Approximation by polynomials in Sobolev spaces with Jacobi weight

TL;DR

This paper investigates polynomial approximation in Sobolev spaces with Jacobi weights, providing sharp error estimates for simultaneous approximation of functions and derivatives, and constructing explicit approximating polynomials.

Contribution

It introduces precise error bounds and explicit polynomial constructions for simultaneous approximation in weighted Sobolev spaces with Jacobi weights.

Findings

Sharp error estimates for polynomial approximation in Sobolev spaces.

Explicit construction of polynomials achieving optimal approximation.

Extension of approximation theory to weighted Sobolev spaces with Jacobi weights.

Abstract

Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\alpha,\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha,\beta}$. This requires simultaneous approximation of a function and its consecutive derivatives up to $s$-th order with $s \le r$. We provide sharp error estimates given in terms of $E_n(f^{(r)})_{L^p(w_{\alpha,\beta})}$, the error of best approximation to $f^{(r)}$ by polynomials in $L^p(w_{\alpha,\beta})$, and an explicit construction of the polynomials that approximate simultaneously with the sharp error estimates.