Approximation by the Bernstein–Durrmeyer Operator on a Simplex

For the Jacobi-type Bernstein–Durrmeyer operator Mn,κ on the simplex T d of R , we proved that for f ∈ L(Wκ ;T ) with 1 <p <∞, K2,Φ ( f,n−1 ) κ,p ≤ c‖f −Mn,κf ‖κ,p ≤ c′K2,Φ ( f,n−1 ) κ,p + c′n−1‖f ‖κ,p, where Wκ denotes the usual Jacobi weight on T d , K2,Φ(f, t)κ,p and ‖ · ‖κ,p denote the second-order Ditzian–Totik K-functional and the L-norm with respect to the weight Wκ 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.