The Baskakov operators for functions of two variables (Q1566951)

The sequences \((A_{m,n})_{m,n \in\mathbb{N}_0}\) and \((B_{m,n})_{m,n \in\mathbb{N}_0}\) 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 \(w_{p,q}(x,y): =(1+x^p)^{-1} (1+y^q)^{-1}\), \(p,q\in\mathbb{N}_0\). 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).