: the Dirichlet problem
• Approximation by polynomials in Sobolev spaces with Jacobi weight