Ismail Aslan

Ismail Aslan
Hacettepe University · Department of Mathematics

Doctor of Philosophy

About

19
Publications
925
Reads
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63
Citations
Additional affiliations
February 2017 - May 2020
Hacettepe University
Position
  • Research Assistant
December 2014 - February 2017
Middle East Technical University
Position
  • Research Assistant
Education
August 2014 - April 2019
TOBB University of Economics and Technology
Field of study
  • Mathematics, Approximation theory, Summability methods
August 2012 - August 2014
September 2006 - October 2011
Hacettepe University
Field of study
  • Mathematics

Publications

Publications (19)
Article
In this study, we construct Kantorovich variant of max-min kind operators, which are non-linear. By using these new operators, we obtain some uniform approximation results in N-dimension (N ≥ 1). Then, we estimate the error with the help of Hölder continuous functions and modulus of continuity. Furthermore, we give some illustrative applications to...
Article
Full-text available
We consider convolution-type nonlinear integral operators endowed with Musielak-Orlicz φ -variation. Ouraim is to get more powerful approximation results with the help of summability methods. In this study, we use φ -absolutely continuous functions for our convergence results. Moreover, we study the order of approximation using suitableLipschitz cl...
Article
In the literature, there are various methods to obtain the fractal sets such as escape time algorithm, L-systems and iterated function system (IFS), etc. In this study, we aim to approximate to the classical fractals by using non-affine contraction mappings. In order to get these non-affine map-pings, we utilize from the sequences of suitable Lipsc...
Article
Full-text available
In this note, we construct a pseudo-linear kind discrete operator based on the continuous and nondecreasing generator function. Then, we obtain an approximation to uniformly continuous functions through this new operator. Furthermore, we calculate the error estimation of this approach with a modulus of continuity based on a generator function. The...
Article
Full-text available
In the present paper, our purpose is to obtain a nonlinear approximation by using convergence in ϕ-variation. Angeloni and Vinti prove some approximation results considering linear sampling-type discrete operators. These types of operators have close relationships with generalized sampling series. By improving Angeloni and Vinti's one, we aim to ge...
Article
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...
Article
Full-text available
We investigate the approximation properties of nonlinear integral operators of the convolution type. In this approximation, we use functions of bounded variation based on the appropriate functionals. To get more general results, we consider Bell-type summability methods in the approximation. Moreover, we examine the rate of approximation. Then, usi...
Article
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...

Questions

Question (1)
Question
In 1-dimension, variation of a function gives us the oscillation of f over the y axis it takes. But in 2-d does it give the area of its shadow or doesn’t have a geometrical meaning?

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Projects

Projects (2)
Project
In this project, it is aimed to approach fractals which are obtained by affine or non-affine contraction mappings, by using new non-affine transformations. For this purpose firstly, non-affine transform sequences will be tried to be obtained with the help of the known contraction mappings using the Lipschitz continuity property of the kernel function sequences used in linearization of non-linear operators. Later, approximations to fractals are expected to be obtained by using iterated function systems and escape time algorithm methods. Construction of new fractals which are obtained by new non-affine contraction mappings is also among the targets of the project. Also, the approaches for some well-known fractals will be visualized with the help of algorithms.
Project
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.