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<h1>On Fatou type convergence of higher derivatives of certain nonlinear singular integral operators</h1>
<p class="authors">
<span class="author">Harun Karsli\(^\ast \) H. Erhan Altin\(^\S \)</span>
</p>
<p class="date">June 28, 2012.</p>
</div>
<p>\(^\ast \)Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280, Golkoy Bolu, Turkey, e-mail: <span class="tt">karsli_h@ibu.edu.tr</span>. </p>
<p>\(^\S \)Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280, Golkoy Bolu, Turkey, e-mail: <span class="tt">erhanaltin@ibu.edu.tr</span>. </p>

<div class="abstract"><p> The present paper concerns with the Fatou type convergence properties of the \(r-th\) and \((r+1)-th\) derivatives of the nonlinear singular integral operators defined as </p>
<div class="displaymath" id="a0000000002">
  \[  \left( I_{\lambda }f\right) (x)=\int \limits _{a}^{b}K_{\lambda }(t-x,f(t))\, {\rm d}t,\, \, \, \, \, \, \, x\in \left( a,b\right) ,  \]
</div>
<p> acting on functions defined on an arbitrary interval \(\left( a,b\right) ,\) where the kernel \(K_{\lambda }\) satisfies some suitable assumptions. The present study is a continuation and extension of the results established in the paper <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span>. </p>
<p><b class="bf">MSC.</b> 41A35, 47G10, 47H30 </p>
<p><b class="bf">Keywords.</b> nonlinear singular integral operator, pointwise convergence, Fatou type convergence. </p>
</div>
<h1 id="a0000000003">1 Introduction</h1>
<p>Let \(\Lambda \) be a nonempty set of indices with a topology and \(\lambda _{0}\) be an accumulation point of \(\Lambda \) in this topology. By \(\mathcal{U(\theta )}\) we denote the family of all neighborhoods of the neutral element \(\theta \) of \(\mathbb {R}\), and \(x_{0}\) is a fixed accumulation point of \(\mathbb {R}.\) We take a family \(\mathcal{K}\) of functions \(K_{\lambda }:\mathbb {R}\, \)x\(\, \mathbb {R}{\rightarrow }\mathbb {R}{,}\) where \(K_{\lambda }(t,0)=0\) for all \(t\in \mathbb {R}\) and \(\lambda \in \Lambda ,\) such that \(K_{\lambda }(t,u)\) is integrable over \(\mathbb {R}\) with respect to \(t,\) in the sense of Lebesgue measure, for all values of the index \(\lambda \) and second variable \(u\). The family \(\mathcal{K}\) will be called a kernel. In addition, if the kernel function \(K_{\lambda }(t,u)\) is continuous in \(\mathbb {R}\) for every \(t\in \mathbb {R}\), then the kernel function is called Carathéodory kernel function. </p>
<p>In this paper we are concerned with the Fatou type pointwise convergence of \(r-th\) and \((r+1)-th\) derivatives of certain family of nonlinear singular integral operators \(\left( I_{\lambda }f\right) \) of the form </p>
<div class="equation" id="1">
<p>
  <div class="equation_content">
    \begin{equation}  \left( I_{\lambda }f\right) (x)=\int \limits _{a}^{b}K_{\lambda }(t-x,f(t))\, {\rm d}t,\, \, \, \, \, \, \, x\in \left( a,b\right) ,\label{1}\end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> acting on functions defined on an arbitrary interval \(\left( a,b\right) ,\) where the kernel function \(K_{\lambda }\) satisfies some suitable assumptions. In these theorems the convergence is restricted to some subsets of the plane, i.e. the Fatou type convergence is discussed, whenever the first parameter tends to an accumulation point \(x_{0}\), at which the function \(f\) has finite \(r-th\) and \((r+1)-th\) derivatives, whereas the second one tends to an accumulation point \(\lambda _{0}\) of a given index set \(\Lambda \).</p>
<p>Further results on convergence of the operators (<a href="#1">1</a>) and its linear cases can be found in <span class="cite">
	[
	<a href="#1" >1</a>
	]
</span>-<span class="cite">
	[
	<a href="#12" >12</a>
	]
</span>. </p>
<p>In particular, we obtain the rate of the Fatou type pointwise convergence for the nonlinear family of singular integral operators (<a href="#1">1</a>) to the point \(x_{0}\), at which the function \(f\) has finite \(r-th\) and \((r+1)-th\) derivatives, as \(\left( x,\lambda \right) \rightarrow \left( x_{0},\lambda _{0}\right) .\) The results presented in this paper are the continuation and extension of those established in <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span> in which the kernel function \(K_{\lambda }\) satisfies </p>
<div class="displaymath" id="a0000000004">
  \[  \left[ \tfrac {\partial }{\partial x}K_{\lambda }(t-x,u)-\tfrac {\partial }{\partial x}K_{\lambda }(t-x,v)\right] =\tfrac {\partial }{\partial x}L_{\lambda }(t-x)\left[ u-v\right]  \]
</div>
<p> for every \(t,u,v\in \) \(\mathbb {R}\) and for any \(\lambda \in \Lambda .\)</p>
<p>Throughout this paper we assume that the function \(K_{\lambda }:\mathbb {R}\) x\(\, \mathbb {R}{\rightarrow }\mathbb {R}\) satisfies the following conditions;<br />\(a\, )\) Let \(L_{\lambda }(t)\) be an integrable function such that for any fixed \(r\in \mathbb {N}\)</p>
<div class="equation" id="c">
<p>
  <div class="equation_content">
    \begin{equation}  \left[ \tfrac {\partial ^{r}}{\partial x^{r}}K_{\lambda }(t-x,u)-\tfrac {\partial ^{r}}{\partial x^{r}}K_{\lambda }(t-x,v)\right] =\tfrac {\partial ^{r}}{\partial x^{r}}L_{\lambda }(t-x)\left[ u-v\right] ,\label{c}\end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> holds for every \(t\) \(u,v\in \mathbb {R}\) and for any \(\lambda \in \Lambda .\)<br />\(b)\) \(\, {\lim \limits _{\lambda \rightarrow \lambda _{0}}\, }\displaystyle {\int \nolimits _{\mathbb {R}\, \backslash \, U}L_{\lambda }(t){\rm d}t}=0, \)for every \(U\in \mathcal{U(}0\mathcal{)}. \)<br />\(\, c\, )\, {\lim \limits _{\lambda \rightarrow \lambda _{0}}}\left[ \sup \limits _{\left\vert t\right\vert \geq \delta }L_{\lambda }(t)\right] {=0,\, \, }\)for every\({\, \, \delta {\gt}0. }\)<br />\(d\, )\, {\lim \limits _{\lambda \rightarrow \lambda _{0}}}\displaystyle {\int \nolimits _{\mathbb {R}}L_{\lambda }(t)\, {\rm d}t}=1.\)<br />According to \(a)\) it is easy to see that \(K_{\lambda }:\mathbb {R}\) x\(\, \mathbb {R}{\rightarrow }\mathbb {R}\) is a kernel function.<br />We introduce a function \(\overset {\sim }{f}\)\(\in L_{1}(\mathbb {R})\) as </p>
<div class="equation" id="2">
<p>
  <div class="equation_content">
    \begin{equation}  \overset {\sim }{f}(t):=\left\{  \begin{array}[c]{lll}f(t), &  t\in \left( a,b\right), \\ \, 0, &  t\notin \left( a,b\right), \end{array} \right. \label{2}\end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> (See <span class="cite">
	[
	<a href="#6" >6</a>
	]
</span>-<span class="cite">
	[
	<a href="#9" >9</a>
	]
</span>). </p>
<p><div class="theorem_thmwrapper " id="a0000000005">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p><span class="cite">
	[
	<a href="#3" >3</a>
	]
</span> Let \(1\leq p{\lt}\infty \, \)and assume that \(K_{\lambda }(t,u)\) is a Carathéodory kernel function. If \(f\in L_{p}\left( a,b\right) \), then \(\left( I_{\lambda }f\right) \in L_{p}\left( a,b\right) \) and </p>
<div class="displaymath" id="a0000000006">
  \[  \left\Vert I_{\lambda }f\right\Vert _{L_{p}\left( a,b\right) }\leq H(\lambda )\left\Vert f\right\Vert _{L_{p}\left( a,b\right) } \]
</div>
<p> for every \(\lambda \in \Lambda .\) </p>
<p>This kind of existence theorem is also valid in general functional spaces (see e.g. <span class="cite">
	[
	<a href="#3" >3</a>
	]
</span>). </p>

  </div>
</div> </p>
<h1 id="a0000000007">2 Convergence of the derivatives</h1>
<p>Let us define, for any constants \(C_{\nu } {\gt} 0 \left( \nu =1,2,...,r\right) ,\) the set</p>
<div class="displaymath" id="k">
  \begin{align}  D_{r}:=\left\{  (x,\lambda )\in I\text{~ }x\text{~ }\Lambda :\left\vert x-x_{0}\right\vert ^{\nu }\int \nolimits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r-\nu }\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert \, {\rm d}t{\lt}C_{\nu }\right\}  ,\label{k}\end{align}
</div>
<p> where \(I=(a,b)\) is an arbitrary interval in \(\mathbb {R}\). <br />Now, we are ready to investigate the approximation for finite \(r\)-th derivatives of the operator \(\left( I_{\lambda }f\right) \) in \(L_{1}(I)\).<b class="bfseries"></b> </p>
<p><div class="theorem_thmwrapper " id="a0000000008">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let the function \(L_{\lambda }(t)\) and its derivatives \(\tfrac {{\rm \partial }^{\nu }}{{\rm \partial } t^{\nu }}L_{\lambda }(t)\), \((\nu =1,2,...,r)\) be continuous with respect to \(t\) on \((-\infty ,\infty )\) and \(L_{\lambda }(t)\) be integrable with respect to \(t\) for each fixed \(\lambda \in \Lambda .\) Suppose that the conditions \({\rm c)}\) and \({\rm d)}\) together with </p>
<div class="equation" id="17">
<p>
  <div class="equation_content">
    \begin{equation}  \sup \limits _{\lambda \in \Lambda }\int \nolimits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r}\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert \, {\rm d}t<\infty \, \, \, \, {and\, \, \, \, \, \, \, }\lim \limits _{\lambda \rightarrow \lambda _{0}}\sup \limits _{0<\delta \leq \left\vert t\right\vert }\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert =0\label{17}\end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p> hold for every \(\delta {\gt}0,\) are satisfied. Suppose that the function \(f\in L_{1}(I)\) has at \(x_{0}\) a finite \(r-th\) derivative.<br />Then </p>
<div class="equation" id="19">
<p>
  <div class="equation_content">
    \begin{equation}  \lim \tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }f\right) (x)=f^{(r)}(x_{0})\label{19}\end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p> as \((x,\lambda )\rightarrow (x_{0},\lambda _{0})\) and \((x,\lambda )\in D_{r}.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000009">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Supposethat<b class="bfseries"> </b>\(a{\lt}x_{0}{\lt}b\) and \(0{\lt}\left\vert x_{0}-x\right\vert {\lt}\delta \) \((\delta {\gt}0).\)<br />We construct a function </p>
<div class="equation" id="20">
<p>
  <div class="equation_content">
    \begin{equation}  g(t)=f(x_{0})+(t-x_{0})f^{\prime }(x_{0})+...+\tfrac {(t-x_{0})^{r}}{r!}\, f^{(r)}(x_{0})\label{20}\end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> so that \(g^{(r)}(t)=\, f^{(r)}(x_{0})\).<br />Firstly we shall prove this theorem for the function \(g(t)\). For this purpose we introduce a function \(\overset {\sim }{g}\)\(\in L_{1}(\mathbb {R})\) as follows; </p>
<div class="equation" id="6">
<p>
  <div class="equation_content">
    \begin{equation}  \overset {\sim }{g}(t):=\left\{  \begin{array}[c]{lll}g(t), &  t\in \left( a,b\right), \\ \, \, 0\, ,&  t\notin \left( a,b\right). \end{array} \right. \label{6}\end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p> Applying the operator \(I_{\lambda }\) to the function \(g(t)\), we have </p>
<div class="displaymath" id="a0000000010">
  \[  \left( I_{\lambda }g\right) (x)=\int \limits _{a}^{b}K_{\lambda }(t-x,g(t)){\rm d}t  \]
</div>
<p> and according to \((\ref{6})\) we can rewrite the last equality as follows;</p>
<div class="equation" id="7">
<p>
  <div class="equation_content">
    \begin{equation}  \left( I_{\lambda }g\right) (x)=\int \limits _{\mathbb {R}}K_{\lambda }(t-x,\overset {\sim }{g}(t)){\rm d}t=\left( I_{\lambda }\overset {\sim }{g}\right) (x)\label{7}\end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> and hence </p>
<div class="displaymath" id="a0000000011">
  \begin{align*}  \tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }g\right) (x) &  =\tfrac {\partial ^{r}}{\partial x^{r}}\int \limits _{\mathbb {R}}K_{\lambda }(t-x,\overset {\sim }{g}(t)){\rm d}t =\int \limits _{R}\overset {\sim }{g}(t)\tfrac {\partial ^{r}}{\partial x^{r}}L_{\lambda }(t-x){\rm d}t\\ &  =(-1)^{r}\int \limits _{\mathbb {R}}\overset {\sim }{g}(t)\tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x){\rm d}t=\int \limits _{\mathbb {R}}\overset {\sim }{g}^{(r)}(t)L_{\lambda }(t-x){\rm d}t. \end{align*}
</div>
<p> In the above, substituting \((\ref{20}),\) we get </p>
<div class="displaymath" id="a0000000012">
  \begin{align*}  \left\vert \tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }g\right) (x)-f^{(r)}(x_{0})\right\vert &  =\left\vert \int \limits _{a}^{b}f^{(r)}(x_{0})L_{\lambda }(t-x){\rm d}t-f^{(r)}(x_{0})\right\vert \\ &  =\left\vert f^{(r)}(x_{0})\right\vert \left\vert \int \limits _{a}^{b}L_{\lambda }(t-x){\rm d}t-1\right\vert . \end{align*}
</div>
<p> Hence, from condition \(d)\) one has</p>
<div class="equation" id="21">
<p>
  <div class="equation_content">
    \begin{equation}  \lim \limits _{(x,\lambda )\rightarrow (x_{0},\lambda _{0})}\tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }g\right) (x)=f^{(r)}(x_{0}).\label{21}\end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> We denote \(I_{\lambda }(x):=\tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }g\right) (x)-\tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }f\right) (x).\)<br />Thanks to (<a href="#21">10</a>), for completing the proof of the theorem, it is sufficient to show that</p>
<div class="displaymath" id="a0000000013">
  \[  \lim \limits _{(x,\lambda )\rightarrow (x_{0},\lambda _{0})}\left\vert I_{\lambda }(x)\right\vert =0,  \]
</div>
<p> which gives (<a href="#19">6</a>). <br />It is easy to see that</p>
<div class="displaymath" id="a0000000014">
  \begin{align*}  \left\vert I_{\lambda }(x)\right\vert &  =\left\vert \int \limits _{\mathbb {R}}\overset {\sim }{f}(t)\tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x){\rm d}t-\int \limits _{\mathbb {R}}\overset {\sim }{g}(t)\tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x){\rm d}t\right\vert \\ &  =\left\vert \int \limits _{a}^{b}\left[ f(t)-g(t)\right] \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x){\rm }{\rm d}t\right\vert \leq \int \limits _{a}^{x_{0}-\delta }\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t\\ &  \quad \! +\! \int \limits _{x_{0}\! -\! \delta }^{x_{0}+\delta }\left\vert f(t)\! -\! g(t)\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t\! +\! \int \limits _{x_{0}\! +\! \delta }^{b}\left\vert f(t)\! -\! g(t)\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t\\ &  \  =:I_{1}(x,\lambda )+I_{2}(x,\lambda )+I_{3}(x,\lambda ). \end{align*}
</div>
<p> To estimate \(I_{1}(x,\lambda )\) and \(I_{3}(x,\lambda ),\) we use the following method:</p>
<div class="displaymath" id="a0000000015">
  \begin{align*}  I_{1}(x,\lambda ) &  =\int \limits _{a}^{x_{0}-\delta }\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t\\ &  \leq \sup \limits _{0{\lt}\delta \leq \left\vert t\right\vert }\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert \int \limits _{a}^{b}\left\vert f(t)-g(t)\right\vert {\rm d}t \end{align*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000016">
  \begin{align*}  I_{3}(x,\lambda ) &  =\int \limits _{x_{0}+\delta }^{b}\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t\\ &  \leq \sup \limits _{0{\lt}\delta \leq \left\vert t\right\vert }\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert \int \limits _{a}^{b}\left\vert f(t)-g(t)\right\vert {\rm d}t. \end{align*}
</div>
<p> Now we shall rewrite \(I_{2}(x,\lambda )\) as follows, </p>
<div class="displaymath" id="a0000000017">
  \[  I_{2}(x,\lambda )=\int \limits _{x_{0}-\delta }^{x_{0}+\delta }\left\vert \tfrac {f(t)-g(t)}{\left( t-x_{0}\right) ^{r}}\right\vert \left\vert \left( t-x_{0}\right) ^{r}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t.  \]
</div>
<p> For every \(\varepsilon {\gt}0\) there exists a \(\delta {\gt}0\) such that; </p>
<div class="displaymath" id="a0000000018">
  \[  I_{2}(x,\lambda )\leq \varepsilon \int \limits _{x_{0}-\delta }^{x_{0}+\delta }\left\vert \left( t-x_{0}\right) ^{r}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert {\rm d}t.  \]
</div>
<p> Let</p>
<div class="displaymath" id="a0000000019">
  \[  I_{2,1}(x,\lambda ):\! =\! \int \limits _{x_{0}-\delta }^{x_{0}\! +\! \delta }\left\vert \left( t\! -\! x_{0}\right) ^{r}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t\! -\! x)\right\vert {\rm d}t\! =\! \int \limits _{x_{0}\! -\! x\! -\! \delta }^{x_{0}\! -\! x\! +\! \delta }\left\vert \left( t+x-x_{0}\right) ^{r}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t.  \]
</div>
<p> It can be rewrite \(I_{2,1}(x,\lambda )\) as follows:</p>
<div class="displaymath" id="a0000000020">
  \begin{align*}  I_{2,1}(x,\lambda ) &  =\int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert \left( t+x-x_{0}\right) ^{r}-t^{r}+t^{r}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t\\ &  \leq \int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert \left( t+x-x_{0}\right) ^{r}-t^{r}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t+\int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r}\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t\\ &  =:I_{2,1,1}(x,\lambda )+I_{2,1,2}(x,\lambda ). \end{align*}
</div>
<p> From \((\ref{17})\) \(I_{2,1,2}(x,\lambda )\) is finite. So it is sufficient to show that \(I_{2,1,1}(x,\lambda )\) is finite.<br />Using the obvious identity </p>
<div class="equation" id="26">
<p>
  <div class="equation_content">
    \begin{equation}  a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})\label{26}\end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p> yields </p>
<div class="displaymath" id="a0000000021">
  \begin{align}  I_{2,1,1}(x,\lambda ) &  =\left\vert x-x_{0}\right\vert \int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert \left( t+x-x_{0}\right) ^{r-1}+...+t^{r-1}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t\nonumber \\ &  \leq \left\vert x-x_{0}\right\vert \int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert \left( t+x-x_{0}\right) ^{r-1}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t\nonumber \\ &  \quad +\left\vert x-x_{0}\right\vert \int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert \left( t+x-x_{0}\right) ^{r-2}t\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t\nonumber \\ &  \quad \vdots \label{27}\\ &  \quad +\left\vert x-x_{0}\right\vert \int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert \left( t+x-x_{0}\right) t^{r-2}\right\vert \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t\nonumber \\ &  \quad +\left\vert x-x_{0}\right\vert \int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r-1}\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert {\rm d}t.\nonumber \end{align}
</div>
<p> Applying the formula (<a href="#26">11</a>) successively to the right-hand side of (<a href="#27">12</a>), we can see that \(I_{2,1,1}(x,\lambda )\) is less than or equal to the linear combinations of </p>
<div class="displaymath" id="a0000000022">
  \[  \left\vert x-x_{0}\right\vert ^{\nu }\int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r-\nu }\, \left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t)\right\vert \, {\rm d}t,\, \, \, \, \, \, \, \, \, \, (\nu =1,2,...,r).  \]
</div>
<p> Taking into account \((\ref{17})\) and \((\ref{k}),\) we show that \(I_{2,1}(x,\lambda )\) is finite\(\, \)on any planar set \(D_{r}\).<br />Hence </p>
<div class="displaymath" id="a0000000023">
  \begin{align*}  \left\vert I_{\lambda }(x)\right\vert &  \leq I_{1}(x,\lambda )+I_{2}(x,\lambda )+I_{3}(x,\lambda )\\ &  \leq \varepsilon I_{2}(x,\lambda )+2\sup \limits _{0{\lt}\delta \leq \left\vert t\right\vert }\left\vert \tfrac {\partial ^{r}}{\partial t^{r}}L_{\lambda }(t-x)\right\vert \int \limits _{a}^{b}\left\vert f(t)-g(t)\right\vert {\rm d}t. \end{align*}
</div>
<p> Since \(f(t)-g(t)\) belongs to \(L_{1}(a,b)\), in view of \((\ref{17}),(\ref{k})\) and the condition \(c),\) we obtain </p>
<div class="displaymath" id="a0000000024">
  \[  \lim \limits _{(x,\lambda )\rightarrow (x_{0},\lambda _{0})}\left\vert I_{\lambda }(x)\right\vert =0,  \]
</div>
<p> i.e., </p>
<div class="displaymath" id="a0000000025">
  \[  \lim \limits _{(x,\lambda )\rightarrow (x_{0},\lambda _{0})}\tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }f\right) (x)=\lim \limits _{(x,\lambda )\rightarrow (x_{0},\lambda _{0})}\tfrac {\partial ^{r}}{\partial x^{r}}\left( I_{\lambda }g\right) (x).  \]
</div>
<p> This completes the proof of the Theorem. <div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Secondly, we will investigate the approximation for finite \((r+1)-th\) derivatives of the operator \(\left( I_{\lambda }f\right) \) in \(L_{1}(I)\). </p>
<p><div class="theorem_thmwrapper " id="a0000000027">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let the function \(L_{\lambda }(t)\) and its derivatives \(\tfrac {\partial ^{\nu }}{\partial t^{\nu }}L_{\lambda }(t)\), \((\nu =1,2,...,r,r+1)\) be continuous with respect to \(t\) on \((-\infty ,\infty )\) and \(L_{\lambda }(t)\) be integrable with respect to \(t\) for each fixed \(\lambda \in \Lambda .\) Suppose that conditions \({\rm c)}\) and \({\rm d)}\) are satisfied. Also we assume that the relations </p>
<div class="displaymath" id="a0000000028">
  \[  \sup \limits _{\lambda \in \Lambda }\int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r+1}\left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t)\right\vert \, {\rm d}t{\lt}\infty \, \, \, \, and{\, \, \, \, \, \, }\lim \limits _{\lambda \rightarrow \lambda _{0}}\sup \limits _{0{\lt}\delta \leq \left\vert t\right\vert }\left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t)\right\vert =0  \]
</div>
<p> hold for every \(\delta {\gt}0.\) Suppose that the function \(f\in L_{1}(I)\) has at \(x_{0}\) finite derivatives \(f_{+}^{(r+1)}(x_{0})\) and \(f_{-}^{(r+1)}(x_{0})\).<br />Then </p>
<div class="displaymath" id="a0000000029">
  \[  \lim \tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }f\right) (x)=Bf_{+}^{(r+1)}(x_{0})+(1-B)f_{-}^{(r+1)}(x_{0})  \]
</div>
<p> as \((x,\lambda )\rightarrow (x_{0},\lambda _{0})\) and \((x,\lambda )\in D_{r+1}\), where </p>
<div class="equation" id="a">
<p>
  <div class="equation_content">
    \begin{equation}  \underset {(x,\lambda )\rightarrow (x_{0},\lambda _{0})}{\lim }\int \limits _{x_{0}}^{\infty }L_{\lambda }(t-x){\rm d}t=B,\, \, \, \, \, \, \, \, \, \, \, \, 0\leq B\leq 1.\label{a}\end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000030">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let<b class="bfseries"> </b>\(a{\lt}x_{0}{\lt}b\) and \(0{\lt}x_{0}-x{\lt}\tfrac {\delta }{2}\) \((\delta {\gt}0).\) Setting</p>
<div class="displaymath" id="35">
  \begin{align}  g(t):=\left\{  \begin{array}[c]{c}f(x_{0})+...+\tfrac {(t-x_{0})^{r}}{r!}\, f^{(r)}(x_{0})+\tfrac {(t-x_{0})^{r+1}}{(r+1)!}f_{-}^{(r+1)}(x_{0}),\  \  a{\lt}t{\lt}x_{0},\\ f(x_{0})+...+\tfrac {(t-x_{0})^{r}}{r!}\, f^{(r)}(x_{0})+\tfrac {(t-x_{0})^{r+1}}{(r+1)!}f_{+}^{(r+1)}(x_{0}),\  \  x_{0}\leq t{\lt}b. \end{array} \right. \label{35}\end{align}
</div>
<p> Note that </p>
<div class="displaymath" id="a0000000031">
  \[  \left( I_{\lambda }g\right) (x)=\int \limits _{a}^{b}K_{\lambda }(t-x,g(t)){\rm d}t=\int \limits _{R}K_{\lambda }(t-x,\overset {\sim }{g}(t)){\rm d}t.  \]
</div>
<p> Differentiating both sides of the last equality \((r+1)\) times with respect to \(x\), one has</p>
<div class="displaymath" id="a0000000032">
  \[  \tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }g\right) (x)=\int \limits _{a}^{b}g^{(r+1)}(t)L_{\lambda }(t-x){\rm d}t.  \]
</div>
<p> By (<a href="#35">14</a>), the last equality can be rewritten in the form </p>
<div class="displaymath" id="a0000000033">
  \[  \tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }g\right) (x)=f_{-}^{(r+1)}(x_{0})\int \limits _{a}^{x_{0}}L_{\lambda }(t-x){\rm d}t+f_{+}^{(r+1)}(x_{0})\int \limits _{x_{0}}^{b}L_{\lambda }(t-x){\rm d}t.  \]
</div>
<p> The hypothesis (<a href="#a">13</a>) yields </p>
<div class="displaymath" id="a0000000034">
  \[  \underset {(x,\lambda )\rightarrow (x_{0},\lambda _{0})}{\lim }\tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }g\right) (x)=B\, f_{+}^{(r+1)}(x_{0})+(1-B)f_{-}^{(r+1)}(x_{0}).  \]
</div>
<p> Define</p>
<div class="displaymath" id="a0000000035">
  \[  \left\vert I_{\lambda }(x)\right\vert :=\left\vert \tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }g\right) (x)-\tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }f\right) (x)\right\vert .  \]
</div>
<p> In order to complete the proof of the theorem, it is sufficient to show</p>
<div class="displaymath" id="a0000000036">
  \[  \underset {(x,\lambda )\rightarrow (x_{0},\lambda _{0})}{\lim }\left\vert I_{\lambda }(x)\right\vert =0.  \]
</div>
<p> One has</p>
<div class="displaymath" id="a0000000037">
  \begin{align}  \left\vert I_{\lambda }(x)\right\vert &  =\left\vert \int \limits _{a}^{b}\left[ f(t)-g(t)\right] \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x){\rm d}t\right\vert \nonumber \\ &  \leq \int \limits _{a}^{x_{0}-\delta }\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \quad +\int \limits _{x_{0}-\delta }^{x_{0}}\left\vert \tfrac {f(t)-g(t)}{\left( t-x_{0}\right) ^{r+1}}\right\vert \left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \quad +\int \limits _{x_{0}+\delta }^{b}\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \quad +\int \limits _{x_{0}}^{x_{0}+\delta }\left\vert \tfrac {f(t)-g(t)}{\left( t-x_{0}\right) ^{r+1}}\right\vert \left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ & =:I_{1}(x,\lambda )+I_{2}(x,\lambda )+I_{3}(x,\lambda )+I_{4}(x,\lambda ).\label{36}\end{align}
</div>
<p> Since \(f\) has at \(x_{0}\) a finite \((r+1)-th\) right and left derivatives, then for every \(\varepsilon {\gt}0\) there exists a \(\delta {\gt}0\) such that</p>
<div class="displaymath" id="a0000000038">
  \begin{align}  I_{2}(x,\lambda ) &  =\int \limits _{x_{0}-\delta }^{x_{0}}\left\vert \tfrac {f(t)-g(t)}{\left( t-x_{0}\right) ^{r+1}}\right\vert \left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \leq \varepsilon \int \limits _{x_{0}-\delta }^{x_{0}}\left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\label{38}\end{align}
</div>
<p> and </p>
<div class="displaymath" id="a0000000039">
  \begin{align}  I_{4}(x,\lambda ) &  =\int \limits _{x_{0}}^{x_{0}+\delta }\left\vert \tfrac {f(t)-g(t)}{\left( t-x_{0}\right) ^{r+1}}\right\vert \left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \leq \varepsilon \int \limits _{x_{0}}^{x_{0}+\delta }\left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t.\label{39}\end{align}
</div>
<p> Thus we get</p>
<div class="displaymath" id="a0000000040">
  \[  I_{2}(x,\lambda )+I_{4}(x,\lambda )\leq \varepsilon \int \limits _{x_{0}-\delta }^{x_{0}+\delta }\left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t.  \]
</div>
<p> Set</p>
<div class="displaymath" id="a0000000041">
  \[  I_{2,1}(x,\lambda )+I_{4,1}(x,\lambda ):=\int \limits _{x_{0}-\delta }^{x_{0}+\delta }\left\vert \left( t-x_{0}\right) ^{r+1}\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t.  \]
</div>
<p> Using the same method as in the proof of Theorem 2 we deduce \(I_{2,1}(x,\lambda )+I_{4,1}(x,\lambda )\) is less than or equal to a linear combination of </p>
<div class="displaymath" id="a0000000042">
  \[  \left\vert x-x_{0}\right\vert ^{\nu }\int \limits _{x_{0}-x-\delta }^{x_{0}-x+\delta }\left\vert t\right\vert ^{r+1-\nu }\left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}K(t,\lambda )\right\vert \, {\rm d}t,\, (\nu =1,...,r+1).  \]
</div>
<p> By virtue of \((\ref{k}),\) the term \(I_{2}(x,\lambda )+I_{4}(x,\lambda )\) is bounded.<br />Now, we consider the integrals \(I_{1}(x,\lambda )\) and \(I_{3}(x,\lambda ),\) respectively.</p>
<div class="displaymath" id="a0000000043">
  \begin{align}  I_{1}(x,\lambda ) &  =\int \limits _{a}^{x_{0}-\delta }\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \leq M\underset {\tfrac {\delta }{2}{\lt}\left\vert u\right\vert }{\sup }\left\vert \tfrac {\partial ^{r+1}}{\partial u^{r+1}}L_{\lambda }(u)\right\vert \label{40}\end{align}
</div>
<p> and </p>
<div class="displaymath" id="a0000000044">
  \begin{align}  I_{3}(x,\lambda ) &  =\int \limits _{x_{0}+\delta }^{b}\left\vert f(t)-g(t)\right\vert \left\vert \tfrac {\partial ^{r+1}}{\partial t^{r+1}}L_{\lambda }(t-x)\right\vert {\rm d}t\nonumber \\ &  \leq M\underset {\tfrac {\delta }{2}{\lt}\left\vert u\right\vert }{\sup }\left\vert \tfrac {\partial ^{r+1}}{\partial u^{r+1}}L_{\lambda }(u)\right\vert .\label{41}\end{align}
</div>
<p> Using (<a href="#38">16</a>), (<a href="#41">19</a>) in (<a href="#36">15</a>) one has </p>
<div class="displaymath" id="a0000000045">
  \begin{align}  \left\vert I_{\lambda }(x)\right\vert &  \leq I_{1}(x,\lambda )+I_{2}(x,\lambda )+I_{3}(x,\lambda )+I_{4}(x,\lambda )\nonumber \\ &  \leq \varepsilon \left[ I_{2,1}(x,\lambda )+I_{4,1}(x,\lambda )\right]\&  \quad +2M\underset {\tfrac {\delta }{2}{\lt}\left\vert u\right\vert }{\sup }\left\vert \tfrac {\partial ^{r+1}}{\partial u^{r+1}}L_{\lambda }(u)\right\vert .\label{42}\end{align}
</div>
<p> Under the hypotheses of the theorem, (<a href="#42">20</a>) yields</p>
<div class="displaymath" id="a0000000046">
  \[  \underset {(x,\lambda )\rightarrow (x_{0},\lambda _{0})}{\lim }\left\vert I_{\lambda }(x)\right\vert =\underset {(x,\lambda )\rightarrow (x_{0},\lambda _{0})}{\lim }\left\vert \tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }g\right) (x)-\tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }f\right) (x)\right\vert =0.  \]
</div>
<p> This completes the proof. <div class="proof_wrapper" id="a0000000047">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000048">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(B=\tfrac {1}{2}\) , then we have </p>
<div class="displaymath" id="a0000000049">
  \[  \underset {(x,\lambda )\rightarrow (x_{0},\lambda _{0})}{\lim }\tfrac {\partial ^{r+1}}{\partial x^{r+1}}\left( I_{\lambda }f\right) (x)=\tfrac {f_{+}^{(r+1)}(x_{0})+f_{-}^{(r+1)}(x_{0})}{2}.\hfil \qed  \]
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000050">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5</span>
  </div>
  <div class="remark_thmcontent">
  <p>Let \(I=(a,b)\) be an arbitrary bounded interval in \(\mathbb {R}\) and \(f\in L_{1}(a,b)\) be a \((b-a)\)-periodic function. In this case the proofs of the Theorems are similar.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000051">3 Examples</h1>
<p><div class="example_thmwrapper " id="a0000000052">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">6</span>
  </div>
  <div class="example_thmcontent">
  <p>A special case of the function \(K_{\lambda }(t,u)\) satisfying the conditions, is the linear case with respect to the second variable, i.e., </p>
<div class="displaymath" id="a0000000053">
  \[  K_{\lambda }(t,u)=Z_{\lambda }(t)\, u.  \]
</div>
<p> This case is widely used in Approximation Theory <span class="cite">
	[
	<a href="#4" >4</a>
	]
</span>.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="a0000000054">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">7</span>
  </div>
  <div class="example_thmcontent">
  <p>We introduce the function </p>
<div class="displaymath" id="a0000000055">
  \[  K_{\lambda }(t,u)=\left\{  \begin{array}[l]{lll}(r+1)\lambda ^{r+1}t^{r}\, u+u, &  t\in \lbrack 0,\tfrac {1}{\lambda }],\\ 0, &  t\notin \lbrack 0,\tfrac {1}{\lambda }], \end{array} \right.  \]
</div>
<p> where \(\Lambda =[1,\infty )\) is a set of indices with natural topology and \(\lambda _{0}=\infty \) is an accumulation point of \(\Lambda \) in this topology.<br />First of all \(K_{\lambda }(t,u)\) is a kernel, i.e., \(K_{\lambda }(t,0)=0\medskip .\)<br />It is seen that for every \(u\in \mathbb {R},\)</p>
<div class="displaymath" id="a0000000056">
  \[  \tfrac {\partial ^{r}}{\partial x^{r}}K_{\lambda }(t-x,u)=\left\{  \begin{array}[l]{lll}(-1)^{r}(r+1)!\lambda ^{r+1}\, u, &  t-x\in \lbrack 0,\tfrac {1}{\lambda }],\\ 0, &  t-x\notin \lbrack 0,\tfrac {1}{\lambda }]. \end{array} \right.  \]
</div>
<p> According to (<a href="#c">2</a>), we obtain </p>
<div class="displaymath" id="a0000000057">
  \[  \tfrac {\partial ^{r}}{\partial x^{r}}L_{\lambda }(t-x)=\left\{  \begin{array}[l]{lll}(-1)^{r}(r+1)!\lambda ^{r+1}, &  t-x\in \lbrack 0,\tfrac {1}{\lambda }],\\ 0 , &  t-x\notin \lbrack 0,\tfrac {1}{\lambda }]. \end{array} \right.  \]
</div>
<p> This implies </p>
<div class="displaymath" id="a0000000058">
  \[  L_{\lambda }(t-x)=\left\{  \begin{array}[l]{lll}(r+1)\lambda ^{r+1}(t-x)^{r}, &  t-x\in \lbrack 0,\tfrac {1}{\lambda }],\\ 0, &  t-x\notin \lbrack 0,\tfrac {1}{\lambda }]. \end{array} \right.  \]
</div>
<p> Moreover </p>
<div class="displaymath" id="a0000000059">
  \[  \int \limits _{\mathbb {R}}L_{\lambda }(t){\rm d}t=\int \limits _{[0,\tfrac {1}{\lambda }]}(r+1)\lambda ^{r+1}t^{r}{\rm d}t=1{\lt}\infty .  \]
</div>
<p> It is easy to see that </p>
<div class="displaymath" id="a0000000060">
  \[  {\lim \limits _{\lambda \rightarrow \infty }\, }\int \limits _{\mathbb {R}\, \backslash \, U}L_{\lambda }(t){\rm d}t=0,  \]
</div>
<p> for every \(U\in \mathcal{U(}0\mathcal{)}\) and </p>
<div class="displaymath" id="a0000000061">
  \[  {\lim \limits _{\lambda \rightarrow \infty }}\left[ \sup \limits _{\left\vert t\right\vert \geq \delta }L_{\lambda }(t)\right] {=0,} \]
</div>
<p> \({\, }\)for every\({\, \, \delta {\gt}0.}\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000062">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgement
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors would like to express their sincere <br />thanks to the referee and to Professor Octavian Agratini for their very careful and intensive study of this manuscript. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><i class="sc">C. Bardaro, H. Karsli</i> and <i class="sc">G. Vinti</i>, <i class="it">On pointwise convergence of linear integral operators with homogeneous kernels</i>, Integral Transforms and Special Functions, <b class="bf">19(6)</b> (2008), pp. 429–439. </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><i class="sc">C. Bardaro, H. Karsli</i> and <i class="sc">G. Vinti</i>, <i class="it">Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems</i>, Applicable Analysis, <b class="bf">90</b> (2011) No. 3–4, pp. 463–474. </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><i class="sc">C. Bardaro, J. Musielak</i> and <i class="sc">G. Vinti</i>, <i class="it">Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications</i>, <b class="bf">9</b> (2003), xii + 201 pp. </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><i class="sc">Butzer P.L.</i> and <i class="sc">R.J. Nessel</i>, <i class="it">Fourier Analysis and Approximation</i>, V.1, Academic Press, New York, London, 1971. </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><i class="sc">A.D. Gadjiev</i>, <i class="it">On convergence of integral operators depending on two parameters</i>, Dokl. Acad. Nauk. Azerb. SSR, <b class="bf">XIX</b> (1963) No. 12, pp. 3–7. </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><i class="sc">H. Karsli</i>, <i class="it">Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters</i>, Applicable Analysis, <b class="bf">85</b> (2006) No. 6-7, pp. 781–791. </p>
</dd>
  <dt><a name="7">7</a></dt>
  <dd><p><i class="sc">H. Karsli</i>, <i class="it">Convergence of the derivatives of nonlinear singular integral operators</i>, J. Math. Anal. Approx. Theory, <b class="bf">2</b> (2007) No. 1, pp. 53–61. </p>
</dd>
  <dt><a name="8">8</a></dt>
  <dd><p><i class="sc">H. Karsli</i>, <i class="it">On approximation properties of a class of convolution type nonlinear singular integral operators</i>, Georgian Math. Jour., <b class="bf">15</b> (2008), No. 1, pp. 77–86. </p>
</dd>
  <dt><a name="9">9</a></dt>
  <dd><p><i class="sc">H. Karsli</i> and <i class="sc">Gupta V.</i>, <i class="it">Rate of convergence by nonlinear integral operators for functions of bounded variation</i>, Calcolo, <b class="bf">45, (2)</b> (2008), pp. 87–99. </p>
</dd>
  <dt><a name="10">10</a></dt>
  <dd><p><i class="sc">J. Musielak</i>, <i class="it">On some approximation problems in modular spaces</i>. In Constructive Function Theory 1981 ( Proc. Int. Conf. Varna, June 1-5, 1981), pp. 455-461, Publ. House Bulgarian Acad. Sci., Sofia 1983. </p>
</dd>
  <dt><a name="11">11</a></dt>
  <dd><p><i class="sc">T. Swiderski</i> and <i class="sc">E. Wachnicki</i>, <i class="it">Nonlinear Singular Integrals depending on two parameters</i>, Commentationes Math., <b class="bf">XL</b> (2000), pp. 181–189. </p>
</dd>
  <dt><a name="12">12</a></dt>
  <dd><p><i class="sc">R. Taberski</i>, <i class="it">Singular integrals depending on two parameters, Rocznicki Polskiego towarzystwa matematycznego</i>, Seria I. Prace matematyczne, <b class="bf">VII</b> (1962), pp. 173–179. </p>
</dd>
</dl>


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