Title: 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.