Weighted approximation by double singular integral operators with radially deﬁned kernels

In this study, we present some results on the weighted pointwise convergence of a family of singular integral operators with radial kernels given in the following form: where K is a set of non-negative numbers with accumulation point k 0 , and the function f is measurable on R 2 in the sense of Lebesgue.


Introduction
The approximation of functions by integral operators with positive definite kernels is widely used in many branches of mathematics, such as approximation theory, representation theory, theory of differential equations, Fourier analysis, and singular integral theory. Besides, it is well known that Fourier analysis is used and has many applications in medicine and engineering; more specifically, magnetic resonance imaging (MRI) and fingerprint identification are the familiar examples in those areas, respectively. Particulary, the great importance of singular integral theory, which originated in Fourier analysis, must be emphasized here. In the construction stage of Fourier series of the functions, the following integral is obtained at the end of consecutive operations, that is where K k t ð Þ denotes a kernel satisfying some conditions similar to usual approximate identities. Singular integrals consist of the different settings of the operators of type (1) with an appropriate singularity assumption on the kernel. For mentioned applications and related other applications concerning the usage of approximation theory in natural and applied sciences, the authors refer to [1][2][3][4][5][6][7][8][9][10][11].
The pointwise approximation problem may be seen as a problem of representing functions at some characteristic points, such as point of continuity, Lebesgue point, generalized Lebesgue point, and l-generalized Lebesgue point. l-generalized Lebesgue point, among others, comes to the fore. Actually, depending on the choice of the function l t ð Þ, definitions of the remaining points can be easily obtained. In practice, there are two major investigation methods related to the pointwise convergence of integral-type operators, such as operators of type (1).
The first method can be described as fixing the variable x within the operator of type (1). In other words, we pick a point in the domain of integration and it represents all other points of the same kind. Therefore, the convergence of the operator is investigated almost everywhere in the domain of integration. This method is used in many works, such as Rydzewska [12], Mamedov [13], Butzer and Nessel [14], and Uysal et al. [15].
The second method also known as Fatou-type convergence can be described as restricting the pointwise convergence to some subsets of the plane [16]. Therefore, a sensitive convergence analysis is obtained. For some studies related to Fatou-type convergence, the authors refer to [16][17][18]. Now, we summarize some of the works in which this method is harnessed.
In [19], Taberski, who indicated the importance of singular integrals in Fourier series in his works, investigated the pointwise approximation of periodic and integrable functions on Àp; p h i; where Àp; p h iis an arbitrary closed, semi-closed, or open interval. The work used a two parameter family of singular integral operators of the form: where K k : R ! R þ 0 denotes a family of periodic kernels satisfying suitable conditions, and K is a given set of nonnegative numbers with accumulation point k 0 .
Taberski [20], which gave enthusiasm to researchers, advanced his analysis to double singular integral operators of the form: iis an arbitrary closed, semiclosed, or open region, H k : R 2 ! R þ 0 stands for a family of kernels, and k 2 K is a set of non-negative numbers with accumulation point k 0 . Indicated paper also contains twodimensional generalization of well-known Natanson's lemma. Then, Siudut [21,22] presented considerable theorems by the aid of these results. Note that Rydzewska [23] also improved her previous work [12] using the results of [20], and she obtained the rate of convergence of the operators of type (3). Later on, Taberski [24] obtained the weighted pointwise approximation of some integral operators using a weight function satisfying some conditions. Moreover, this study was seen as a continuation and twodimensional analogue of [25]. In recent papers [26][27][28], the kernel functions within the operators of type (3) were defined as radial functions and the domains of integration were replaced by an arbitrary region a; b h iÂ c; d h i: As concerns the study of integral operators in several settings, the reader may also see [29][30][31][32][33][34][35]. This study is a continuation and further generalization of [26]. Besides, the current manuscript deals with Fatou-type pointwise convergence of a family of singular integral operators with radial kernels given in the following form: where H k t À x; s À y ð Þ¼K k ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðt À xÞ 2 þ ðs À yÞ 2 q Þ; and K is a set of non-negative numbers with accumulation point k 0 . Here, f 2 L 1;u ðR 2 Þ and L 1;u ðR 2 Þ are the space of all The paper is organized as follows: In Sect. 2, we introduce the fundamental definitions. In Sect. 3, we prove the pointwise convergence of L k f ; x; y ð Þ to f x 0 ; y 0 ð Þ. In Sect. 4, we establish the rate of convergence of the operators of type 4 ð Þ.

Preliminaries
In this section, we introduce the main definitions used in this paper. [36]. Now, we give another characterization of l-generalized Lebesgue point using the l-generalized Lebesgue point definition given in [23].
Definition 2 Let d 0 [ 0 be an arbitrary fixed real number. A l-generalized Lebesgue point of a locally integrable function g : 0\h\d 0 and q 1 ðtÞ is an integrable and non-negative function on 0; d 0 ½ , and similarly, l 2 ðkÞ ¼ R k 0 q 2 ðsÞds [ 0; 0\k\d 0 and q 2 ðsÞ is an integrable and non-negative function on 0; d 0 ½ : Example 1 Let f : R 2 ! R is given as follows: and u : R 2 ! R þ is given by uðt; sÞ ¼ ð1 þ t j jÞð1 þ s j jÞ. Therefore, we have Using the definition of l-generalized Lebesgue point and taking q 1 ðtÞ ¼ t we refer the reader to see [37].
0 be a family of radial kernels, which are integrable on R 2 and the weight function u : R 2 ! R þ ; which is bounded on arbitrary bounded subsets of R 2 ; satisfies the following inequality:  d. H k t; s ð Þ is monotonically increasing with respect to s on ðÀ1; 0, and similarly, H k t; s ð Þ is monotonically increasing with respect to t on ðÀ1; 0 for any k 2 K: Analogously, H k t; s ð Þ is bimonotonically increasing with respect to (t,s) on ½0; 1Þ Â ½0; 1Þ and ðÀ1; 0 Â ðÀ1; 0 and bimonotonically decreasing with respect to (t,s) on ½0; 1Þ Â ðÀ1; 0 and ðÀ1; 0 Â ½0; 1Þ for any k 2 K: e. uK k k k L 1 ðR 2 Þ M\1; for all k 2 K: f. For fixed ðt 0 ; s 0 Þ 2 R 2 ; H k t 0 ; s 0 ð Þ tends to infinity, as k tends to k 0 : Note that throughout this paper, we suppose that the function H k t; s ð Þ belongs to class A u .

Convergence at characteristic points
The following lemma gives the existence of the operators defined by (4).
Lemma 1 If f 2 L 1;u ðR 2 Þ, then the operator L k f ; x; y ð Þ defines a continuous transformation acting on L 1;u ðR 2 Þ: Proof Since L k ðf ; x; yÞ is linear, it is sufficient to show that the expression given by The following expression defines a norm in the space L 1;u ðR 2 Þ [24]. Using inequality (5) and Fubini's theorem [14], we have Thus, the proof is completed. h The following theorem gives a Fatou-type pointwise convergence of the integral operators of type (4) at lgeneralized Lebesgue point of f 2 L 1;u ðR 2 Þ: Proof Let x 0 À x j j\ d 2 and y 0 À y j j\ d 2 ; for any 0\d\d 0 : Furthermore, let 0\x 0 À x\ d 2 and 0\y 0 À y\ d 2 for any 0\d\d 0 . Since x 0 ; y 0 ð Þ2R 2 is a l-generalized Lebesgue point of function f 2 L 1;u R 2 À Á ; for all given e [ 0, there exists d [ 0, such that for all h and k satisfying 0\h; k d; we have the following inequality: Write Adding and subtracting the expression given by f x 0 ;y 0 uðt; sÞH k t À x; s À y ð Þ ds dt to the right-hand side of the above equality, we have uðt; sÞH k t À x; s À y ð Þ ds dt À u x 0 ; y 0 ð Þ Since H k is a radial function, we may write In view of condition (a) of class A u ; I 2 ! 0 as x; y; k ð Þ tends to x 0 ; y 0 ; k 0 ð Þ : The integral I 1 can be written in the form: where In view of definition of H k ; and using inequality (5), we have Now, using the initial assumptions given as 0\ x 0 À x j j\ d 2 and 0\ y 0 À y j j\ d 2 ; we may define the following set: : Taking into account the geometric representations of the sets B d and A d gives the inclusion relation R 2 nB d R 2 nC d ; where : Therefore, we have the following inequality: Consequently, by conditions (b) and (c) of class A u ; and using boundedness of u, I 11 ! 0 as x; y; k ð Þ! x 0 ; y 0 ; k 0 ð Þ : Now, we prove that I 12 tends to zero, as x; y; k ð Þtends to x 0 ; y 0 ; k 0 ð Þ . Since uðt; sÞ is bounded on B d ; it is easy to see that the following inequality In view of inequality (7), the following expression where 0\t À x 0 d and 0\y 0 À s d; holds. From Theorem 2.6 in [20], we can write where LS denotes Lebesgue-Stieltjes integral. Applying integration by parts (see Theorem 2.2, p. 100 in [20]) to the Lebesgue-Stieltjes integral, we have jdH k x 0 À x þ d; s À y ð Þ j j þ V x 0 þ d; y 0 À d ð Þ j j H k x 0 þ d À x; y 0 À d À y ð Þ : Math Sci (2016) 10:149-157 153 If we apply inequality (8) to the last inequality and make change of variables, then we have I 121 j j e Z x 0 Àx x 0 þdÀx Z y 0 ÀdÀy y 0 Ày l 1 t þ x À x 0 ð Þ l 2 y 0 À s À y ð ÞdH k t; s ð Þ j j þ el 2 ðdÞ Z x 0 Àx x 0 þdÀx l 1 t þ x À x 0 ð ÞdH k t; y 0 À d À y ð Þ j j þ el 1 ðdÞ Z y 0 ÀdÀy y 0 Ày l 2 y 0 À s À y ð ÞdH k x 0 þ d À x; s ð Þ j j þ el 1 ðdÞl 2 ðdÞH k x 0 þ d À x; y 0 À d À y ð Þ : Let us define the following variations: Þ ; x 0 À x t\x 0 þ d À x y 0 À d À y\s y 0 À y 0; otherwise: 8 > > > < > > > : ðH k u; y 0 À d À y ð Þ Þ ; x 0 À x t\x 0 þ d À x 0; otherwise:

< :
Taking above variations into account and applying bivariate integration by parts method to the last inequality, we have I 121 j j Àe Z x 0 Àx x 0 þdÀx Z y 0 ÀdÀy y 0 Ày remaining part of the proof is obvious by the hypotheses. Thus, the proof is completed. h

Rate of convergence
In this section, we give a theorem concerning the rate of pointwise convergence. ffiffi ffi k p