Approximation Order for Multivariate Durrmeyer Operators with Jacobi Weights

and Applied Analysis 3 Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are many paperswhich studied theirproperties 1–7 . In 1991, Zhang studied the characterization of convergence forMn,1 f ;x with Jacobi weights. In 1992, Zhou 5 considered multivariate Bernstein-Durrmeyer operators Mn,d f ;x and obtained a characterization of convergence. In 2002, Xuan et al. studied the equivalent characterization of convergence for Mn,d f ;x with Jacobi weights and obtained the following result. Theorem 1.1. For ωf ∈ L S , 0 < r < 1, the following results are equivalent: i ‖ω Mn,df − f ‖p O n−r ; ii K2 φ f, t ω O t r . In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper bound and lower bound of approximation function by Mn,d f ;x on simplex. The main results are as follows. Theorem 1.2. If ωf ∈ L S , then ∥∥ω Mn,df − f ∥∥p ≤ C { ω2 φ ( f, 1 √ n )