# Oktay DumanTOBB University of Economics and Technology · Department of Mathematics

Oktay Duman

Professor (Ph.D.)

## About

145

Publications

8,262

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2,286

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Introduction

Additional affiliations

June 2015 - September 2015

June 2010 - present

April 2005 - June 2010

## Publications

Publications (145)

In this paper, we introduce the Kantorovich version of complex Shepard operators in order to approximate functions whose pth powers are integrable on the unit square. We also give an application which explains why we need such operators. Furthermore, we study the effects of some regular summability methods on this Lp-approximation.

In this paper, we construct the complex Shepard operators to approximate continuous and complex-valued functions on the unit square. We also examine the effects of regular summability methods on the approximation by these operators. Some applications verifying our results are provided. To illustrate the approximation theorems graphically we conside...

In this paper, we approximate to functions in N-dimension by means of nonlinear integral operators of the convolution type. Our approximation is based on not only the uniform norm but also the variation semi-norm in Tonelli's sense. We also study the rates of convergence. To get more general results we mainly use regular summability methods in the...

In this paper, by using regular summability methods we modify the Bernstein–Chlodovsky operators in order get more general and powerful results than the classical aspects. We study Korovkin-type approximation theory on weighted spaces. As a special case, it is possible to Cesàro approximate (arithmetic mean convergence) to the test function e2(x)=x...

In this paper, by using nonnegative regular summability methods we improve and generalize the approximation properties of max-min operators which have been investigated systematically in our recent study. We also discuss the rate of convergence in the approximation. Applications and concluding remarks at the end of the paper explain why we need suc...

In the present paper, by considering nonlinear integral operators and using their approximations via regular summability methods, we obtain characterizations for some function spaces including the space of absolutely continuous functions, the space of uniformly continuous functions, and their other variants. We observe that Bell-type summability me...

We study the approximation properties of Cardaliaguet-Euvrard type neural network operators. We first modify the operators in order to get the uniform convergence, later we use regular summability matrix methods in the approximation by means of these operators to get more general results than the classical ones. We also display some examples and sh...

In this paper, we study the approximation properties of nonlinear integral operators of convolution-type by using summability process. In the approximation, we investigate the convergence with respect to both the variation semi-norm and the classical supremum norm. We also compute the rate of approximation on some appropriate function classes. At t...

This work is related to inequalities in the approximation theory. Mainly, we study numerical solutions of delay Volterra integral equations by using a collocation method based on sigmoidal function approximation. Error estimation and convergence analysis are provided. At the end of the paper we display numerical simulations verifying our results.

In this study, we obtain a general approximation theorem for max-min operators including many significant applications. We also study the error estimation in this approximation by using Hölder continuous functions. The main motivation for this work is the paper by Bede et al. (2008) [12]. As a special case of our results, we explain how to approxim...

In this study, approximation properties of the Mellin-type nonlin-ear integral operators defined on multivariate functions are investigated. In order to get more general results than the classical aspects, we mainly use the summability methods defined by Bell. Considering the Haar measure with variation semi-norm in Tonelli's sense, we approach to...

Hardy’s well-known Tauberian theorem for number sequences states that if a sequence x=xk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=\left( x_{k}\right) $$\end{doc...

In this note, using the idea of the late Professor Gadjiev (Mat Zametki 20(5):781–786, 1976), we give new, direct and easy proofs of the Korovkin theorems for positive linear operators acting on weighted spaces. Recent improvements and new applications are also presented.

In this paper, we first obtain a Tauberian condition for statistical convergence on time scales. We also find necessary and sufficient conditions for the equivalence of statistical convergence and lacunary statistical convergence on time scales. Some significant applications are also presented.

In this paper, we approximate a continuous function in a polydisc by means of multivariate complex singular operators which preserve the analytic functions. In this singular approximation, we mainly use a regular summability method (process) from the summability theory. We show that our results are non-trivial generalizations of the classical appro...

The summability process introduced by Bell (Proc Am Math Soc 38: 548–552, 1973) is a more general and also weaker method than ordinary convergence. Recent studies have demonstrated that using this convergence in classical approximation theory provides many advantages. In this paper, we study the summability process to approximate a function and its...

In this article, we consider modified Bernstein-Kantorovich operators and investigate their approximation properties. We show that the order of approximation to a function by these operators is at least as good as that of ones classically used. We obtain a simultaneous approximation result for our operators. Also, we prove two direct approximation...

In this study, we focus on the approximation to continuous functions by max-product operators in the sense of summation process. We also study error estimation corresponding to this approximation. At the end, we present an application to max-product Bernstein operators.

In this paper, we construct a certain family of nonlinear operators in order to approximate a function by these nonlinear operators. For construction, we use the linear q-Bernstein polynomials and also the max-product algebra.

This special volume is a collection of outstanding more applied articles presented in AMAT 2015 held in Ankara, May 28-31, 2015, at TOBB Economics and Technology University.
The collection is suitable for Applied and Computational Mathematics and Engineering practitioners, also for related graduate students and researchers. Furthermore it will be a...

This special volume is a collection of outstanding theoretical articles presented at the conference AMAT 2015, held in Ankara, Turkey from May 28-31, 2015, at TOBB University of Economics and Technology.
The collection is suitable for a range of applications: from researchers and practitioners of applied and computational mathematics, to students...

In this paper, we prove a general Korovkin-type approximation theorem for the Mastroianni operators using a regular summability process with non-negative entries. We also obtain some useful estimates via the modulus of continuity and the second modulus of smoothness. Furthermore, we construct a sequence of Szász–Mirakjan type operators satisfying a...

In this paper, we use summability methods on the approximation to derivatives of functions by a family of linear operators acting on weighted spaces. This point of view enables us to overcome the lack of ordinary convergence in the approximation. To support this idea, at the end of the paper, we will give a sequence of positive linear operators obe...

The aim of this paper is to obtain some approximation theorems for a sequence of singular operators that do not have to be positive in general. In the approximation, we mainly use a general matrix summability process introduced by Bell [Order summability and almost convergence, Proc. Amer. Math. Soc. 38 (1973) 548-552], which includes many well-kno...

In this paper, when approximating a continuos non-negative function on the unit interval, we present an alternative way to the classical Bernstein polynomials. Our new operators become nonlinear, however, for some classes of functions, they provide better error estimations than the Bernstein polynomials. Furthermore, we obtain a simultaneous approx...

In this paper, we introduce the concepts of lacunary statistical
convergence and strongly lacunary Cesàro summability of delta
measurable functions on time scales and obtain some inclusion results
between them. We also display some examples containing discrete and
continuous cases.

We generalize and develop the Korovkin-type
approximation theory by using an appropriate abstract space. We show that
our approximation is more applicable than the classical one. At the end, we
display some applications.

In this paper, we introduce the concept of statistical convergence of delta measurable real-valued functions defined on time scales. The classical cases of our definition include many well-known convergence methods and also suggest many new ones. We obtain various characterizations on statistical convergence.

In this paper we show that it is possible to approximate a continuous and 2 pi-periodic function on the disk centered at origin with radius pi by means of double Poisson-Cauchy singular integral operators which do not need to be positive in general. Our results cover not only the classical approximation but also the statistical approximation proces...

We study some ideal convergence results of 𝑘-positive
linear operators defined on an appropriate subspace of the space of all analytic
functions on a bounded simply connected domain in the complex plane. We
also show that our approximation results with respect to ideal convergence are
more general than the classical ones.

We study some ideal convergence results of k-positive linear operators defined on an
appropriate subspace of the space of all analytic functions on a bounded simply connected
domain in the complex plane. We also show that our approximation results with respect to
ideal convergence are more general than the classical ones

We study the statistical approximation properties of real and complex Post-Widder operators based on q-integers. We also obtain a Voronovskaja-type formula in statistical sense for these operators.

In this paper, using the notion of A-statistical convergence from the summability theory, we obtain a Korovkin-type theorem for the approximation by means of matrix-valued linear positive operators. We show that our theorem is more applicable than the result introduced by S. Serra-Capizzano [A Korovkin based approximation of multilevel Toeplitz mat...

In this paper, by including high order derivatives of functions being approximated, we introduce a general family of the linear positive operators constructed by means of the Chan–Chyan–Srivastava multivariable polynomials and study a Korovkin-type approximation result with the help of the concept of A-statistical convergence, where A is any non-ne...

In this paper, we introduce the complex Gauss–Weierstrass integral operators defined on a space of analytic functions in two variables on the Cartesian product of two unit disks. Then, we study the geometric properties and statistical approximation process of our operators.

In this paper, we obtain a statistical Voronovskaya-type theorem for the Szász-Mirakjan-Kantorovich (SMK) operators by using
the notion of A-statistical convergence, where A is a non-negative regular summability matrix.
Keywords
A-statistical convergence–Szász-Mirakjan operators–Korovkin-type approximation theorem–Voronovskaya-type theorem

In this work, we further develop the Korovkin-type approximation theory by utilizing a fuzzy logic approach and principles
of neoclassical analysis, which is a new branch of fuzzy mathematics and extends possibilities provided by the classical analysis.
In the conventional setting, the Korovkin-type approximation theory is developed for continuous...

In this paper, we introduce a general modification of the classical Baskakov operators which do not need to preserve the test
function x
2. Then, we study an approximation theorem, a Voronovskaya theorem, and various local approximation results for our modified
Baskakov operators.
Mathematics Subject Classification (2010)41A25–41A36

This study is the continuation of our earlier works [3, 4]. Here, we mainly investigate the global approximation behavior of modified Szasz-Mirakjan operators presented in the papers mentioned above.

The conventional continuity of a function was further advanced by the concept of approximate continuity introduced by Denjoy to solve some problems of differentiation and integration. According to this new type of continuity, the classical continuity conditions may be true not necessarily everywhere, but almost everywhere with respect to some measu...

In this paper, we give some statistical approximation results for the double smooth Picard singular integral operators defined on L-p-spaces, which are not positive in general. Also, displaying a nontrivial example we demonstrate that our statistical L-p-approximation is stronger than the ordinary one.

In this paper, a q-based generalization of Meyer-Konig and Zeller (MKZ) operators in several variables are introduced. A Korovkin-type approximation theorem via A-statistical convergence is obtained and their various A-statistical approximation properties are investigated when A is any non-negative regular summability matrix. (C) 2010 Mathematics S...

At first we construct a sequence of bivariate smooth Picard singular integral operators which do not have to be positive in
general. After giving some useful estimates, we mainly prove that it is possible to approximate a function by these operators
in statistical sense even though they do not obey the positivity condition of the statistical Korovk...

In this chapter, we relax the positivity condition of linear operators in the Korovkin-type approximation theory via the concept
of statistical convergence. Especially, we prove some Korovkin-type approximation theorems providing the statistical convergence
to derivatives of functions by means of a class of linear operators. This chapter relies on...

In this chapter, we consider non-negative regular summability matrix transformations in the approximation by fuzzy positive
linear operators, where the test functions are trigonometric. So, we mainly obtain a trigonometric fuzzy Korovkin theorem
by means of A-statistical convergence. We also compute the rates of A-statistical convergence of a seque...

In this chapter, we study statistical Lp-approximation properties of the bivariate Gauss-Weierstrass singular integral operators which are not positive in general.
Furthermore, we introduce a non-trivial example showing that the statistical Lp-approximation is more powerful than the ordinary case. This chapter relies on [23].

In this chapter, we obtain some Korovkin-type approximation theorems for multivariate stochastic processes with the help of
the concept of A-statistical convergence. A non-trivial example showing the importance of this method of approximation is also introduced.
This chapter relies on [26].

In this chapter, we get some statistical Korovkin-type approximation theorems including fractional derivatives of functions.
Furthermore, we demonstrate that these results are more applicable than the classical ones. This chapter relies on [21].

In this chapter, we investigate some statistical approximation properties of the bivariate complex Picard integral operators.
Furthermore, we show that the statistical approach is more applicable than the well-known aspects. This chapter relies on
[24].

In this chapter, with the help of the notion of A-statistical convergence, we get some statistical variants of Baskakov’s results on the Korovkin-type approximation theorems.
This chapter relies on [16].

In this chapter, we prove some Korovkin-type approximation theorems providing the statistical weighted convergence to derivatives
of functions by means of a class of linear operators acting on weighted spaces. We also discuss the contribution of these
results to the approximation theory. This chapter relies on [19].

In this chapter, we present the complex Gauss-Weierstrass integral operators defined on a space of analytic functions in two
variables on the Cartesian product of two unit disks. Then, we investigate some geometric properties and statistical approximation
process of these operators. This chapter relies on [30].

The main idea of statistical convergence is to demand convergence only for a majority of elements of a sequence. This method of convergence has been investigated in many fundamental areas of mathematics such as: measure theory, approximation theory, fuzzy logic theory, summability theory, and so on. In this monograph we consider this concept in app...

In this chapter, we obtain a statistical fuzzy Korovkin-type approximation result with high rate of convergence. Main tools used in this work are statistical convergence and higher order continuously differentiable functions in the fuzzy sense. An application is also given, which demonstrates that the statistical fuzzy approximation is stronger tha...

In this paper, we obtain fuzzy approximations to fuzzy differentiable functions by means of fuzzy linear operators whose positivity condition and classical limits fail. In order to get more powerful results than the classical approach we investigate the effects of matrix summability methods on the fuzzy approximation. So, we mainly use the notion o...

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 r...

In this work, we study the statistical approximation properties of the double-complex Picard integral operators. We also show that our statistical approach is more applicable than the classical one.

In this paper, we estimate the rates of pointwise approximation of certain King-type positive linear operators for functions with derivative of bounded variation. We also extend our results to the statistical approximation process via the concept of statistical convergence.

In this article, we study Korovkin-type approximation theorems for multivariate stochastic processes via the concept of A-statistical convergence. A non-trivial example expressing the importance of our results is also presented.

In this paper, we investigate the problem of statistical approximation to a function by means of positive linear operators
defined on a modular space. Especially, in order to get more powerful results than the classical aspects we mainly use the
concept of statistical convergence. A non-trivial application is also presented.
KeywordsPositive linea...

In this article, we obtain various Opial-type inequalities on time scales via the notion of the diamond-alpha derivative which
is a general concept covering both delta and nabla derivatives on time scales.
KeywordsDynamic equation-Opial’s inequality-Time scale-Delta derivative-Nabla derivative-Diamond-alpha derivative
Mathematics Subject Classific...

We first construct a sequence of double smooth Picard singular integral operators which do not have to be positive in general. After giving some useful estimates, we mainly show that it is possible to approximate a function by these operators in statistical sense even though they do not obey the positivity condition of the statistical Korovkin theo...

In this paper, introducing a general modification of the classical Szász-Mirakjan-Kantorovich (SMK) operators, we study their global approximation behavior. Some special cases are also presented.

In this paper, we obtain some fuzzy Korovkin-type results based on statistical rates. Our results cover not only the fuzzy Korovkin theory but also the statistical fuzzy Korovkin theory. Important applications and remarks are also presented.

In this paper, we study statistical Lp-approximation properties of the double Gauss–Weierstrass singular integral operators which do not need to be positive. Also, we present a non-trivial example showing that our statistical Lp-approximation is stronger than the ordinary one.

In this paper, we study a general Korovkin-type approximation theory by using the notion of ideal convergence which includes many convergence methods, such as, the usual convergence, statistical convergence, A-statistical convergence, etc. We mainly compute the rate of ideal convergence of sequences of positive linear operators.

In this study, using the notion of statistical convergence, we obtain various statistical approximation theorems for a general sequence of max-product approximating operators, including Shepard type operators, although its classical limit fails. We also compute the corresponding statistical rates of the approximation.

In this paper, for a general modification of the classical Szasz-Mirakjan-Kantorovich operators, we obtain many local approximation results including the classical cases. In particular, we obtain a Korovkin theorem, a Voronovskaya theorem, and some local estimates for these operators.

In this chapter, we obtain some statistical approximation results for the bivariate smooth Picard singular integral operators defined on Lp
-spaces, which do not need to be positive in general. Also, giving a non-trivial example we show that the statistical Lp
-approximation is stronger than the ordinary one. This chapter relies on [29].

In this chapter, we develop the classical trigonometric Korovkin theory by using the concept of statistical convergence from
the summability theory and also by considering the fractional derivatives of trigonometric functions. We also show that these
results are more applicable than the classical ones. This chapter relies on [27].

In this chapter, we study the statistical approximation properties of a sequence of bivariate smooth Gauss-Weierstrass singular
integral operators which are not positive in general. We also show that the statistical approximation results are stronger
than the classical uniform approximations. This chapter relies on [28].

In this paper, in order to converge faster to a function being approximated we modify two different Bernstein-Durrmeyer type operators introduced in [5] and [7] such that linear functions are preserved.

In this paper, considering A-statistical convergence instead of Pringsheim's sense for double sequences, we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued Bogel-type continuous and periodic functions on the whole real two-dimensional space. A strong application is also...

This paper presents the stability analysis of equilibrium points of a general continuous time population dynamics model involving predation subject to an Allee effect which occurs at low population density. The mathematical results and numerical simulations show that the system subject to an Allee effect takes much longer time to reach its stable s...

In this article, using the notion of statistical convergence, we relax the hypotheses of the well-known theorems from classical complex analysis, such as Weierstrass' Theorem, Montel's Theorem and Hurwitz's Theorem. So, we obtain more powerful results than the classical ones in complex analysis.

This paper is mainly connected with the approximation properties of Meyer-König and Zeller (MKZ) type operators. We first introduce a general sequence of MKZ operators based on q-integers and then obtain a Korovkin-type approximation theorem for these operators. We also compute their rates of convergence by means of modulus of continuity and the el...

This paper presents the stability analysis of equilibrium points of a general discrete-time population dynamics involving predation with and without Allee effects which occur at low population density. The mathematical analysis and numerical simulations show that the Allee effect has a stabilizing role on the local stability of the positive equilib...

In this paper, we study the stability of a discrete-time predator–prey system with and without Allee effect. By ana-lyzing both systems, we first obtain local stability conditions of the equilibrium points without the Allee effect and then exhibit the impact of the Allee effect on stability when it is imposed on prey population. We also show the st...

In this article, we obtain strong Korovkin-type approximation theorems for stochastic processes by using the concept of A-statistical convergence from the summability theory.

In this article, we present a statistical fuzzy Korovkin type approximation result with high rate of convergence. Main tools used in this work are statistical convergence and higher order continuously differentiable functions in the fuzzy sense. An example is also displayed, which shows that our statistical fuzzy approximation is stronger than the...

This paper presents the stability analysis of equilibrium points of a continuous population dynamics with delay under the Allee effect which occurs at low population density. The mathematical results and numerical simulations show the stabilizing role of the Allee effects on the stability of the equilibrium point of this population dynamics. Editor...

We obtain some oscillation criteria for a class of neutral difference equations with time delays. We also investigate the behavior of the eventually positive solutions of these equations. To verify our results we give various numerical simulations by using the MATLAB programming. Editorial remark: There are doubts about a proper peer-reviewing proc...

In this study, without preserving some test functions, we present a new approach in obtaining a better error estimation in the approximation by means of positive linear operators. We also show that our method can be applied to many well-known approximation operators.

Approximately continuous functions were first introduced in connection with problems of differentiation and integration. The main idea of the approximate continuity of a function f is that the continuity conditions are true not necessarily everywhere but only almost everywhere with respect to some measure, e.g., Borel measure or Lebesgue measure. A...

In this paper, we relax the positivity condition of linear operators in the Korovkin-type approximation theory via the concept of statistical convergence. Especially, we obtain various Korovkin-type approximation theorems providing the statistical convergence to derivatives of functions by means of a class of linear operators.

## Projects

Project (1)

In this project, convolution and Mellin type integral operators will be studied. As it is known, beside the approximation theory, convolution and Mellin type operators is quite useful in image processing, optical physics, signal processing, seismic engineering and etc. On the other hand, summability methods give alternative solutions when the classical estimation does not hold in approximation theory. In the literature, there are many summability methods such as power series method, Abel’s method, Borel’s method, Cesaro summability and almost convergence. However, when compared with the others, Bell’s method is quite general and it consists Cesaro summability, almost convergence and order summability. Although like the other methods, Bell’s method has many successful applications on positive linear operators, to the best of our knowledge, there are no applications of it to the nonlinear operators except the project coordinator’s ones. Thereby, a huge lack of applications of Bell type methods attract the attention. The main aim of this project is to fill this gap in the literature. Beside the theoretical estimations, it is also planned to include applications for the real life in this project.
Image processing technics take part not only for engineers but also for the mathematicians. Images resolutions are enhanced and clarified with the help of approximation theory. Thanks to this, it is both used for making diagnosing of the disease easy and clarifying the computer tomography. It has many applications too in the other different fields. One another aim of this project is to give concrete examples for the question “what does it do?”.
We can sum up the basic research problems in the following main three titles for now.
1. Applicability of summation process to nonlinear operators,
2. Characterizations of absolutely continuous functions,
3. Application of approximation theory to the image processing.