Best polynomial approximation on the triangle

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.