EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Convergence of Singular Integral Operators in Weighted Lebesgue Spaces Mine Menekse Yilmaz1, Gumrah Uysal2,∗ 1 Department of Mathematics, Faculty of Arts and Science, Gaziantep University, Gaziantep, Turkey 2 Department of Computer Technologies, Division of Technology of Information Security, Karabuk University, Karabuk, Turkey Abstract. In this paper, the pointwise approximation to functions f ∈ L1,w 〈a, b〉 by the convo- lution type singular integral operators given in the following form: Lλ (f ;x) = b∫ a f (t)Kλ (t− x) dt, x ∈ 〈a, b〉 , λ ∈ Λ ⊂ R+ 0 where 〈a, b〉 stands for arbitrary closed, semi closed or open bounded interval in R or R itself, L1,w 〈a, b〉 denotes the space of all measurable but non-integrable functions f for which ∣∣∣ fw ∣∣∣ is integrable on 〈a, b〉 and w : R→ R+ is a corresponding weight function, at a µ-generalized Lebesgue point and the rate of convergence at this point are studied. 2010 Mathematics Subject Classifications: 41A35, 41A25, 45P05 Key Words and Phrases: Generalized Lebesgue point, Weighted pointwise convergence, Rate of convergence 1. Introduction In paper [16], Taberski analyzed the pointwise convergence of integrable functions and the approximation properties of derivatives of integrable functions in L1 〈−π, π〉 by a two parameter family of convolution type singular integral operators of the form: Tλ (f ;x) = π∫ −π f (t)Kλ (t− x) dt, x ∈ 〈−π, π〉 , λ ∈ Λ ⊂ R+ 0 , (1) where Kλ (t) is the kernel fulfilling appropriate assumptions and λ ∈ Λ and Λ is a given set of non-negative numbers with accumulation point λ0. Later on, the weighted ∗Corresponding author. Email addresses: menekse@gantep.edu.tr (M. Menekse Yilmaz), guysal@karabuk.edu.tr (G. Uysal) http://www.ejpam.com 335 c© 2017 EJPAM All rights reserved. M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 336 pointwise convergence and rate of convergence of a family of m-singular integral operators in f ∈ Lp (R) were investigated by Mamedov [14]. Then, Gadjiev [9] and Rydzewska [15] studied the pointwise convergence theorems and the order of pointwise convergence theorems for operators type (1) at a generalized Lebesgue point and µ−generalized Lebesgue point of f ∈ L1 (−π, π) based on Taberski’s analysis, respectively. Further, in [10] and [12] Karsli extended the results of Taberski [16], Gadjiev [9] and Rydzewska’s [15] studies by considering the more general integral operators of the form: Tλ (f ;x) = b∫ a f (t)Kλ (t− x) dt, x ∈ 〈a, b〉 , λ ∈ Λ ⊂ R+ 0 (2) for functions in L1 〈a, b〉 , where 〈a, b〉 is an arbitrary interval in R such as [a, b] , (a, b) , [a, b) or (a, b] . In [11, 13], Karsli analyzed the pointwise convergence theorems and the rate of point- wise convergence theorems for a family of nonlinear singular integral operators at a µ−generalized Lebesgue point and at a generalized Lebesgue point of f ∈ L1 〈a, b〉, re- spectively. Bardaro and Cocchieri [2] evaluated the degree of pointwise convergence of Fejer-Type singular integrals at the generalized Lebesgue points of the functions f ∈ L1 (R). In an another study, Bardaro [3] studied similar convergence results about moment type oper- ators. Besides, the pointwise convergence of family of nonlinear Mellin type convolution operators at Lebesgue points was also analyzed by Bardaro and Mantellini [4]. Bardaro, Karsli and Vinti [5] obtained some approximation results related to the point- wise convergence and the rate of pointwise convergence for non-convolution type linear op- erators at a Lebesgue point based on Bardaro and Mantellini’s study [4]. After that, they got similar results for its nonlinear counterpart in [6] while in an another study [7], the pointwise convergence and the rate of pointwise convergence results for a family of Mellin type nonlinear m-singular integral operators at m-Lebesgue points of f were investigated. Almali [1] studied the problem of pointwise convergence of non-convolution type inte- gral operators at Lebesgue points of some classes of measurable functions. In this paper, we also investigated the pointwise convergence and the rate of conver- gence of the operators similar to the study of Karsli [12]. However, the difference between this study and Karsli [12] is that while Karsli [12] considers the pointwise convergence of the integral operators to the functions f ∈ L1 〈a, b〉, our study covers the case f /∈ L1 〈a, b〉. The main contribution of this paper is obtaining the pointwise convergence of the convolution type singular integral operators of the form: Lλ (f ;x) = b∫ a f (t)Kλ (t− x) dt, x ∈ 〈a, b〉 , λ ∈ Λ ⊂ R+ 0 , (3) where the symbol 〈a, b〉 stands for an arbitrary closed, semi closed or open bounded in- terval in R or R itself, to the function f ∈ L1,w 〈a, b〉 where L1,w 〈a, b〉 is the space of M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 337 all measurable and non-integrable functions f for which ∣∣∣ fw ∣∣∣ is integrable on 〈a, b〉 and w : R → R+ is a corresponding weight function, at a common µ-generalized Lebesgue point of f w and w. The paper is organized as follows: In Section 2, we introduce the fundamental def- initions. In Section 3, we prove the existence of the operators type (3). In Section 4, we present two theorems concerning the pointwise convergence of Lλ (f ;x, y) to f (x0, y0) whenever (x0, y0) is a common µ−generalized Lebesgue point of f w and w. In section 5, we give two theorems concerning the rate of pointwise convergence. 2. Preliminaries Definition 1. A point x0 ∈ 〈a, b〉 is called µ−generalized Lebesgue point of the function f ∈ L1 〈a, b〉 , if lim h→0 1 µ (h) h∫ 0 |f(x0 + t)− f(x0)| dt = 0, where the function µ : R → R is increasing and absolutely continuous on [0, b− a] and µ(0) = 0 [15, 12]. Example 1. Consider the function f ∈ L1(R) defined by f(t) = { e−t, if t ∈ (0, 1] 0, if t ∈ R\(0, 1]. One can compute that x0 = 0 is not a Lebesgue point of f. On the other hand, by taking µ(t) = √ t we see that x0 = 0 is a µ−generalized Lebesgue point of f. Now, we define a new class for the weighted approximation. Definition 2. (Class Aw) Let Λ ⊂ R+ 0 be an index set and λ0 is an accumulation point of it. Further, let Kλ : R→ R be an integrable function for each λ ∈ Λ and ϕ(t) = sup x∈〈a,b〉 [ w(t+x) w(x) ] , ∀t ∈ 〈a, b〉 . It is said that Kλ(t) belongs to class Aw, if it satisfies the following conditions: a. ‖Kλϕ‖L1(R) ≤M <∞, ∀λ ∈ Λ. b. lim λ→λ0 [ sup |t|>ξ |Kλ (t)| ] = 0, ∀ξ > 0. c. lim λ→λ0 [ ∫ |t|>ξ |Kλ (t)| dt ] = 0, ∀ξ > 0. d. For a given δ0 > 0, |Kλ(t)| is non-decreasing function with respect to t on 〈−δ0, 0] and non-increasing function with respect to t on [0, δ0〉 . M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 338 e. At some x ∈ R, Kλ(x) tends to infinity as λ tends to λ0. f. lim λ→λ0 ∣∣∣∣∫ R Kλ (t) dt− 1 ∣∣∣∣ = 0. Throughout this paper Kλ belongs to class Aw. 3. Existence of the Operators Main results in this work are based on the following theorem. Theorem 1. Suppose that f ∈ L1,w 〈a, b〉 . Then the operator Lλ (f ;x) defines a continu- ous transformation acting on L1,w 〈a, b〉 . Proof. Let 〈a, b〉 be an arbitrary closed, semi closed or open bounded interval in R. By the linearity of the operator Lλ(f ;x), it is sufficient to show that the expression: ‖Lλ‖1,w = sup f 6=0 ‖Lλ(f ;x)‖L1,w〈a,b〉 ‖f‖L1,w〈a,b〉 remains finite. Here, the norm of f ∈ L1,w 〈a, b〉 is given by the following equality (see, for example [14]): ‖f‖L1,w〈a,b〉 = b∫ a ∣∣∣∣ f(x) w(x) ∣∣∣∣ dx. Let us define a new function g by g(t) := { f(t), if t ∈ 〈a, b〉 0, if t ∈ R\ 〈a, b〉 . Now, using Fubini’s Theorem [8] we can write ‖Lλ(f ;x)‖L1,w〈a,b〉 = b∫ a 1 w(x) ∣∣∣∣∣∣ b∫ a f(t)Kλ(t− x)dt ∣∣∣∣∣∣ dx = b∫ a 1 w(x) ∣∣∣∣∣∣ ∞∫ −∞ g(t+ x) w(t+ x) w(t+ x) Kλ(t)dt ∣∣∣∣∣∣ dx = b∫ a 1 w(x) ∣∣∣∣∣∣ ∞∫ −∞ w(t+ x) g(t+ x) w(t+ x) Kλ(t)dt ∣∣∣∣∣∣ dx ≤ b∫ a 1 w(x)  ∞∫ −∞ w(t+ x) ∣∣∣∣ g(t+ x) w(t+ x) ∣∣∣∣ |Kλ(t)| dt  dx M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 339 = ∞∫ −∞ |Kλ(t)|  b∫ a w(t+ x) w(x) ∣∣∣∣ g(t+ x) w(t+ x) ∣∣∣∣ dx  dt ≤ M ‖f‖L1,w〈a,b〉 . Now, the proof of the indicated case is completed. The assertion can be proved with the above method for the case 〈a, b〉 = R. Thus the proof is completed. 4. Convergence at Characteristic Points In this section, two theorems concerning pointwise convergence of the operators of type (3) will be presented. In the following theorem we suppose that 〈a, b〉 is an arbitrary closed, semi closed or open bounded interval in R. Theorem 2. Suppose that w(t) and |Kλ(t− x)| are almost everywhere differentiable func- tions on R with respect to variable t such that the following inequality: d dt w(t) d dt |Kλ(t− x)| > 0, for any fixed x ∈ 〈a, b〉 (4) holds. If x0 ∈ 〈a, b〉 is a common µ−generalized Lebesgue point of functions f ∈ L1,w 〈a, b〉 and w ∈ L1 〈a, b〉 , then lim (x,λ)→(x0,λ0) Lλ(f ;x) = f(x0) on any planar set Z on which the function x0+δ∫ x0−δ |Kλ(t− x)|w(t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2 |Kλ (0)|w (x)µ (|x0 − x|) , 0 < δ < δ0, where δ0 is a fixed positive real number, is bounded as (x, λ) tends to (x0, λ0). Proof. Let x0 + δ < b, x0− δ > a and |x0 − x| < δ 2 for a given δ > 0. The proof will be stated for the case 0 < x0 − x < δ 2 . The proof of the reverse case is similar. Set I = |Lλ (f ;x)− f (x)|. Using Theorem 2 in [12] we may write the integral I as follows: I = ∣∣∣∣∣∣ b∫ a f (t)Kλ (t− x) dt− f(x0) ∣∣∣∣∣∣ = ∣∣∣∣∣∣ b∫ a [ f (t) w (t) − f (x0) w (x0) ] w (t)Kλ (t− x) dt+ f (x0) w (x0)  b∫ a w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 340 ≤ b∫ a ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣w (t) |Kλ (t− x)| dt+ ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∣∣∣∣∣∣ b∫ a w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ =  x0−δ∫ a + x0∫ x0−δ + x0+δ∫ x0 + b∫ x0+δ  ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣w (t) |Kλ (t− x)| dt + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∣∣∣∣∣∣ b∫ a w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ = I1 + I2 + I3 + I4 + I5. It is sufficient to show that Ii → 0 (i = 1, 5) as (x, λ)→ (x0, λ0) provided (x, λ) ∈ Z. Let us consider the integral I5. By Theorem 2 in [12], we get lim (x,λ)→(x0,λ0) b∫ a w (t)Kλ (t− x) dt = w (x0) . Therefore, we have lim (x,λ)→(x0,λ0) I5 = 0. Now, we consider the integrals I1 and I4. From hypothesis (4) and condition (d) of class Aw, we have I1 = x0−δ∫ a ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣w (t) |Kλ (t− x)| dt ≤ sup t∈〈a,x0−δ〉 w (t) sup t∈〈a,x0−δ〉 |Kλ (t− x)| x0−δ∫ a ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣ dt Hence we obtain I1 ≤ w (x0 − δ) sup |ξ|> δ 2 |Kλ (ξ)| b∫ a ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣ dt ≤ w (x0 − δ) sup |ξ|> δ 2 |Kλ (ξ)| { ‖f‖L1,w〈a,b〉 + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ (b− a) } . Using similar strategy, we have the following inequality for the integral I4 I4 ≤ w (x0 + δ) sup |ξ|> δ 2 |Kλ (ξ)| { ‖f‖L1,w〈a,b〉 + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ (b− a) } . M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 341 Combining these integrals, we get the following inequality: I1 + I4 ≤ {w (x0 − δ) + w (x0 + δ)} sup |ξ|> δ 2 |Kλ (ξ)| { ‖f‖L1,w〈a,b〉 + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ (b− a) } . In view of condition (b) of class Aw, I1 + I4 → 0 as λ→ λ0. Now, we consider the integral I2. By the definition of µ−generalized Lebesgue point, for every ε > 0 there exists a corresponding number δ > 0 such that the expression: x0∫ x0−h ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣ dt < εµ (h) (5) holds for all 0 < h ≤ δ < δ0. Define the function F (t) by F (t) = x0∫ t ∣∣∣∣ f (y) w (y) − f (x0) w (x0) ∣∣∣∣ dy. (6) From (6), we have dF (t) = − ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣ dt. (7) From (5) and (6) , for all t satisfying 0 < x0 − t ≤ δ < δ0 we have F (t) ≤ εµ (x0 − t) . (8) By virtue of (6) and (7) we have I2 = x0∫ x0−δ ∣∣∣∣ f (t) w (t) − f (x0) w (x0) ∣∣∣∣ |Kλ (t− x)|w (t) dt = x0∫ x0−δ |Kλ (t− x)|w (t) d [−F (t)] . Using integration by parts and applying (8), we have the following inequality: |I2| ≤ εµ (δ) |Kλ (x0 − δ − x)|w (x0 − δ) + ε x0∫ x0−δ µ (x0 − t) |d |Kλ (t− x)|w (t)| . It is easy to see that |I2| ≤ εµ (δ) |Kλ (x0 − δ − x)|w (x0 − δ) M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 342 +ε x0−x∫ x0−x−δ µ (x0 − x− t) ∣∣∣∣∣∣d  t∨ x0−x−δ |Kλ (u)|w (u+ x) ∣∣∣∣∣∣ . If we use hypothesis (4) and integration by parts, then we have the following expression: |I2| ≤ −ε x0−x∫ x0−x−δ  t∨ x0−x−δ |Kλ (u)|w (u+ x)  {µ (x0 − x− t)}′t dt. Using condition (d) of class Aw, we obtain |I2| ≤ ε x0∫ x0−δ |Kλ (t− x)|w (t) ∣∣{µ (x0 − t)}′t ∣∣ dt+ 2 |Kλ (0)|w (x)µ (x0 − x) . Using preceding method, we can estimate the integral I3 as |I3| ≤ ε x0+δ∫ x0 |Kλ (t− x)|w (t) ∣∣{µ (t− x0)}′t ∣∣ dt. Combining |I2| and |I3| , we obtain |I2|+ |I3| ≤ ε x0+δ∫ x0−δ |Kλ (t− x)|w (t) ∣∣{µ (|t− x0|)}′t ∣∣ dt+ 2ε |Kλ (0)|w (x)µ (|x0 − x|) . Note that the above inequality is obtained for the case 0 < x − x0 < δ 2 . Therefore, if the points (x, λ) ∈ Z are sufficiently near to (x0, λ0) , we have |I2|+ |I3| tends to zero. Thus, the proof is completed. In the next theorem we suppose that 〈a, b〉 = R. Theorem 3. Suppose that w(t) and |Kλ(t− x)| are almost everywhere differentiable func- tions on R with respect to variable t such that the following inequality: d dt w(t) d dt |Kλ(t− x)| > 0, for any fixed x ∈ R. (4) holds. If x0 ∈ R is a common µ−generalized Lebesgue point of functions f ∈ L1,w (R) and w ∈ L1 (R) , then lim (x,λ)→(x0,λ0) Lλ(f ;x) = f(x0) on any planar set Z on which the function x0+δ∫ x0−δ |Kλ(t− x)|w(t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2 |Kλ (0)|w (x)µ (|x0 − x|) , 0 < δ < δ0, where δ0 is a fixed positive real number, is bounded as (x, λ) tends to (x0, λ0). M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 343 Proof. Making similar calculations as in Theorem 2 we have |Lλ(f ;x)− f(x0)| ≤ sup |ξ|> δ 2 |Kλ (ξ)| {w (x0 − δ) + w (x0 + δ)} ‖f‖L1,w(R) + {w (x0 − δ) + w (x0 + δ)} ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∫ |ξ|> δ 2 |Kλ (ξ)| dξ +ε x0+δ∫ x0−δ |Kλ(t− x)|w(t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2ε |Kλ (0)|w (x)µ (|x0 − x|) + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∣∣∣∣∣∣ ∞∫ −∞ w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ . Using conditions (b) and (c) of classAw and Theorem 3 in [12] we obtain |Lλ(f ;x)− f(x0)| → 0 as (x, λ)→ (x0, λ0) . Hence the proof is completed. Example 2. The following Gauss-Weierstrass type kernel and weight functions satisfy the hypothesis of Theorem 3, respectively. Let Λ = (0,∞), λ0 = 0 and Kλ : R → R+ for each λ ∈ Λ is given by Kλ(t) = 1 λ √ π e− t2 λ2 and w : R→ R+ is given by w(t) = { 1, if t = 0 1√ |t|(1+|t|) , otherwise. For detailed analysis of the above functions, authors refer to [1]. 5. Rate of Convergence In this section, two theorems concerning rate of pointwise convergence will be given. Theorem 4. Suppose that the hypothesis of Theorem 2 is satisfied. Let ∆(x, λ, δ) = x0+δ∫ x0−δ |Kλ (t− x)|w (t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2 |Kλ (0)|w (x)µ (|x0 − x|) , where 0 < δ ≤ δ0, and the following conditions are satisfied: i. ∆(x, λ, δ)→ 0 as (x, λ)→ (x0, λ0) for some δ > 0. M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 344 ii. For every ξ > 0 |Kλ(ξ)| = o(∆(x, λ, δ)) as (x, λ)→ (x0, λ0). iii. ∣∣∣∣∣∣ b∫ a w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ = o(∆(x, λ, δ)) as (x, λ)→ (x0, λ0). Then at each common µ−generalized Lebesgue point of functions f ∈ L1,w 〈a, b〉 and w ∈ L1 〈a, b〉 we have as (x, λ)→ (x0, λ0) |Lλ (f ;x)− f (x0)| = o(∆(x, λ, δ)). Proof. Under the hypothesis of Theorem 2, we may write |Lλ (f ;x)− f (x)| ≤ {w (x0 − δ) + w (x0 + δ)} × sup |ξ|> δ 2 |Kλ (ξ)| { ‖f‖L1,w〈a,b〉 + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ (b− a) } +ε x0+δ∫ x0−δ |Kλ (t− x)|w (t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2ε |Kλ (0)|w (x)µ (|x0 − x|) + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∣∣∣∣∣∣ b∫ a w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ . From (i)-(iii) we have the desired result i.e.: |Lλ (f ;x)− f (x0)| = o(∆(x, λ, δ)). Theorem 5. Suppose that the hypothesis of Theorem 3 is satisfied. Let ∆(x, λ, δ) = x0+δ∫ x0−δ |Kλ (t− x)|w (t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2 |Kλ (0)|w (x)µ (|x0 − x|) where 0 < δ ≤ δ0, and the following conditions are satisfied: i. ∆(x, y, λ, δ)→ 0 as (x, λ)→ (x0, λ0) for some δ > 0. M. M. Yilmaz, G. Uysal, / Eur. J. Pure Appl. Math, 10 (2) (2017), 335-347 345 ii. For every ξ > 0 |Kλ(ξ)| = o(∆(x, λ, δ)) as (x, λ)→ (x0, λ0). iii. For every ξ > 0 lim λ→λ0  ∫ |t|>ξ |Kλ (t)| dt  = o(∆(x, λ, δ)) as (x, λ)→ (x0, λ0). iv. ∣∣∣∣∣∣ ∞∫ −∞ w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ = o(∆(x, λ, δ)) as (x, λ)→ (x0, λ0). Then at each common µ−generalized Lebesgue point of functions f ∈ L1,w(R) and w ∈ L1(R) we have as (x, λ)→ (x0, λ0) |Lλ (f ;x)− f (x0)| = o(∆(x, λ, δ)). Proof. Under the hypothesis of Theorem 3, we write |Lλ (f ;x)− f (x0)| ≤ sup |ξ|> δ 2 |Kλ (ξ)| {w (x0 − δ) + w (x0 + δ)} ‖f‖L1,w(R) + {w (x0 − δ) + w (x0 + δ)} ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∫ |ξ|> δ 2 |Kλ (ξ)| dξ + ε x0+δ∫ x0−δ |Kλ(t− x)|w(t) ∣∣{µ(|x0 − t|)}′t ∣∣ dt+ 2ε |Kλ (0)|w (x)µ (|x0 − x|) + ∣∣∣∣ f (x0) w (x0) ∣∣∣∣ ∣∣∣∣∣∣ ∞∫ −∞ w (t)Kλ (t− x) dt− w (x0) ∣∣∣∣∣∣ and from (i)-(iv) we have the desired result i.e.: |Lλ (f ;x)− f (x0)| = o(∆(x, λ, δ)). Acknowledgements The authors thank the referees for their valuable comments and suggestions. REFERENCES 346 References [1] S E Almali. Convergence and the Order of Convergence of Family of Nonconvolution Type In- tegral Operators at Characteristic points. Ph. D. Thesis, Ankara University, Graduate School of Applied Science, Ankara, 2002. [2] C Bardaro and C G Cocchieri. On the Degree of Approximation for a Class of Singular Integrals. Rend. Mat., 7(4):481–490, 1984. [3] C Bardaro. 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