On the approximation by convolution type double singular integral operators

In this paper, we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators in two-dimensional setting in the following form: \begin{equation*} L_{\lambda }\left( f;x,y\right) =\underset{D}{\iint }f\left( t,s\right) K_{\lambda }\left( t-x,s-y\right) dsdt,$$\left( x,y\right) \in D, \end{equation*} where $D=⟨a,b⟩\times ⟨c,d⟩$ is an arbitrary closed, semi-closed or open rectangle in ${\mathbb{R}}^2$ and $% \lambda \in \Lambda ,$ $\mathrm{\Lambda }$ is a set of non-negative indices with accumulation point ${\lambda }_0$. Also, we provide an example to support these theoretical results. In contrast to previous works, the kernel function $K_{\lambda }(t,s)$ does not have to be even, positive or 2$\pi -$periodic.