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Approximation by (p, q)-Baskakov-Durrmeyer-Stancu Operators
(Springer Basel Ag, 2018) Acar, Tuncer; Mohiuddine, S. A.; Mursaleen, Mohammad
The present paper deals with the Stancu-type generalization of (p, q)-Baskakov-Durrmeyer operators. We investigate local approximation, weighted approximation properties of new operators and present the rate of convergence by means of suitable modulus of continuity. At the end of the paper, we introduce a new modification of (p, q)-Baskakov-Durrmeyer-Stancu operators with King approach.
Approximation by Bivariate (p, q)-Bernstein-Kantorovich Operators
(Springer International Publishing Ag, 2018) Acar, Tuncer; Aral, Ali; Mohiuddine, S. A.
In the present paper, we introduce Kantorovich modifications of (p, q)-Bernstein operators for bivariate functions using a new (p, q)-integral. We first estimate the moments and central moments. We give the uniform convergence of new operators, rate of convergence in terms of modulus of continuity. The approximations behaviours of the operators for functions having continuous partial derivatives and for functions belong to Lipschitz class are investigated as well.
Approximation by generalized Baskakov-Durrmeyer-Stancu type operators
(Springer-Verlag Italia Srl, 2016) Kumar, A. Sathish; Acar, Tuncer
In this paper, we introduce Stancu type modification of generalized Baskakov-Durrmeyer operators and study their approximation properties. First, we derive the recurrence relation and central moments of these operators and then we study the local approximation, weighted approximation results for the new operators. The last section is devoted to A-statistical convergence behaviours of these operators by using the Korovkin type approximation of statistical convergence.
Approximation by k-th order modifications of Szász-Mirakyan operators
(Akademiai Kiado Rt, 2016) Acar, Tuncer; Aral, Ali; Rasa, Ioan
In this paper, we study the k-th order Kantorovich type modication of Szasz-Mirakyan operators. We first establish explicit formulas giving the images of monomials and the moments up to order six. Using this modification, we present a quantitative Voronovskaya theorem for differentiated Szasz-Mirakyan operators in weighted spaces. The approximation properties such as rate of convergence and simultaneous approximation by the new constructions are also obtained.
Approximation by modified Szasz-Durrmeyer operators
(Springer, 2016) Acar, Tuncer; Ulusoy, Gulsum
The main goal of this paper is to introduce Durrmeyer modifications for the generalized Szasz-Mirakyan operators defined in (Aral et al., in Results Math 65:441-452, 2014). The construction of the new operators is based on a function which is continuously differentiable times on such that and Involving the weighted modulus of continuity constructed using the function , approximation properties of the operators are explored: uniform convergence over unbounded intervals is established and a quantitative Voronovskaya theorem is given. Moreover, we obtain direct approximation properties of the operators in terms of the moduli of smoothness. Our results show that the new operators are sensitive to the rate of convergence to f, depending on the selection of For the particular case , the previous results for classical Szasz-Durrmeyer operators are captured.
Approximation by sampling Kantorovich series in weighted spaces of functions
(Tubitak Scientific & Technological Research Council Turkey, 2022) Acar, Tuncer; Alagoz, Osman; Aral, Ali; Costarelli, Danilo; Turgay, Metin; Vinti, Gianluca
This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for functions that are not necessarily uniformly continuous and bounded on R. In that context we also prove quantitative estimates for the rate of convergence of the family of the above operators in terms of weighted modulus of continuity. Finally, pointwise convergence results in quantitative form by means of Voronovskaja type theorems have been also established.
Approximation properties of two dimensional Bernstein-Stancu-Chlodowsky operators
(Univ Studi Catania, Dipt Matematica, 2013) Acar, Tuncer; Aral, Ali
In this paper, as a generalization of Bernstein-Stancu type operators of two variable, we introduce a new positive linear operator C-n (alpha,alpha,beta,beta,) (f; x, y) called Bernstein-Stancu-Chlodowsky on a triangular domain, with mobile boundaries, which extends to [0,infinity) x [0,infinity) as n -> infinity. We give some shape properties that are preserved and also obtain weighted approximation properties of these operators.
Approximation Results for Hadamard-Type Exponential Sampling Kantorovich Series
(Springer Basel Ag, 2023) Kurşun, Sadettin; Aral, Ali; Acar, Tuncer
The present paper deals with construction of a new family of exponential sampling Kantorovich operators based on a suitable fractional-type integral operators. We study convergence properties of newly constructed operators and give a quantitative form of the rate of convergence thanks to logarithmic modulus of continuity. To obtain an asymptotic formula in the sense of Voronovskaja, we consider locally regular functions. The rest of the paper devoted to approximations of newly constructed operators in logarithmic weighted space of functions. By utilizing a suitable weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem via remainder of Mellin-Taylor's formula. Furthermore, some examples of kernels which satisfy certain assumptions are presented and the results are examined by illustrative numerical tables and graphical representations.
Asymptotic Formulas for Generalized Szasz-Mirakyan Operators
(Elsevier Science Inc, 2015) Acar, Tuncer
In the present paper, we consider the general Szasz-Mirakyan operators and investigate their asymptotic behaviours. We obtain quantitative Voronovskaya and quantitative Gruss type Voronovskaya theorems using the weighted modulus of continuity. The particular cases are presented for classical Szasz-Mirakyan operators. (C) 2015 Elsevier Inc. All rights reserved.
Bezier variant of the Bernstein-Durrmeyer type operators
(Springer Basel Ag, 2017) Acar, Tuncer; Agrawal, P. N.; Neer, Trapti
In the present paper, we introduce the Bezier-variant of Durrmeyer modification of the Bernstein operators based on a function , which is infinite times continuously differentiable and strictly increasing function on [0, 1] such that and . We give the rate of approximation of these operators in terms of usual modulus of continuity and K-functional. Next, we establish the quantitative Voronovskaja type theorem. In the last section we obtain the rate of convergence for functions having derivative of bounded variation.
Bir ve iki değişkenli bernstein-stancu-chlodowsky polinomlarının yaklaşım özellikleri
(2013) Aral, Ali; Acar, Tuncer; Yıldız, Duygu Döndü
[Abstract Not Available]
Blending Type Approximation By Bezier-Summation-Integral Type Operators
(Ankara Univ, Fac Sci, 2018) Acar, Tuncer; Kajla, Arun
In this note we construct the Bezier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In the last section, we study the quantitative Voronovskaja type theorem.
Construction of a new family of Bernstein-Kantorovich operators
(Wiley, 2017) Mohiuddine, S. A.; Acar, Tuncer; Alotaibi, Abdullah
In the present paper, we construct a new sequence of Bernstein-Kantorovich operators depending on a parameter alpha. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first- and second-order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein-Kantorovich operators and their approximation behaviors.
Convergence of generalized sampling series in weighted spaces
(De Gruyter Poland Sp Z O O, 2022) Acar, Tuncer; Alagoz, Osman; Aral, Ali; Costarelli, Danilo; Turgay, Metin; Vinti, Gianluca
The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronovskaja-type theorem is obtained.
Differentiated Bernstein Type Operators
(PADOVA UNIV PRESS, 2020) Aral, Ali; Acar, Tuncer; Ozsarac, Firat
The present paper deals with the derivatives of Bernstein type operators preserving some exponential functions. We investigate the uniform convergence of the differentiated operators. The rate of convergence by means of a modulus of continuity is studied, an upper estimate theorem for the difference of new constructed differentiated Bernstein type operators is presented.
Durrmeyer tipli operatörlerin özellikleri
(Kırıkkale Üniversitesi, 2011) Acar, Tuncer; Aral, Ali
Bu çalışma dört bölümden oluşmaktadır. Birinci bölüm giriş için ayrılmıştır.İkinci bölümde bazı temel tanımlar ve kavramlar verilmiştir.Üçüncü bölümde genelleştirilmiş Szasz operatörleri ve türevi sınırlı salınımlı olan fonksiyonlar için yakınsaklık hızı incelenmiştir.Dördüncü bölümde Szasz operatörlerinin bir başka genelleştirilmesi tanımlanmış ve bu operatörlerinde türevi sınırlı salınımlı fonksiyonlar için yakınsaklık hızı incelenmiştir.
Genelleştirilmiş Gadjiev operatörlerinin yaklaşım özellikleri
(Kırıkkale Üniversitesi, 2015) Acar, Tuncer; Aral, Ali
Bu tez beş bölümden oluşmaktadır. Birinci bölüm, giriş kısmına ayrılmıştır. İkinci bölümde tezde gerekli olan kavramlar ve tanımlar verilmiştir. Üçüncü bölümde, Bernstein-Chlodowsky polinomlarının Gadjiev tipli genelleştirmesi tanımlanmakta ve ağırlıklı uzaylarda yaklaşım özellikleri incelenmektedir. Ayrıca, yeni tanımlanan operatörlerin türevlerinin yaklaşım özellikleri de çalışılmıştır. Dördüncü bölümde, üçüncü bölümde tanımlanan operatörlerin iki değişkenli versiyonu tüm kenarları hareketli olan üçgensel bölgeler üzerinde tanımlanmış, bazı şekil koruyan özellikleri ve ağırlıklı yaklaşım özellikleri incelenmiştir. Beşinci bölümde ise hareketli aralıklar üzerinde Bernstein-Durrmeyer operatörleri tanımlanmakta ve yaklaşım hızı, noktasal yakınsaklığı incelenmekte, daha iyi yaklaşım sonuçları veren genelleştirmeleri çalışılmıştır.
Generalized Kantorovich forms of exponential sampling series
(Springer Basel Ag, 2022) Aral, Ali; Acar, Tuncer; Kursun, Sadettin
In this paper, we introduce a new family of operators by generalizing Kantorovich type of exponential sampling series by replacing integral means over exponentially spaced intervals with its more general analogue, Mellin Gauss Weierstrass singular integrals. Pointwise convergence of the family of operators is presented and a quantitative form of the convergence using a logarithmic modulus of continuity is given. Moreover, considering locally regular functions, an asymptotic formula in the sense of Voronovskaja is obtained. By introducing a new modulus of continuity for functions belonging to logarithmic weighted space of functions, a rate of convergence is obtained. Some examples of kernels satisfying the obtained results are presented.
Iterated Boolean Sums of Bernstein Type Operators
(TAYLOR & FRANCIS INC, 2020) Acar, Tuncer; Aral, Ali; Rasa, Ioan
The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present some qualitative and quantitative results concerning the approximation by such Boolean sums. The general results are illustrated by examples.
Korovkin-Type Theorems in Weighted Lp-Spaces via Summation Process
(Hindawi Ltd, 2013) Acar, Tuncer; Dirik, Fadime
Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted L-p spaces, 1 <= p < infinity for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of A-summability which is a stronger convergence method than ordinary convergence.

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