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    APPROXIMATION BY BASKAKOV-DURRMEYER OPERATORS BASED ON (p, q)-INTEGERS
    (WALTER DE GRUYTER GMBH, 2018) Acar, Tuncer; Aral, Ali; Mursaleen, Mohammad
    In the present paper, we introduce a new sequence of linear positive operators based on (p, q)-integers. To approximate functions over unbounded intervals, we introduce Baskakov-Durrmeyer type operators using the (p, q)-Gamma function. We investigate rate of convergence of new operators in terms of modulus of continuities and obtain their approximation behavior for the functions belonging to Lipschitz class. At the end, we present a modification of new operators preserving the test function x. (C) 2018 Mathematical Institute Slovak Academy of Sciences
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    Approximation by bivariate (p, q)-Baskakov-Kantorovich operators
    (WALTER DE GRUYTER GMBH, 2018) Ilarslan, Hatice Gul Ince; Acar, Tuncer
    The present paper deals with the bivariate (p, q)-Baskakov-Kantorovich operators and their approximation properties. First we construct the operators and obtain some auxiliary results such as calculations of moments and central moments, etc. Our main results consist of uniform convergence of the operators via the Korovkin theorem and rate of convergence in terms of modulus of continuity.
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    Approximation of Functions by Genuine Bernstein-Durrmeyer Type Operators
    (ELEMENT, 2018) Acar, Tuncer; Acu, Ana-Maria; Manav, Nesibe
    Very recently, in [4] Chen et. al introduced and considered a new generalization of Bernstein polynomials depending on a patameter alpha. As classical Bernstein operators, these operators also provide interpolation at the end points of [0,1] and they have the linear precision property which means those reproduce the linear functions. In this paper we introduce genuine alpha-Bernstein-Durrmeyer operators. Some approximation results, which include local approximation, error estimation in terms of Ditzian-Totik modulus of smoothness are obtained. Also, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms.
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    Bezier-Bernstein-Durrmeyer type operators
    (SPRINGER-VERLAG ITALIA SRL, 2019) Kajla, Arun; Acar, Tuncer
    In this note, we construct the Bezier variant of the Bernstein-Durrmeyer type operators. We present local results, a direct approximation theorem by using the Ditzian-Totik modulus of smoothness and a quantitative Voronovskaja type theorem with the help of the Ditzian-Totik modulus of continuity. The rate of convergence for differential functions whose derivatives are of bounded variation is also established. Finally, we show that the numerical examples which illustrate the authenticity of the theoretical results and the effectiveness of the defined operators.
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    BLENDING TYPE APPROXIMATION BY GENERALIZED BERNSTEIN-DURRMEYER TYPE OPERATORS
    (UNIV MISKOLC INST MATH, 2018) Kajla, Arun; Acar, Tuncer
    In this article we construct a Durrmeyer modification of the operators introduced by Chen et al. in [10] based on a non-negative real parameter. We establish local approximation, global approximation, Voronovskaja type asymptotic theorem. The rate of convergence for differentiable functions whose derivatives are of bounded variation is also obtained.
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    Degree of Approximation for Bivariate Generalized Bernstein Type Operators
    (SPRINGER BASEL AG, 2018) Acar, Tuncer; Kajla, Arun
    In this paper we study an extension of the bivariate generalized Bernstein operators based on a non-negative real parameters. For these operators we obtain the order of approximation using Peetre's K-functional, a Voronovskaja type theorem and the degree of approximation by means of the Lipschitz class. Further, we consider the Generalized Boolean Sum operators of generalized Bernstein type and we study the degree of approximation in terms of the mixed modulus of continuity. Finally, we show the comparisons by some illustrative graphics in Maple for the convergence of the operators to certain functions.
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    DURRMEYER TYPE (p,q)-BASKAKOV OPERATORS PRESERVING LINEAR FUNCTIONS
    (ELEMENT, 2018) Mohiuddine, S. A.; Acar, Tuncer; Alotaibi, Abdullah
    The present paper deals with the construction of Baskakov Durrmeyer operators, which preserve the linear functions, in (p,q)-calculus. More precisely, using (p,q)-Gamma function we introduce genuine mixed type Baskakov Durrmeyer operators having Baskakov and Szasz basis functions. After construction of the operators and calculations of their moments and central moments, rate of convergence of the operators by means of appropriate modulus of continuity, approximation behaviors for functions belong to Lipschitz class and weighted approximation are explored.
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    Genuine modified bernstein-durrmeyer operators
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2018) Mohiuddine, Syed Abdul; Acar, Tuncer; Alghamdi, Mohammed A.
    The present paper deals with genuine Bernstein-Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre K-functional and corresponding modulus of smoothness, quantitative Voronovskaya type theorem and Gruss-Voronovskaya type theorem in quantitative mean are discussed. Finally, the graphic for new operators with special cases and for some values of n is also presented.
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    A new modification of Durrmeyer type mixed hybrid operators
    (NORTH UNIV BAIA MARE, 2018) Kajla, Arun; Acar, Tuncer
    In 2008 V. Mihesan constructed a general class of linear positive operators generalizing the Szasz operators. In this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.
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    On sequences of JP King-type operators
    (HINDAWI LTD, 2019) Acar, Tuncer; Montano, Mirella Cappelletti; Garrancho, Pedro; Leonessa, Vita
    This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and x2 on [0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King's approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szasz-Mirakyan operators.
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    Operator methods in approximation theory
    (HINDAWI LTD, 2019) Leonessa, Vita; Acar, Tuncer; Montano, Mirella Cappelletti; Garrancho, Pedro
    [Abstract not Available]
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    Power series of positive linear operators
    (SPRINGER BASEL AG, 2019) Acar, Tuncer; Aral, Ali; Rasa, Ioan
    We describe a unifying approach for studying the power series of the positive linear operators from a certain class. For the same operators, we give simpler proofs of some known ergodic theorems.
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    QUANTITATIVE ESTIMATES FOR A NEW COMPLEX Q-DURRMEYER TYPE OPERATORS ON COMPACT DISKS
    (UNIV POLITEHNICA BUCHAREST, SCI BULL, 2018) Kumar, A. Sathish; Agrawal, Purshottam N.; Acar, Tuncer
    In the present article, the upper bound and Voronovskaya type result with quantitative estimate and the exact degree of approximation for a new complex q-Bernstein-Durrmeyer operators attached to analytic functions on compact disks are obtained. In this way, we put in evidence the over convergence phenomenon for the q-Bernstein-Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.
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    Reconstruction of Baskakov operators preserving some exponential functions
    (WILEY, 2019) Ozsarac, Firat; Acar, Tuncer
    The present paper deals with a new modification of Baskakov operators in which the functions exp(mu t) and exp(2 mu t), mu > 0 are preserved. Approximation properties of the operators are captured, ie, uniform convergence and rate of convergence of the operators in terms of modulus of continuity, approximation behaviors of the operators exponential weighted spaces, and pointwise convergence of the operators by means of the Voronovskaya theorem. Advantages of the operators for some special functions are presented.
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    Some approximation properties by a class of bivariate operators
    (WILEY, 2019) Acu, Ana-Maria; Acar, Tuncer; Muraru, Carmen-Violeta; Radu, Voichita Adriana
    Starting with the well-known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bogel space of continuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBS-type operator are compared.