Best polynomial approximation on the triangle

TL;DR

This paper establishes sharp estimates for the best polynomial approximation error on a triangle with weighted measures, linking it to derivatives of the function and Sobolev spaces.

Contribution

It provides a new sharp estimate for polynomial approximation errors on a triangle, connecting them to derivatives and Sobolev space characterizations.

Findings

Sharp estimate of approximation error in terms of derivatives

Characterization via weighted K-functional

Extension to Sobolev space approximation

Abstract

Let $E_n(f)_{\alpha,\beta,\gamma}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_{\alpha,\beta,\gamma})$ on the triangle $\{(x,y): x, y \ge 0, x+y \le 1\}$, where $\varpi_{\alpha,\beta,\gamma}(x,y) := x^\alpha y ^\beta (1-x-y)^\gamma$ for $\alpha,\beta,\gamma > -1$. Our main result gives a sharp estimate of $E_n(f)_{\alpha,\beta,\gamma}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. The result also leads to a characterization of $E_n(f)_{\alpha,\beta,\gamma}$ by a weighted $K$-functional.