The generalization of bivariate MKZ operators by multiple generating functions (Q879153)

In \textit{J. Altin}, \textit{O. Doğru} and \textit{F. Taşdelen} [J. Math. Anal. Appl. 312, No. 1, 181--194 (2005; Zbl 1076.41008)] it is introduced for functions \(f: [0,1)\to \mathbb{R}\) a sequence of linear positive operators defined by \[ L_n(f, x)={1\over h_n(x, t)} \sum^\infty_{k=0} f\Biggl({a_{k,n}\over a_{k,n}+ b_n}\Biggr) c_{k,n}(t) x^k,\quad x\in [0,1), \] where \(0\leq{a_{k,n}\over a_{k,n}+ b_n}\leq A\), \(A\in (0,1)\), and \(\{h_n(x, t)\}\) is the generating functions for the sequence of functions \(\{c_{k,n}(t)\}\) in the form \[ h_n(x, t)= \sum^\infty_{k=0} c_{k,n}(t) x^k \] for all \(t\in I\) (\(I\) is an arbitrary subinterval of \(\mathbb{R})\). In the present paper is considered the bivariate version of the operators \(L_n\). Some Korovkin-type approximation properties of these bivariate operators are established and applications to partial differential equations are given.