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A study on pointwise approximation by double singular integral operators

Journal of Inequalities and Applications20152015:94

https://doi.org/10.1186/s13660-015-0615-6

Received: 28 October 2014

Accepted: 27 February 2015

Published: 7 March 2015

Abstract

In the present work we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators with radial kernel in two-dimensional setting in the following form: \(L_{\lambda} ( f;x,y ) =\iint_{D}f ( t,s ) H_{\lambda} ( t-x,s-y ) \,dt\,ds\), \(( x,y ) \in D\), where \(D= \langle a,b \rangle\times \langle c,d \rangle\) is an arbitrary closed, semi-closed or open region in \(\mathbb{R}^{2}\) and \(\lambda\in\Lambda\), Λ is a set of non-negative numbers with accumulation point \(\lambda_{0}\). Also we provide an example to justify the theoretical results.

Keywords

μ-generalized Lebesgue pointradial kernelrate of convergencebimonotonicitybounded bivariation

MSC

41A3541A25

1 Introduction

Taberski [1] analyzed both the pointwise convergence of functions in \(L_{1} ( -\pi,\pi ) \), where \(L_{1} ( -\pi,\pi ) \) is the collection of all measurable functions f for which \(\vert f\vert \) is integrable on \(( -\pi,\pi ) \) and the approximation properties of their derivatives by a two parameter family of convolution type singular integral operators \(U_{\lambda} ( f;x ) \) of the form
$$ U_{\lambda} ( f;x ) = \int^{\pi}_{-\pi}f( t ) K_{\lambda} ( t-x )\,dt, \quad x\in ( -\pi ,\pi ) . $$
(1.1)
Here, \(K_{\lambda} ( t ) \) denotes a kernel fulfilling appropriate conditions with \(\lambda\in\Lambda\), where Λ is a given set of non-negative numbers with accumulation point \(\lambda_{0}\). Following this work, Gadjiev [2] proved the pointwise convergence of operators of type (1.1) at a generalized Lebesgue point and established the pertinent convergence order. Rydzewska [3] extended these results to approximation at a μ-generalized Lebesgue point. Karsli and Ibikli [4, 5] proceeded to the study of the more general integral operators defined by
$$ T_{\lambda} ( f;x ) =\int^{b}_{a}f ( t ) K_{\lambda} ( t-x )\,dt,\quad x\in \langle a,b \rangle,\lambda\in\Lambda\in \mathbb{R}, $$
(1.2)
with functions in \(L_{1} \langle a,b \rangle\) where \(\langle a,b \rangle\) is an arbitrary interval in such as \([ a,b ] \), \(( a,b ) \), \([ a,b ) \) or \(( a,b ] \).

The convergence of the other operators have been studied at characteristic points such as a generalized Lebesgue point, m-Lebesgue point, and so on, by other workers: a family of nonlinear singular integral operators [6, 7], a family of nonlinear m-singular integral operators [8], Fejer-Type singular integrals [9], moment type operators [10], a family of nonlinear Mellin type convolution operators [11], nonlinear integral operators with homogeneous kernels [12] and a family of Mellin type nonlinear m-singular integral operators [13].

Taberski [14] stepped up his analysis to two-dimensional singular integrals of the form
$$ T_{\lambda} ( f;x,y ) =\iint_{Q}f ( t,s ) K_{\lambda} ( t-x,s-y ) \,dt\,ds,\quad ( x,y ) \in Q, $$
(1.3)
where Q denotes a given rectangle. His findings were later used by Siudut [15, 16] rendering significant results. Yilmaz et al. [17] replaced \(K_{\lambda}\) in (1.3) by a radial function \(H_{\lambda}\) as follows:
$$ L_{\lambda} ( f;x,y ) =\int_{-\pi}^{\pi} \int_{-\pi}^{\pi}f ( t,s ) H_{\lambda } ( t-x,s-y ) \,dt\,ds, \quad( x,y ) \in \langle-\pi,\pi \rangle\times \langle-\pi,\pi \rangle. $$
(1.4)
The new operator approaches \(f ( x_{0},y_{0} ) \) as \(( x,y,\lambda ) \) tends to \(( x_{0},y_{0},\lambda_{0} ) \). In [18], the function \(f\in L_{1} ( \langle-\pi,\pi \rangle\times \langle-\pi,\pi \rangle ) \) became \(f\in L_{p} ( D ) \) where \(D= \langle a,b \rangle \times \langle c,d \rangle\) is an arbitrary closed, semi-closed or open region in \(\mathbb{R}^{2}\).
The current manuscript presents a continuation and further generalization of [18]. The main purpose is to investigate the pointwise convergence and the rate of convergence of the operators in the following form:
$$ L_{\lambda} ( f;x,y ) =\iint_{D}f ( t,s ) H_{\lambda} ( t-x,s-y ) \,ds\,dt,\quad ( x,y ) \in D, $$
(1.5)
where \(D= \langle a,b \rangle\times \langle c,d \rangle\) is an arbitrary closed, semi-closed or open region in \(\mathbb{R}^{2}\), at a μ-generalized Lebesgue point of \(f\in L_{1} ( D ) \) as \(( x,y,\lambda ) \rightarrow ( x_{0},y_{0},\lambda_{0} ) \). Here \(L_{1} ( D ) \) is the collection of all measurable functions f for which \(\vert f\vert \) is integrable on D and the kernel function \(H_{\lambda} ( s,t ) \) is a radial function. As concerns the study of linear singular operators in several settings, the reader may see also e.g. [1923].

The paper is organized as follows: In Section 2, we introduce the fundamental definitions. In Section 3, we give a theorem concerning the existence of the operator of type (1.5). In Section 4, we prove two theorems about the pointwise convergence of \(L_{\lambda } ( f;x,y ) \) to \(f ( x_{0},y_{0} ) \) whenever \(( x_{0},y_{0} ) \) is a μ-generalized Lebesgue point of f in bounded region and unbounded region. In Section 5, we establish the rate of convergence of operators of type (1.5) to \(f ( x_{0},y_{0} ) \) as \(( x,y,\lambda ) \) tends to \(( x_{0},y_{0},\lambda_{0} ) \) and the paper is ended with an example to support our results.

2 Preliminaries

In this section we introduce the main definitions used in this paper.

Definition 1

A function \(H\in L_{1} ( \mathbb{R}^{2} ) \) is said to be radial, if there exists a function \(K:\mathbb{R} _{0}^{+}\rightarrow \mathbb{R} \) such that \(H ( t,s ) =K ( \sqrt{t^{2}+s^{2}} ) \) a.e. [24].

Definition 2

A point \(( x_{0},y_{0} ) \in D\) is called a μ-generalized Lebesgue point of function \(f\in L_{1} ( D ) \) if
$$ \lim_{ ( h,k ) \rightarrow ( 0,0 ) }\frac {1}{\mu_{1}(h)\mu_{2}(k)}\int_{0}^{h} \int_{0}^{k}\bigl\vert f ( t+x_{0},s+y_{0} ) -f ( x_{0},y_{0} ) \bigr\vert \,dt\,ds=0, $$
where \(\mu_{1}(t):\mathbb{R} \rightarrow \mathbb{R} \), absolutely continuous on \([ -\delta_{0},\delta_{0} ] \), increasing on \([ 0,\delta_{0} ] \) and \(\mu_{1}(0)=0\) and also \(\mu_{2}(s):\mathbb{R} \rightarrow \mathbb{R} \), absolutely continuous on \([ -\delta_{0},\delta_{0} ] \), increasing on \([ 0,\delta_{0} ] \) and \(\mu_{2}(0)=0\). Here \(0< h,k<\delta_{0}\) [25].

The following two examples are simple applications to a generalized Lebesgue point and μ-generalized Lebesgue point of some functions that belong to \(L_{1}(\mathbb{R} ^{2})\).

Example 1

Let \(g:\mathbb{R} ^{2}\rightarrow \mathbb{R} \) be given by
$$ g(t,s)= \left \{ \begin{array}{l@{\quad}l} 1, & \mbox{if } (t,s)=(0,0),\\ \frac{1}{\sqrt{\vert t\vert }(1+\vert t\vert )\sqrt{\vert s\vert }(1+\vert s\vert )}, & \mbox{if } (t,s)\in \mathbb{R} ^{2}\backslash(0,0). \end{array} \right . $$
Now, if \(\mu_{1}(t)=t^{\frac{1}{4}}e^{t}\) and \(\mu_{2}(s)=s^{\frac {1}{4}}e^{s}\), then the origin is a μ-generalized Lebesgue point of \(g\in L_{1}(\mathbb{R} ^{2})\) but not a generalized Lebesgue point.

Example 2

Let \(f:\mathbb{R} ^{2}\rightarrow \mathbb{R} \) be given by
$$ f(t,s)= \left \{ \begin{array}{@{}l@{\quad}l} e^{-(t+s)},& \mbox{if } (t,s)\in(0,1]\times(0,1],\\ 0, & \mbox{if }(t,s)\in \mathbb{R} ^{2} \backslash(0,1]\times(0,1].\end{array} \right . $$
If we take \(\mu_{1}(t)=t^{\frac{1}{4}+1}\) and \(\mu_{2}(s)=s^{\frac {1}{4}+1}\), then the origin is a μ-generalized Lebesgue point of \(f\in L_{1}(\mathbb{R} ^{2})\). On the other hand, if we take \(\alpha=\frac{1}{4}\) and \(p=1\), then the origin is also a generalized Lebesgue point. Clearly, this example shows that generalized Lebesgue points are also μ-generalized Lebesgue points.

Definition 3

(Class A)

Let \(H_{\lambda }:\mathbb{R}^{2}\times\Lambda\rightarrow\mathbb{R}\) be a radial function i.e., there exists a function \(K_{\lambda}:\mathbb{R} _{0}^{+}\times\Lambda\rightarrow \mathbb{R} \) such that the following equality holds for \((t,s)\in \mathbb{R} ^{2}\) a.e.:
$$ H_{\lambda}(t,s):=K_{\lambda} \bigl( \sqrt{t^{2}+s^{2}} \bigr), $$
where Λ is a given set of non-negative numbers with accumulation point \(\lambda_{0}\).
\(H_{\lambda} ( t,s ) \) belongs to class A, if the following conditions are satisfied:
  1. (a)

    \(H_{\lambda}(t,s)=K_{\lambda} ( \sqrt {t^{2}+s^{2}} ) \) is even, non-negative and integrable as a function of \((s,t)\) on \(\mathbb{R} ^{2}\) for each fixed \(\lambda\in\Lambda\).

     
  2. (b)

    For fixed \((x_{0},y_{0})\in D\), \(K_{\lambda} ( \sqrt{x_{0}^{2}+y_{0}^{2}} ) \) tends to infinity as λ tends to \(\lambda_{0}\).

     
  3. (c)

    \(\lim_{ ( x,y,\lambda ) \rightarrow ( x_{0},y_{0},\lambda_{0} ) }\iint _{\mathbb{R}^{2}}K_{\lambda} ( \sqrt{(t-x)^{2}+(s-y)^{2}} ) \,dt\,ds=1\).

     
  4. (d)

    \(\lim_{\lambda\rightarrow\lambda_{0}} [ \sup_{\xi\leq\sqrt{t^{2}+s^{2}}}K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) ] =0\), \(\forall\xi>0\).

     
  5. (e)

    \(\lim_{\lambda\rightarrow\lambda_{0}}\iint_{\xi\leq\sqrt{t^{2}+s^{2}}}K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) \,dt\,ds=0\), \(\forall\xi>0\).

     
  6. (f)

    \(K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) \) is monotonically increasing with respect to t on \((-\infty ,0]\) and similarly \(K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) \) is monotonically increasing with respect to s on \((-\infty,0]\) for any \(\lambda\in\Lambda\). Analogously, \(K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) \) is bimonotonically increasing with respect to \((t,s)\) on \([0,\infty )\times [0,\infty)\) and \((-\infty,0]\times(-\infty ,0]\) and bimonotonically decreasing with respect to \((t,s)\) on \([0,\infty)\times(-\infty,0]\) and \((-\infty,0]\times [0,\infty)\) for any \(\lambda\in\Lambda\).

     

Throughout this paper we assume that the kernel \(H_{\lambda} ( t,s ) \) belongs to class A.

3 Existence of the operator

Lemma 1

If \(f\in L_{1}(D)\), then the operator \(L_{\lambda} ( f;x,y ) \) defines a continuous transformation over \(L_{1}(D)\) [26].

4 Pointwise convergence

The following theorem gives a pointwise approximation of the integral operators of type (1.5) to the function f at a μ-generalized Lebesgue point of \(f\in L_{1}(D)\) where \(D= \langle a,b \rangle \times \langle c,d \rangle\) is a bounded region in \(\mathbb{R}^{2}\), which is closed, semi-closed or open.

Theorem 1

If \(( x_{0},y_{0} ) \) is a μ-generalized Lebesgue point of \(f\in L_{1} ( D ) \), then
$$ \lim_{ ( x,y,\lambda ) \rightarrow ( x_{0},y_{0},\lambda _{0} ) }L_{\lambda} ( f;x,y ) =f ( x_{0},y_{0} ) $$
on any set Z on which the functions
$$ \int_{x_{0}-\delta}^{x_{0}+\delta} \int _{y_{0}-\delta}^{y_{0}+\delta}K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \bigl\vert \bigl\{ \mu_{1}\bigl(\vert t-x_{0}\vert \bigr) \bigr\} _{t}^{{\prime }}\bigr\vert \bigl\vert \bigl\{ \mu_{2} \bigl(\vert s-y_{0}\vert \bigr) \bigr\} _{s}^{{\prime}} \bigr\vert \,dt\,ds $$
(4.1)
and
$$ K_{\lambda} ( 0 ) \mu_{1}\bigl(\vert x-x_{0}\vert \bigr) \quad\textit{and}\quad K_{\lambda} ( 0 ) \mu_{2}\bigl(\vert y-y_{0}\vert \bigr) $$
(4.2)
are bounded as \(( x,y,\lambda ) \) tends to \(( x_{0},y_{0},\lambda_{0} ) \).

Proof

Suppose that \(( x_{0},y_{0} ) \in D\) is a μ-generalized Lebesgue point of \(f\in L_{1} ( D ) \). Therefore, for all given \(\varepsilon>0\), there exists \(\delta>0\) such that for all h, k satisfying \(0< h,k\leq\delta\), the following inequality holds:
$$ \int_{x_{0}}^{x_{0}+\delta} \int _{y_{0}-\delta}^{y_{0}}\bigl\vert f ( t,s ) -f ( x_{0},y_{0} ) \bigr\vert \,dt\,ds< \varepsilon \mu_{1}(h)\mu_{2}(k). $$
(4.3)
If we follow the same strategy as used in the proof of Theorem 4.1 in [18], then we obtain
$$\begin{aligned} \bigl\vert L_{\lambda} ( f;x,y ) -f ( x_{0},y_{0} ) \bigr\vert \leq&\iint_{D}\bigl\vert f ( t,s ) -f ( x_{0},y_{0} ) \bigr\vert K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds\\ &{}+\bigl\vert f ( x_{0},y_{0} ) \bigr\vert \biggl\vert \iint_{\mathbb{R}^{2}}K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds-1\biggr\vert \\ &{}+\bigl\vert f ( x_{0},y_{0} ) \bigr\vert \iint_{\mathbb {R}^{2}\backslash D}K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds\\ =&I_{1}+I_{2}+I_{3}. \end{aligned}$$
In view of conditions (c) and (d) of class A, \(I_{2}\rightarrow0\), and \(I_{3}\rightarrow0\) as \(\lambda\rightarrow\lambda_{0}\), respectively,
$$\begin{aligned} I_{1} =& \biggl\{ \iint_{D\backslash B_{\delta}}+\iint _{B_{\delta}} \biggr\} \bigl\vert f ( t,s ) -f ( x_{0},y_{0} ) \bigr\vert K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds\\ =&I_{11}+I_{12}, \end{aligned}$$
where \(B_{\delta}:= \{ ( s,t ) : ( s-x_{0} ) ^{2}+ ( t-y_{0} ) ^{2}<\delta^{2}, ( x_{0},y_{0} ) \in D \} \).
Since \(K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) \) is monotonically decreasing on \(D\backslash B_{\delta}\), the inequality
$$ I_{11}\leq K_{\lambda} \bigl( (\sqrt{2}-1)\delta/\sqrt{2} \bigr) \bigl( \Vert f\Vert _{L_{1} ( D ) }+\bigl\vert f ( x_{0},y_{0} ) \bigr\vert \vert b-a\vert \vert d-c\vert \bigr) $$
holds. Hence by condition (d) of class A, \(I_{11}\rightarrow0\) as \((x,y,\lambda ) \rightarrow (x_{0},y_{0},\lambda_{0} )\).
Now, we prove that \(I_{12}\) tends to zero as \(( x,y,\lambda ) \) tends to \(( x_{0},y_{0},\lambda_{0} ) \). It is easy to see that the following inequality holds for \(I_{12}\), i.e.:
$$\begin{aligned} I_{12} \leq& \biggl\{ \int_{x_{0}}^{x_{0}+\delta}\int_{y_{0}-\delta}^{y_{0}} +\int_{x_{0}-\delta}^{x_{0}} \int_{y_{0}-\delta}^{y_{0}} \biggr\} \bigl\vert f ( t,s ) -f ( x_{0},y_{0} ) \bigr\vert K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds \\ &{}+ \biggl\{ \int_{x_{0}-\delta}^{x_{0}} \int _{y_{0}}^{y_{0}+\delta}+ \int_{x_{0}}^{x_{0}+\delta} \int_{y_{0}}^{y_{0}+\delta} \biggr\} \bigl\vert f ( t,s ) -f ( x_{0},y_{0} ) \bigr\vert K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds \\ =&I_{121}+I_{122}+I_{123}+I_{124}. \end{aligned}$$
Let us consider the integral \(I_{121}\). In view of (4.3), for every \(\varepsilon>0\) there exists \(\delta>0\) such that
$$ \int_{x_{0}}^{x_{0}+h} \int_{y_{0}-k}^{y_{0}} \bigl\vert f ( t,s ) -f ( x_{0},y_{0} ) \bigr\vert \,dt\,ds< \varepsilon\mu_{1}(h)\mu_{2}(k) $$
holds for all \(0< h,k\leq\delta\).
Let us define a new function by
$$ F ( t,s ) := \int^{t}_{x_{0}} \int _{s}^{y_{0}} \bigl\vert f ( u,v ) -f ( x_{0},y_{0} ) \bigr\vert \,du\,dv. $$
(4.4)
For all t and s satisfying \(0< t-x_{0}\leq\delta\) and \(0< y_{0}-s\leq \delta\) we have
$$ \bigl\vert F ( t,s ) \bigr\vert \leq\varepsilon\mu _{1} ( t-x_{0} ) \mu_{2} ( y_{0}-s ) . $$
(4.5)
In view of (4.4) and (4.5) and applying the method of bivariate integration by parts to \(I_{121}\) (see Theorem 2.2, p.100 in [14]) we have
$$\begin{aligned} I_{121} \leq&\varepsilon \int_{x_{0}}^{x_{0}+\delta}\int_{y_{0}-\delta}^{y_{0}}\mu_{1} ( t-x_{0} ) \mu_{2} ( y_{0}-s ) \bigl\vert dK_{\lambda } \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \bigr\vert \\ &{}+\varepsilon\mu_{2} ( \delta ) \int_{x_{0}}^{x_{0}+\delta} \mu_{1} ( t-x_{0} ) \bigl\vert dK_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( y_{0}-\delta -y ) ^{2}} \bigr) \bigr\vert \\ &{}+\varepsilon\mu_{1} ( \delta ) \int_{y_{0}-\delta}^{y_{0}} \mu_{2} ( y_{0}-s ) \bigl\vert dK_{\lambda } \bigl( \sqrt{ ( x_{0}+\delta-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \bigr\vert \\ &{}+\varepsilon\mu_{1} ( \delta ) \mu_{2} ( \delta ) K_{\lambda} \bigl( \sqrt{ ( x_{0}+\delta-x ) ^{2}+ ( y_{0}-\delta-y ) ^{2}} \bigr) . \end{aligned}$$
Let us define the variations:
$$\begin{aligned}& B_{1} ( u,v ) := \left \{ \begin{array}{@{}l@{\quad}l} \bigvee_{u}^{x_{0}+\delta-x} \bigvee_{y_{0}-\delta-y}^{v} ( K_{\lambda} ( \sqrt{t^{2}+s^{2}} ) ) , & x_{0}-x\leq u< x_{0}+\delta-x, \\ &y_{0}-\delta-y<v\leq y_{0}-y, \\ 0,& \mbox{otherwise}, \end{array} \right .\\& B_{2} ( u ) :=\left \{ \begin{array}{@{}l@{\quad}l} \bigvee_{u}^{x_{0}+\delta-x} ( K_{\lambda} ( \sqrt{t^{2}+ ( y_{0}-\delta-y ) ^{2}} ) ) ,& x_{0}-x\leq u<x_{0}+\delta-x ,\\ 0,& \mbox{otherwise}, \end{array} \right .\\& B_{3} ( v ) := \left \{ \begin{array}{@{}l@{\quad}l} \bigvee_{y_{0}-\delta-y}^{v} ( K_{\lambda } ( \sqrt{ ( x_{0}-x+\delta ) ^{2}+s^{2}} ) ) ,& y_{0}-\delta-y<v\leq y_{0}-y ,\\ 0,&\mbox{otherwise}. \end{array} \right . \end{aligned}$$
Taking the above variations into account and applying the method of bivariate integration by parts to the last inequality, we have
$$\begin{aligned} I_{121} \leq&-\varepsilon \int_{x_{0}-x}^{x_{0}-x+\delta} \int_{y_{0}-y-\delta}^{y_{0}-y} \bigl[ B_{1} ( t,s ) +B_{2} ( t ) +B_{3} ( s ) +K_{\lambda} \bigl( \sqrt{ ( x_{0}-x+\delta ) ^{2}+ ( y_{0}-\delta-y ) ^{2}} \bigr) \bigr] \\ &{}\times \bigl\{ \mu_{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime } \bigl\{ \mu_{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds \\ =&\varepsilon ( i_{1}+i_{2}+i_{3}+i_{4} ) . \end{aligned}$$

Remark 1

If the function \(g:\mathbb{R} ^{2}\rightarrow \mathbb{R} \) is bimonotonic on \([\alpha_{1},\alpha_{2}]\times{}[\beta _{1},\beta_{2}]\subset \mathbb{R} ^{2}\) then the equality given by
$$\begin{aligned} V\bigl(g;[\alpha_{1},\alpha_{2}]\times{}[ \beta_{1},\beta_{2}]\bigr) =&\bigvee _{\alpha_{1}}^{\alpha_{2}} \bigvee_{\beta_{1}}^{\beta_{2}} \bigl( g(t,s) \bigr) =\bigl\vert g(\alpha_{1},\beta_{1})-g( \alpha_{1},\beta _{2})-g(\alpha _{2}, \beta_{1})+g(\alpha_{2},\beta_{2})\bigr\vert \end{aligned}$$
holds [14, 27].
Splitting \(i_{1}\) into two parts yields
$$\begin{aligned} i_{1} =&- \biggl\{ \int_{x_{0}-x}^{x_{0}-x+\delta}\int_{0}^{y_{0}-y}+ \int_{x_{0}-x}^{x_{0}-x+\delta} \int_{y_{0}-y-\delta}^{0} \biggr\} B_{1} ( t,s ) \bigl\{ \mu_{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime} \bigl\{ \mu_{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds\\ =&i_{11}+i_{12}. \end{aligned}$$
Using Remark 1 and condition (f) of class A, we can write for \(i_{11}\)
$$\begin{aligned} i_{11} =&- \int_{x_{0}-x}^{x_{0}-x+\delta} \int _{0}^{y_{0}-y} \Biggl[ \Biggl\{ \bigvee _{t}^{x_{0}+\delta-x} \bigvee_{y_{0}-\delta-y}^{0}+\bigvee_{t}^{x_{0}+\delta-x} \bigvee _{0}^{s} \Biggr\} K_{\lambda} \bigl( \sqrt{u^{2}+ v ^{2}} \bigr) \Biggr] \\ &{}\times \bigl\{ \mu_{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime } \bigl\{ \mu_{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds \\ =&\int_{x_{0}-x}^{x_{0}-x+\delta} \int_{0}^{y_{0}-y} \bigl( K_{\lambda} \bigl( \sqrt{t^{2}+ ( y_{0}-\delta-y ) ^{2}} \bigr) -K_{\lambda} \bigl( \sqrt {s^{2}+ ( x_{0}+\delta-x ) ^{2}} \bigr) -2K_{\lambda} \bigl( \vert t\vert \bigr) \\ &{} +K_{\lambda} \bigl( \sqrt{t^{2}+s^{2}} \bigr) +2K_{\lambda } \bigl( \vert x_{0}+\delta-x\vert \bigr) -K_{\lambda} \bigl( \sqrt{ ( y_{0}-\delta-y ) ^{2}+ ( x_{0}+\delta-x ) ^{2}} \bigr) \bigr) \\ &{}\times \bigl\{ \mu_{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime } \bigl\{ \mu_{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds. \end{aligned}$$
Using the same method for \(i_{12}\), we have
$$\begin{aligned} i_{12} =&\int_{x_{0}-x}^{x_{0}-x+\delta} \int _{y_{0}-y-\delta}^{0} \bigl( K_{\lambda} \bigl( \sqrt{t^{2}+ ( y_{0}-\delta-y ) ^{2}} \bigr) +K_{\lambda} \bigl( \sqrt{s^{2}+ ( x_{0}+\delta-x ) ^{2}} \bigr) \\ &{} -K_{\lambda} \bigl( \sqrt{t^{2}+s^{2}} \bigr) -K_{\lambda } \bigl( \sqrt{ ( y_{0}-\delta-y ) ^{2}+ ( x_{0}+\delta -x ) ^{2}} \bigr) \bigr) \\ &{}\times \bigl\{ \mu_{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime } \bigl\{ \mu_{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds. \end{aligned}$$
Making similar calculations for \(i_{2}\) and \(i_{3}\) and collecting the obtained terms, we may write
$$\begin{aligned} i_{1}+i_{2}+i_{3}+i_{4} =&- \int _{x_{0}-x}^{x_{0}-x+\delta} \int_{y_{0}-y-\delta}^{0}K_{\lambda} \bigl( \sqrt{t^{2}+s^{2}} \bigr) \bigl\{ \mu_{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime} \bigl\{ \mu_{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds \\ &{}+\int_{x_{0}-x}^{x_{0}-x+\delta} \int_{0}^{y_{0}-y} \bigl( K_{\lambda} \bigl( \sqrt{t^{2}+s^{2}} \bigr) -2K_{\lambda} \bigl( \vert t\vert \bigr) \bigr)\\ &{}\times \bigl\{ \mu _{1} ( t-x_{0}+x ) \bigr\} _{t}^{\prime} \bigl\{ \mu _{2} ( y_{0}-s-y ) \bigr\} _{s}^{\prime}\,dt\,ds. \end{aligned}$$
Hence the following inequality holds for \(I_{121}\):
$$\begin{aligned} I_{121} \leq&\varepsilon \int_{x_{0}}^{x_{0}+\delta}\int_{y_{0}-\delta}^{y_{0}}K_{\lambda} \bigl( \sqrt {(t-x)^{2}+(s-y)^{2}} \bigr) \bigl\vert \bigl\{ \mu_{1} ( t-x_{0} ) \bigr\} _{t}^{\prime} \bigr\vert \bigl\vert \bigl\{ \mu _{2} ( y_{0}-s ) \bigr\} _{s}^{\prime}\bigr\vert \,dt\,ds \\ &{}+2\varepsilon K_{\lambda} ( 0 ) \mu_{1} ( \delta ) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) . \end{aligned}$$
By a similar argument to the evaluation of the integral \(I_{121}\), we can easily obtain the following inequalities for \(I_{122}\), \(I_{123}\), and \(I_{124}\):
$$\begin{aligned}& \begin{aligned}[b] I_{122} \leq{}&\varepsilon\int_{x_{0}-\delta}^{x_{0}}\int_{y_{0}-\delta}^{y_{0}}K_{\lambda} \bigl( \sqrt {(t-x)^{2}+(s-y)^{2}} \bigr) \bigl\vert \bigl\{ \mu_{1} ( x_{0}-t ) \bigr\} _{t}^{\prime} \bigr\vert \bigl\vert \bigl\{ \mu _{2} ( y_{0}-s ) \bigr\} _{s}^{\prime}\bigr\vert \,dt\,ds\\ &{}+2\varepsilon K_{\lambda} ( 0 ) \bigl( \mu_{1} ( \delta ) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) +\mu _{2} ( \delta ) \mu_{1} \bigl( \vert x_{0}-x \vert \bigr) \bigr) \\ &{}+4\varepsilon K_{\lambda} ( 0 ) \mu _{1} \bigl( \vert x_{0}-x\vert \bigr) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) , \end{aligned}\\& \begin{aligned}[b] I_{123} \leq{}&\varepsilon\int_{x_{0}-\delta}^{x_{0}}\int_{y_{0}}^{y_{0}+\delta}K_{\lambda} \bigl( \sqrt {(t-x)^{2}+(s-y)^{2}} \bigr) \bigl\vert \bigl\{ \mu_{1} ( x_{0}-t ) \bigr\} _{t}^{\prime} \bigr\vert \bigl\vert \bigl\{ \mu _{2} ( s-y_{0} ) \bigr\} _{s}^{\prime}\bigr\vert \,dt\,ds \\ &{}+2\varepsilon K_{\lambda} ( 0 ) \mu_{2} ( \delta ) \mu_{1} \bigl( \vert x_{0}-x\vert \bigr) , \end{aligned}\\& I_{124} \leq\varepsilon\int_{x_{0}}^{x_{0}+\delta}\int_{y_{0}}^{y_{0}+\delta}K_{\lambda} \bigl( \sqrt {(t-x)^{2}+(s-y)^{2}} \bigr) \bigl\vert \bigl\{ \mu_{1} ( t-x_{0} ) \bigr\} _{t}^{\prime} \bigr\vert \bigl\vert \bigl\{ \mu _{2} ( s-y_{0} ) \bigr\} _{s}^{\prime}\bigr\vert \,dt\,ds. \end{aligned}$$
Hence the following inequality is obtained for \(I_{12}\) i.e.:
$$\begin{aligned} I_{12} \leq&\varepsilon \int_{x_{0}-\delta}^{x_{0}+\delta} \int_{y_{0}-\delta}^{y_{0}+\delta}K_{\lambda } \bigl( \sqrt{(t-x)^{2}+(s-y)^{2}} \bigr) \bigl\vert \bigl\{ \mu _{1} \bigl( \vert x_{0}-t\vert \bigr) \bigr\} _{t}^{\prime }\bigr\vert \bigl\vert \bigl\{ \mu_{2} \bigl( \vert y_{0}-s\vert \bigr) \bigr\} _{s}^{\prime} \bigr\vert \,dt\,ds \\ &{}+4\varepsilon K_{\lambda} ( 0 ) \bigl( \mu_{1} ( \delta ) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) +\mu _{2} ( \delta ) \mu_{1} \bigl( \vert x_{0}-x \vert \bigr) \bigr) \\ &{}+4\varepsilon K_{\lambda} ( 0 ) \mu _{1} \bigl( \vert x_{0}-x\vert \bigr) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) . \end{aligned}$$
The remaining part of the proof is obvious by the hypotheses (4.1) and (4.2). Hence \(I_{12}\rightarrow0\) as \(\lambda\rightarrow \lambda_{0}\). Thus the proof is completed. □

The following theorem gives a pointwise approximation of the integral operators of type (1.5) to the function f at a μ-generalized Lebesgue point of \(f\in L_{1}(\mathbb{R} ^{2})\).

Theorem 2

Suppose that the hypothesis of Theorem  1 is satisfied for \(D=\mathbb{R} ^{2}\). If \(( x_{0},y_{0} ) \) is a μ-generalized Lebesgue point of \(f\in L_{1}(\mathbb{R} ^{2})\) then
$$ \lim_{ ( x,y,\lambda ) \rightarrow ( x_{0},y_{0},\lambda _{0} ) }L_{\lambda} ( f;x,y ) =f ( x_{0},y_{0} ) . $$

Proof

The proof of this theorem is quite similar to the proof of Theorem 4.2 in [18] and thus is omitted. □

5 Rate of convergence

In this section, we give a theorem concerning the rate of pointwise convergence.

Theorem 3

Suppose that the hypotheses of Theorem  1 and Theorem  2 are satisfied. Let
$$\begin{aligned} \Delta(\lambda,\delta,x,y)={}& \int_{x_{0}-\delta}^{x_{0}+\delta} \int _{y_{0}-\delta}^{y_{0}+\delta}K_{\lambda } \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \\ &{}\times\bigl\vert \bigl\{ \mu_{1}\bigl(\vert t-x_{0}\vert \bigr) \bigr\} _{t}^{{\prime}}\bigr\vert \bigl\vert \bigl\{ \mu_{2} \bigl(\vert s-y_{0}\vert \bigr) \bigr\} _{s}^{{\prime}} \bigr\vert \,dt\,ds \end{aligned}$$
for \(\delta>0\) and the following assumptions be satisfied:
  1. (i)

    \(\Delta(\lambda,\delta,x,y)\rightarrow0\) as \((x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0})\) for some \(\delta>0\).

     
  2. (ii)
    For every \(\xi>0\)
    $$ K_{\lambda}(\xi)=o\bigl(\Delta(\lambda,\delta,x,y)\bigr) $$
    as \((x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0})\).
     
  3. (iii)
    For every \(\xi>0\)
    $$ \iint_{\xi\leq\sqrt{s^{2}+t^{2}}}K_{\lambda } \bigl( \sqrt{t^{2}+s^{2}} \bigr) \,dt\,ds=o\bigl(\Delta(\lambda, \delta,x,y)\bigr) $$
    as \((x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0})\).
     
Then at each μ-generalized Lebesgue point of \(f\in L_{1}(\mathbb{R} ^{2})\) we have as \((x,y,\lambda)\rightarrow (x_{0},y_{0},\lambda _{0})\)
$$ \bigl\vert L_{\lambda} ( f;x,y ) -f ( x_{0},y_{0} ) \bigr\vert =o\bigl(\Delta(\lambda,\delta,x,y)\bigr). $$

Proof

Under the hypotheses of Theorem 1 and Theorem 2 we can write
$$\begin{aligned} &\bigl\vert L_{\lambda} ( f;x,y ) -f ( x_{0},y_{0} ) \bigr\vert \\ &\quad\leq\varepsilon\int_{x_{0}-\delta}^{x_{0}+\delta}\int _{y_{0}-\delta}^{y_{0}+\delta}K_{\lambda } \bigl( \sqrt{(t-x)^{2}+(s-y)^{2}} \bigr) \bigl\vert \bigl\{ \mu_{1} \bigl( \vert x_{0}-t\vert \bigr) \bigr\} _{t}^{\prime}\bigr\vert \bigl\vert \bigl\{ \mu_{2} \bigl( \vert y_{0}-s\vert \bigr) \bigr\} _{s}^{\prime} \bigr\vert \,dt\,ds \\ &\qquad{}+4\varepsilon K_{\lambda} ( 0 ) \bigl( \mu_{1} ( \delta ) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) +\mu _{2} ( \delta ) \mu_{1} \bigl( \vert x_{0}-x \vert \bigr) \bigr)\\ &\qquad{} +4\varepsilon K_{\lambda} ( 0 ) \mu _{1} \bigl( \vert x_{0}-x\vert \bigr) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) \\ &\qquad{}+K_{\lambda} \bigl( (\sqrt{2}-1)\delta/\sqrt{2} \bigr) \Vert f\Vert _{L_{1} ( \mathbb{R} ^{2} ) }+\bigl\vert f ( x_{0},y_{0} ) \bigr\vert \iint_{(\sqrt{2}-1)\delta/\sqrt{2}\leq\sqrt{s^{2}+t^{2}}}K_{\lambda} \bigl( \sqrt{t^{2}+s^{2}} \bigr) \,dt\,ds \\ &\qquad{}+\bigl\vert f ( x_{0},y_{0} ) \bigr\vert \biggl\vert \iint_{\mathbb{R}^{2}}K_{\lambda} \bigl( \sqrt{ ( t-x ) ^{2}+ ( s-y ) ^{2}} \bigr) \,dt\,ds-1\biggr\vert . \end{aligned}$$
From (i)-(iii) and using conditions of class A, we have the desired result i.e.,
$$ \bigl\vert L_{\lambda} ( f;x,y ) -f ( x_{0},y_{0} ) \bigr\vert =o\bigl(\Delta(\lambda,\delta,x,y)\bigr). $$
 □

Example 3

Let \(\Lambda=(0,\infty)\), \(\lambda_{0}=0\), and
$$ H_{\lambda}(t,s)=\frac{1}{4\pi\lambda}e^{\frac {-(t^{2}+s^{2})}{4\lambda}}. $$
To verify that \(H_{\lambda}(t,s)\) satisfies the hypotheses of Theorem 1 and Theorem 2 see [15].
Let \((x_{0},y_{0})=(0,0)\), \(\mu_{1}(t)=t\) and \(\mu_{2}(s)=s\). Hence we obtain
$$\begin{aligned} \Delta(\lambda,\delta,x,y) =& \int_{-\delta}^{+\delta} \int _{-\delta}^{+\delta}\frac{1}{4\pi\lambda }e^{\frac{-((t-x)^{2}+(s-y)^{2})}{4\lambda}}\,dt\,ds \\ =&\frac{1}{2} \biggl( Erf \biggl( \frac{\delta-x}{2\sqrt{\lambda }} \biggr) +Erf \biggl( \frac{x}{2\sqrt{\lambda}} \biggr) \biggr) \biggl( Erf \biggl( \frac{\delta-y}{2\sqrt{\lambda}} \biggr) +Erf \biggl( \frac{y}{2\sqrt {\lambda }} \biggr) \biggr) . \end{aligned}$$
In order to find for which \(\delta>0\) the condition (i) in Theorem 3 is satisfied, let \(\Delta(\lambda,\delta ,x,y)\rightarrow0\) as \((x,y,\lambda)\rightarrow (0,0,0)\). Hence
$$ \lim_{(x,y,\lambda)\rightarrow(0,0,0)}\Delta(\lambda ,\delta ,x,y)=0 $$
if and only if \(\delta^{2}=o(\lambda)\). Consequently, the following equality holds:
$$ \Delta(\lambda,\delta,x,y)=O(\lambda). $$
Finally, in order to get finite limit values from the expressions
$$\begin{aligned}& \lim_{(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0})}K_{\lambda} ( 0 ) \mu_{1} \bigl( \vert x_{0}-x\vert \bigr) =\lim_{(x,y,\lambda)\rightarrow(0,0,0)} \frac {1}{4\pi \lambda}e^{\frac{-(x^{2}+y^{2})}{4\lambda}} \vert x\vert , \\& \lim_{(x,y,\lambda)\rightarrow(x_{0},y_{0},\lambda_{0})}K_{\lambda} ( 0 ) \mu_{2} \bigl( \vert y_{0}-y\vert \bigr) =\lim_{(x,y,\lambda)\rightarrow(0,0,0)} \frac {1}{4\pi \lambda}e^{\frac{-(x^{2}+y^{2})}{4\lambda}} \vert y\vert , \end{aligned}$$
the rates of convergence \(\frac{1}{4\pi\lambda}e^{\frac{-(x^{2}+y^{2})}{4\lambda}}\rightarrow\infty\) and \(\vert x\vert \rightarrow0\) and also \(\frac{1}{4\pi\lambda }e^{\frac{-(x^{2}+y^{2})}{4\lambda}}\rightarrow\infty\) and \(\vert y\vert \rightarrow0\) must be equivalent. Note that \(\vert x\vert =\vert y\vert =O(\lambda)\).
Hence
$$ \bigl\vert L_{\lambda} ( f;x,y ) -f ( x_{0},y_{0} ) \bigr\vert =o \bigl( \Delta(\lambda,\delta,x,y) \bigr) =o ( \lambda ) . $$

Notes

Declarations

Acknowledgements

The authors thank the referees for their valuable comments and suggestions for the improvement of the manuscript.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Karabuk University, Karabuk, Turkey
(2)
Department of Mathematics, Faculty of Arts and Science, Gaziantep University, Gaziantep, Turkey
(3)
Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey

References

  1. Taberski, R: Singular integrals depending on two parameters. Rocznicki Polskiego towarzystwa matematycznego, Seria I. Prace matematyczne, VII (1962) Google Scholar
  2. Gadjiev, AD: The order of convergence of singular integrals which depend on two parameters. In: Special Problems of Functional Analysis and Their Applications to the Theory of Differential Equations and the Theory of Functions, pp. 40-44. Izdat. Akad. Nauk Azerbaĭdžan. SSR., Baku (1968) Google Scholar
  3. Rydzewska, B: Approximation des fonctions par des intégrales singulières ordinaires. Fasc. Math. 7, 71-81 (1973) MATHMathSciNetGoogle Scholar
  4. Karsli, H, Ibikli, E: Approximation properties of convolution type singular integral operators depending on two parameters and of their derivatives in \(L_{1}(a,b)\). In: Proc. 16th Int. Conf. Jangjeon Math. Soc., vol. 16, pp. 66-76 (2005) Google Scholar
  5. Karsli, H, Ibikli, E: On convergence of convolution type singular integral operators depending on two parameters. Fasc. Math. 38, 25-39 (2007) MATHMathSciNetGoogle Scholar
  6. Karsli, H: Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters. Appl. Anal. 85(6-7), 781-791 (2006) View ArticleMATHMathSciNetGoogle Scholar
  7. Karsli, H: On the approximation properties of a class of convolution type nonlinear singular integral operators. Georgian Math. J. 15(1), 77-86 (2008) MATHMathSciNetGoogle Scholar
  8. Karsli, H: Fatou type convergence of nonlinear m-singular integral operators. Appl. Math. Comput. 246, 221-228 (2014) View ArticleMathSciNetGoogle Scholar
  9. Bardaro, C, Gori Cocchieri, C: On the degree of approximation for a class of singular integrals. Rend. Mat. 4(4), 481-490 (1984) (in Italian) MATHMathSciNetGoogle Scholar
  10. Bardaro, C: On approximation properties for some classes of linear operators of convolution type. Atti Semin. Mat. Fis. Univ. Modena 33(2), 329-356 (1984) MATHMathSciNetGoogle Scholar
  11. Bardaro, C, Mantellini, I: Pointwise convergence theorems for nonlinear Mellin convolution operators. Int. J. Pure Appl. Math. 27(4), 431-447 (2006) MATHMathSciNetGoogle Scholar
  12. Bardaro, C, Vinti, G, Karsli, H: Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems. Appl. Anal. 90(3-4), 463-474 (2011) View ArticleMATHMathSciNetGoogle Scholar
  13. Bardaro, C, Karsli, H, Vinti, G: On pointwise convergence of Mellin type nonlinear m-singular integral operators. Commun. Appl. Nonlinear Anal. 20(2), 25-39 (2013) MATHMathSciNetGoogle Scholar
  14. Taberski, R: On double integrals and Fourier series. Ann. Pol. Math. 15, 97-115 (1964) MATHMathSciNetGoogle Scholar
  15. Siudut, S: On the convergence of double singular integrals. Comment. Math. Prace Mat. 28(1), 143-146 (1988) MATHMathSciNetGoogle Scholar
  16. Siudut, S: A theorem of Romanovski type for double singular integrals. Comment. Math. Prace Mat. 29, 277-289 (1989) MathSciNetGoogle Scholar
  17. Yilmaz, MM, Serenbay, SK, Ibikli, E: On singular integrals depending on three parameters. Appl. Math. Comput. 218(3), 1132-1135 (2011) View ArticleMATHMathSciNetGoogle Scholar
  18. Yilmaz, MM, Uysal, G, Ibikli, E: A note on rate of convergence of double singular integral operators. Adv. Differ. Equ. 2014, 287 (2014) View ArticleGoogle Scholar
  19. Angeloni, L, Vinti, G: Convergence and rate of approximation for linear integral operators in BV-φ spaces in multidimensional setting. J. Math. Anal. Appl. 349, 317-334 (2009) View ArticleMATHMathSciNetGoogle Scholar
  20. Angeloni, L, Vinti, G: Approximation with respect to Goffman-Serrin variation by means of non-convolution type integral operators. Numer. Funct. Anal. Optim. 31, 519-548 (2010) View ArticleMATHMathSciNetGoogle Scholar
  21. Bardaro, C, Karsli, H, Vinti, G: On pointwise convergence of linear integral operators with homogeneous kernel. Integral Transforms Spec. Funct. 19(6), 429-439 (2008) View ArticleMATHMathSciNetGoogle Scholar
  22. Costarelli, D, Vinti, G: Approximation by multivariate generalized sampling Kantorovich operator in the setting of Orlicz spaces. Boll. UMI 4, 445-468 (2011) MATHMathSciNetGoogle Scholar
  23. Vinti, G, Zampogni, L: A unifying approach to convergence of linear sampling type operators in Orlicz spaces. Adv. Differ. Equ. 16, 573-600 (2011) MATHMathSciNetGoogle Scholar
  24. Bochner, S, Chandrasekharan, K: Fourier Transforms. ix+219 pp. Annals of Mathematics Studies, vol. 19. Princeton University Press, Princeton (1949) MATHGoogle Scholar
  25. Serenbay, SK, Dalmanoglu, O, Ibikli, E: On convergence of singular integral operators with radial kernels. In: Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol. 83, pp. 295-308 (2014) View ArticleGoogle Scholar
  26. Yilmaz, MM: On convergence of singular integral operators depending on three parameters with radial kernels. Int. J. Math. Anal. 4(39), 1923-1928 (2010) MATHGoogle Scholar
  27. Ghorpade, SR, Limaye, BV: A Course in Multivariable Calculus and Analysis. xii+475 pp. Springer, New York (2010) View ArticleMATHGoogle Scholar

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© Uysal et al.; licensee Springer. 2015

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