Best polynomial approximation on the triangle

Abstract

Let $E_n(f{)}_{\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2({\varpi }_{\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3})$ on the triangle $\{(x,y):x,y\ge 0,x+y\le 1\}$, where ${\varpi }_{\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3}(x,y):=x^{\hat{I}}\pm y^2(1-x-y{)}^3$ for $\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3>-1$. Our main result gives a sharp estimate of $E_n(f{)}_{\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3}$ 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 $E_n(f{)}_{\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3}$ by a weighted $K$-functional.