On the approximation by convolution type double singular integral operators

Abstract

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_{λ}\left( f;x,y\right) =\underset{D}{\iint }f\left( t,s\right) K_{λ}\left( t-x,s-y\right) dsdt,\text{ }\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 $\Lambda$ 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 $K_{\lambda }\left(t,s\right)$ does not have to be even, positive or 2$\pi -$periodic.