Approximation by the Bernstein–Durrmeyer Operator on a Simplex

Abstract

For the Jacobi-type Bernstein–Durrmeyer operator M _n,κ on the simplex T ^d of ℝ^d, we proved that for f∈L ^p(W _κ ;T ^d) with 1<p<∞,

$$K_{2,\varPhi}\bigl(f,n^{-1}\bigr)_{\kappa,p}\leq c\|f-M_{n,\kappa}f\|_{\kappa,p}\leq c'K_{2,\varPhi}\bigl(f,n^{-1}\bigr)_{\kappa ,p}+c'n^{-1}\|f\|_{\kappa,p},$$

where W _κ denotes the usual Jacobi weight on T ^d, K _2,Φ (f,t ²)_κ,p and ‖⋅‖_κ,p denote the second-order Ditzian–Totik K-functional and the L ^p-norm with respect to the weight W _κ on T ^d, respectively, and the constants c and c′ are independent of f and n. This confirms a conjecture of Berens, Schmid, and Yuan Xu (J. Approx. Theory 68(3), 247–261, 1992). Also, a related conjecture of Ditzian (Acta Sci. Math. (Szeged) 60(1–2), 225–243, 1995; J. Math. Anal. Appl. 194(2), 548–559, 1995) was settled in our proof of this result.