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The Baskakov operators for functions of two variables. (English) Zbl 0955.41018
The sequences $(A_{m,n})_{m,n \in\bbfN_0}$ and $(B_{m,n})_{m,n \in\bbfN_0}$ of bivariate (tensor product) Baskakov and Baskakov-Kantorovič operators are known to form a pointwise approximation process on spaces of continuous functions $f$ in two variables with $w_{p,q}f$ uniformly continuous and bounded on $[0,\infty) \times [0,\infty)$, where $w_{p,q}(x,y): =(1+x^p)^{-1} (1+y^q)^{-1}$, $p,q\in\bbfN_0$. Assuming additionally $C^2$-smoothness and $C^1$-smoothness respectively the authors prove a pointwise Voronovskaja type result (Theorem 3) and the pointwise convergence of the partial derivatives of $A_{m,n}f$ and $B_{m,n}f$ to the corresponding partial derivative of $f$ (Theorem 4).

41A36Approximation by positive operators
41A35Approximation by operators (in particular, by integral operators)
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