Best polynomial approximation on the triangle

Abstract

Let En(f)α,β,γ denote the error of best approximation by polynomials of degree at most n in the space L2(ϖα,β,γ) on the triangle {(x,y):x,y≥0,x+y≤1}, where ϖα,β,γ(x,y)≔xαyβ(1−x−y)γ for α,β,γ>−1. Our main result gives a sharp estimate of En(f)α,β,γ 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 En(f)α,β,γ by a weighted K-functional.