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 ((omega) over bar (alpha,beta,gamma)) on the triangle {(x, y) : x, y >= 0, x + y <= 1}, where (omega) over bar (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. (C) 2019 Elsevier Inc. All rights reserved.