Voronovskaja type theorem for some nonpositive Kantorovich type operators

Vasian, Bianca Ioana

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Issue no: Vol 40/2024 no. 1 Tags: Simultaneous approximation, Bernstein operators, Nonpositive operators, Kantorovich type operators, Voronovskaya result

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In this paper we will study a Voronovskaja type theorem and a simultaneous approximation result for a new class of generalized Bernstein operators. The new operators are obtained using a generalization of Kantorovich’s method, namely, we will introduce a sequence of operators $K_n^l=D^l\circ B_{n+l}\circ I^l$ , where $B_{n+l}$ are Bernstein operators, $D^{l}f=f^{(l)}+a_{l-1}f^{(n-1)}+\dots+a_1f'+a_0f$ is a differential operator with constant coefficients $a_j,\ j\in\{0,\dots,l-1\}$ and $I^{l}$ a corresponding antiderivative operator such that $D^{l}\circ I^{l}=Id$ .