On weighted statistical convergence based on \((p,q)\)-integers and related approximation theorems for functions of two variables (Q302021)

Let \(0<q<p\leq 1\) and \(m\in \mathbb{N}\). The operator \(\Delta_{p,q}^{[m]}\) is defined by NEWLINE\[NEWLINE \Delta_{p,q}^{[m]}(x_k)=\sum_{i=0}^{m}(-1)^{i}\begin{bmatrix} m \\ n \end{bmatrix}_{p,q}x_{k-i}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\begin{bmatrix} m \\ n \end{bmatrix}_{p,q}=\frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}NEWLINE\]NEWLINE for all \(n,k\in \mathbb{N}\) with \(n\geq k\) and NEWLINE\[NEWLINE[0]_{p,q}!=1, \, [n]_{p,q}!=[1]_{p,q}[2]_{p,q}\dots [n]_{p,q}, \, n\geq 1.NEWLINE\]NEWLINE The author presents the sum \(\Lambda_{p,q}^{[m]}(x_n)\) by using the difference operator \(\Delta_{p,q}^{[m]}\) of natural order \(m\) with respect to \((p, q)\)-integers. The concepts of \(\Lambda_{p,q}^{[m]}\)-statistical convergence, statistical \(\Lambda_{p,q}^{[m]}\)-summability and strong \(\Lambda_{p,q}^{[m]}\)-summability of order \(\gamma\) are introduced. Moreover, Korovkin and Voronovskaja type approximation results for functions of two variables are obtained.