We supplement recent results on a class of Bernstein-Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein-Durrmeyer operators.
In the present note we prove general asymptotic and Voronovskaya theorems for simultaneous approximation. These generalize the Voronovskaya type theorems obtained recently by Floater for the Bernstein operators, and previously by Heilmann and Müller for the Durrmeyer operators.Mathematics Subject Classification (2010). Primary 41A28; Secondary 41A36, 41A35, 41A10.
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