Approximation by the Bernstein-Durrmeyer operator on a simplex
The authors investigate the Jacobi-type Bernstein-Durrmeyer operator \(M_{n,k}\) on the simplex \(T^d\) of \(\mathbb{R}^d.\) They prove that for \(f\in L^p(W_k;T^d)\) with \(1<p<\infty\), \[ K_{2,\Phi}(f,n^{-1})_{k,p}\leq c\|f-M_{n,k}|_{k,p}\leq c'K_{2,\Phi}(f,n^{-1})_{k,p}+c'n^{-1}\|f\|_{k,p} \] holds, where \(W_k\) denotes the uasual Jacobi weight on \(T^d\), \(K_{2,\Phi}(f,n^{-1})_{k,p}\) and \(\|.\|_{k,p}\) denote the second-order Ditzian-Totik K-functional and the \(L^p\)-norm with respect to the weight \(W_k\) on \(T^d\), respectively, and the constants \(c\) and \(c'\) are independent of \(f\) and \(n\). This is a very good contribution in the approximation theory, all the more it confirms a conjecture of Berens et al.
scientific article; zbMATH DE number 5719079
scientific article