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For the Bernstein left quasi-interpolants B n ( 2 r − 1 ) f , we prove that for some k ω φ 2 r ( f , 1 n ) ≤ C ( ‖ B n ( 2 r − 1 ) f − f ‖ + ‖ B k n ( 2 r − 1 ) f − f ‖ ) . This is a strong converse inequality of type B.

For the left Bernstein-Durrmeyer quasi-interpolants $M_{n}^{(2r-1)}f$ , we prove that, for some l, $$\omega_{\varphi}^{2r}\biggl(f,\frac{1}{\sqrt{n}} \biggr)_{p}\leqslant C \bigl(\bigl\Vert M_{n}^{(2r-1)}f-f \bigr\Vert _{p}+\bigl\Vert M_{ln}^{(2r-1)}f-f\bigr\Vert _{p} \bigr) \quad (1< p\leq\infty). $$ This is a strong converse inequality of type B.

In this paper we give the direct approximation theorem, the inverse theorem, and the equivalence theorem for Szasz-Durrmeyer-Bezier operators in the space ( ) with Ditzian-Totik modulus. MSC:41A25, 41A27, 41A36.

We give the direct and inverse approximation theorems for a new type of Bernstein-Durrmeyer operators with the modulus of smoothness. MSC:41A25, 41A27, 41A36.