Best polynomial approximation on the triangle

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.