Best polynomial approximation on the unit ball

TL;DR

This paper establishes new bounds for the best polynomial approximation error on the unit ball in terms of Laplacian and Laplace-Beltrami derivatives, advancing approximation theory in weighted Sobolev spaces.

Contribution

It provides novel inequalities linking polynomial approximation errors with derivatives involving Laplacian operators on the unit ball.

Findings

Derived bounds for approximation errors involving Laplacian derivatives.

Extended results to include odd order derivatives.

Improved understanding of polynomial approximation in weighted spaces.

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 $\varpi_\mu(x) = (1-\|x\|^2)^\mu$ 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.