# On the approximation by convolution type double singular integral operators

## TL;DR

This paper establishes pointwise convergence and rates for a family of two-dimensional singular integral operators with flexible kernel functions, expanding the theoretical understanding of their approximation properties without restrictive kernel conditions.

## Contribution

It proves convergence and rate results for convolution-type double singular integral operators with general kernels, not requiring evenness, positivity, or periodicity, which broadens previous theoretical frameworks.

## Key findings

- Proved pointwise convergence of the operators.
- Established rates of convergence under general conditions.
- Provided an example illustrating the theoretical results.

## 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_{\lambda }\left( f;x,y\right) =\underset{D}{\iint }f\left( t,s\right) K_{\lambda }\left( t-x,s-y\right) dsdt,\text{ }\left( x,y\right) \in D, \end{equation*} where $D=\left \langle a,b\right \rangle \times \left \langle c,d\right \rangle $ is an arbitrary closed, semi-closed or open rectangle in $\mathbb{R}^{2}$ and $% \lambda \in \Lambda ,$ $\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.