Best polynomial approximation on the triangle

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