A note on nonlinear singular integral operators depending on two parameters

In this paper, the pointwise approximation of nonlinear singular integral operators of the form:% begin{equation*} T_{lambda }left( f;xright) =underset{D}{int }K_{lambda }left( t-x;f(t)right) dt,text{ }xin D,text{ }lambda in Lambda end{equation*}% where$ D=$ is open, semi-open or closed arbitrary bounded interval in $% %TCIMACRO{U{211d} }% %BeginExpansion mathbb{R} %EndExpansion $ or $D=% %TCIMACRO{U{211d} }% %BeginExpansion mathbb{R} %EndExpansion $, $Lambda neq emptyset $ be the set of indices, at a common continuity point and Lebesgue point of the functions $fin L_{1,varphi }(D)$ and $% varphi in L_{1}(D).$ Here $L_{1,varphi }(D)$ is the space of all measurable functions for which $left vert frac{f}{varphi }right vert $ is integrable on $D.$ Also we investigate the rate of convergence at this point.