Best polynomial approximation in Lw2(S) for the simplex S

Abstract This paper deals with a study of E n ( f ) L w 2 ( S ) , the rate of approximation by polynomials of total degree n in terms of the norm of L w 2 ( S ) where S is the simplex S = { ( x 1 , … , x d ) : x i ≥ 0 , x 0 = 1 − ∑ i = 1 d x i ≥ 0 } and w = w γ = x 0 γ 0 … x d γ d , γ i > − 1 . Direct and converse results are achieved relating E n ( f ) L w 2 ( S ) to the operators φ ξ r ( x ) ( ∂ ∂ ξ ) r f and corresponding K -functionals, where ξ ∈ E ( S ) and E ( S ) are the edges of S . For the special case that S is the triangle somewhat weaker results were recently achieved in Feng et al. (2019).