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Direct and converse results for operators of Baskakov-Durrmeyer type. (English) Zbl 0669.41014

We consider the nth so-called operators of Baskakov-Durrmeyer type, which result from the classical Baskakov-type operators with weights \(p_{nk}\), if the discrete values f(k/n) are replaced by the integral terms \((n-c)\int^{\infty}_{0}p_{nk}(t)f(t)dt.\) The main difference between these operators and their classical and Kantorovich-variants respectively is that they commute. We prove direct and converse theorems also for linear combinations of the operators and results of Zamansky- Sunouchi type. As an important tool for measuring the smoothness of a function we use the Ditzian-Totik modulus of smoothness and its equivalence to appropriate K-functionals.

MSC:

41A36 Approximation by positive operators
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