Yazar "Dirik, Fadime" seçeneğine göre listele

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    A Korovkin type approximation theorem for double sequences of positive linear operators of two variables in A-statistical sense
    (Bulletin of the Korean Mathematical Society, 2010) Demirci, Kamil; Dirik, Fadime
    In this paper, we obtain a Korovkin type approximation theorem for double sequences of positive linear operators of two variables from Hw(K) to C(K) via A-statistical convergence. Also, we construct an example such that our new approximation result works but its classical case does not work. Furthermore, we study the rates of A -statistical convergence by means of the modulus of continuity.
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    A KOROVKIN-TYPE APPROXIMATION THEOREM FOR DOUBLE SEQUENCES OF POSITIVE LINEAR OPERATORS OF TWO VARIABLES IN STATISTICAL A-SUMMABILITY SENSE
    (Univ Miskolc Inst Math, 2014) Orhan, Sevda; Dirik, Fadime; Demirci, Kamil
    In this paper, using the concept of statistical A- summability which is stronger than classical convergence and A- statistical convergence, we obtain a Korovkin- type approximation theorem for double sequences of positive linear operators of two variables from H w. K/ to C B. K/. Also, we give an example such that our new approximation result works but its classical and A- statistical cases do not work.
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    A Korovkin-type theorem for double sequences of positive linear operators via power series method
    (Springer, 2018) Sahin, Pinar Okcu; Dirik, Fadime
    In this paper, using power series method we obtain a Korovkin type theorem for double sequences of real valued functions defined on a compact subset of (the real two-dimensional space). We also present an example that satisfies our theorem. Finally, we calculate the rate of convergence.
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    ABSTRACT KOROVKIN THEORY FOR DOUBLE SEQUENCES VIA POWER SERIES METHOD IN MODULAR SPACES
    (Element, 2019) Dirik, Fadime; Yildiz, Sevda; Demirci, Kamil
    In the present paper, we obtain an abstract version of the Korovkin type approximation theorems for double sequences of positive linear operators on modular spaces in the sense of power series method. We present an example that satisfies our theorem but not satisfies the classical one and also, we study an extension to non-positive operators.
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    Almost A-statistical convergence and approximation theorems
    (Univ Nis, Fac Sci Math, 2025) Demirci, Kamil; Dirik, Fadime; Yildiz, Sevda
    In this paper, we define almost A-statistical convergence via almost regular matrices, which extend the notion of regular matrices, and find its relationship with almost A-summability. Then we study its use in a Korovkin-type approximation theorem. We also construct an example such that our new result works but its classical and statistical versions do not work and a figure will be presented to support our result. Finally, we compute the corresponding rate of almost A-statistical convergence of positive linear operators in two different ways, we have newly defined.
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    An Extension of Korovkin Theorem via P-Statistical A-Summation Process
    (Padova Univ Press, 2023) Demirci, Kamil; Dirik, Fadime; Yildiz, Sevda
    In the present work, we study and prove Korovkin-type approximation theorems for linear operators defined on derivatives of functions by means of A-summation process via statistical convergence with respect to power series method. We give an example that our theorem is stronger. Also, we study the rate of convergence of these operators. Finally, we summarize our results and we show the importance of the study.
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    Approximation for periodic functions via statistical ?-convergence
    (Mathematical Communications, 2011) Demirci, Kamil; Dirik, Fadime
    In this study, using the concept of statistical ?-convergence which is stronger than convergence and statistical convergence we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on $C^{\ast }$ which is the space of all $2\pi $-periodic and continuous functions on $\mathbb{R}$, the set of all real numbers. We also study the rates of statistical ?-convergence of approximating positive linear operators.
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    APPROXIMATION IN STATISTICAL SENSE TO B-CONTINUOUS FUNCTIONS BY POSITIVE LINEAR OPERATORS
    (Akademiai Kiado Rt, 2010) Dirik, Fadime; Duman, Oktay; Demirci, Kamil
    In the present work, using the concept of A-statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.
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    Approximation in statistical sense to n-variate B-continuous functions by positive linear operators
    (Versita, 2010) Dirik, Fadime; Demirci, Kamil
    Our primary interest in the present paper is to prove a Korovkintype approximation theorem for sequences of positive linear operators defined on the space of all real valued n-variate B-continuous functions on a compact subset of the real n-dimensional space via statistical convergence. Also, we display an example such that our method of convergence is stronger than the usual convergence.
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    Approximation in triangular statistical sense to B-continuous functions by positive linear operators
    (Sciendo, 2017) Demirci, Kamil; Dirik, Fadime; Okçu, Pınar
    The main object of this paper is to prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line using a new type of statistical convergence of double sequences called triangular-A-statistical convergence for double real sequences. We give an illustrative example in support of our result. Finally, we investigate a rates of triangular-A-statistical convergence of positive linear operators. © 2018, Universitatii Al.I.Cuza din Iasi. All rights reserved.
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    Approximation Results on an Infinite Interval Based on Power Series Statistical Sense
    (Padova Univ Press, 2025) Yildiz, Sevda; Demirci, Kamil; Dirik, Fadime
    This paper introduces a new approximation theorem, type of Korovkin, for positive linear operators (pLO) defined on the Banach space C-* [0, infinity) comprising all real-valued continuous functions on [0, infinity) that converge to a finite limit as their argument approaches infinity. By applying statistical convergence with respect to power series methods and employing the test functions 1, exp(-u) and exp(-2u), we derive a novel approximation result. Our findings demonstrate that the proposed method outperforms classical and statistical approaches, as illustrated by a concrete example. Furthermore, we explore the rate of convergence associated with this new approximation theorem.
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    Approximation Results via Power Series Method for Sequences of Monotone and Sublinear Operators
    (Springer Basel Ag, 2025) Demirci, Kamil; Dirik, Fadime; Yildiz, Sevda
    In this paper, we use the power series methods to study Korovkin-type approximation theorems for sequences of operators that are monotone and sublinear. It is important to point out that any positive linear operator is monotone sublinear but the converse is not true. We also calculate the rate of convergence in terms of modulus of continuity. Finally, we provide some examples that satisfy our theorems and give quantitative estimates. We also define Bernstein-Chlodovsky-Kantorovich-Choquet (BCKC) polynomial operators similar to those given before and obtain approximation estimate.
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    Approximation theorems using the method of $\mathcal{I}_{2}$-statistical convergence
    (Amasya University, 2024) Dirik, Fadime; Demirci, Kamil; Yildiz, Sevda
    In this study, we utilize the concept of $\mathcal{I}$-statistical convergence for double sequences to establish a general approximation theorem of Korovkin-type for double sequences of positive linear operators $(PLOs)$ mapping from $H_{\omega }\left( X\right) $ to $C_{B}\left( X\right) $ where $% X=\left
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    Approximation theorems via power series statistical convergence and applications for sequences of monotone and sublinear operators
    (Springer-Verlag Italia Srl, 2025) Yildiz, Sevda; Demirci, Kamil; Dirik, Fadime
    In this paper, we address the problem of extending Korovkin-type approximation theorems to sequences of monotone and sublinear operators via the concept of power series statistical convergence (statistical convergence with respect to power series methods), which is incompatible with statistical convergence and in general a non-matrix method. It is important to emphasise that any positive linear operator is monotone sublinear, but the opposite is not correct. We establish several Korovkin-type theorems under these generalized settings and demonstrate their applicability with concrete examples.
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    Approximation via a product (P, Ka)-convergence method on Hω(K)
    (Walter De Gruyter Gmbh, 2026) Cinar, Selin; Yildiz, Sevda; Dirik, Fadime
    This paper presents a Korovkin-type approximation theorem for positive linear operators defined on the space H omega K ${H}_{\omega }\left(K\right)$ with K = 0 , infinity $K=\left[0,\infty \right)$ . The main result is formulated using the concept of (P, K a )-convergence, which is defined as the product of P-statistical convergence and K a -convergence. We provide a constructive example of a sequence of operators that satisfies the conditions of the theorem. The behaviour of the operator in the example has been further illustrated through graphs. Furthermore, we extend our results to the multidimensional case. Finally, we use a modulus of smoothness to calculate the rate of (P, K a )-convergence for this sequence of operators.
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    Approximation via equi-statistical convergence in the sense of power series method
    (Springer-Verlag Italia Srl, 2022) Demirci, Kamil; Dirik, Fadime; Yildiz, Sevda
    In this study, we define a new type of convergence using the notions of statistical convergence in the sense of the power series method and equi-statistical convergence then, we give an example that supports this new definition and after that we use it to prove a Korovkin-type approximation theorem. This theorem is a non-trivial generalization of Korovkin-type approximation theorems that have been studied in earlier papers. Also, we present an example that satisfies our approximation theorem which hasn't satisfied the one studied before. Moreover, we calculate the rate of equi-statistical convergence in the sense of the power series method and then, we prove a Voronovskaya-type approximation theorem. Finally, we summarize our results in the conclusion section.
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    Approximation via Power Series Method in Two-Dimensional Weighted Spaces
    (Malaysian Mathematical Sciences Soc, 2020) Demirci, Kamil; Yildiz, Sevda; Dirik, Fadime
    In this work, we obtain a Korovkin-type approximation theorem for double sequences of real-valued functions by using the power series method in two-dimensional weighted spaces. We also study the rate of convergence by using the weighted modulus of continuity, and in the last section, we present an application that satisfies our new Korovkin-type approximation theorem but does not satisfy classical one.
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    Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method
    (Springer Heidelberg, 2023) Demirci, Kamil; Dirik, Fadime; Yildiz, Sevda
    In the present paper, we generalize the notion of P-statistical convergence and we first define the notion of P-statistical relative uniform convergence of sequences of functions at a point. We demonstrate an approximation theorem for a sequence of functions. Also, we give an example, showing that our result is strict generalization of the corresponding classical ones. In the final section, we study the rates of convergence.
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    B-STATISTICAL APPROXIMATION FOR PERIODIC FUNCTIONS
    (Akademiai Kiado Rt, 2010) Dirik, Fadime; Demirci, Kamil
    In this study, using the concept of B -statistical convergence for sequence of in finite matrices B = (B(i)) with B(i) = (b(nk) (i)) we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on C* which is the space of all 2 pi-periodic and continuous functions on R, the set of all real numbers. Also we study the rates of B -statistical convergence of approximating positive linear operators.
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    DEFERRED NORLUND STATISTICAL RELATIVE UNIFORM CONVERGENCE AND KOROVKIN-TYPE APPROXIMATION THEOREM
    (Ankara Univ, Fac Sci, 2021) Demirci, Kamil; Dirik, Fadime; Yildiz, Sevda
    In this paper, we define the concept of statistical relative uniform convergence of the deferred Norlund mean and we prove a general Korovkin-type approximation theorem by using this convergence method. As an application, we use classical Bernstein polynomials for defining an operator that satisfies our new approximation theorem but does not satisfy the theorem given before. Additionally, we estimate the rate of convergence of approximating positive linear operators by means of the modulus of continuity.