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Öğe Applications of q-calculus in operator theory(Springer New York, 2013) Aral A.; Gupta V.; Agarwal R.P.The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations. q-Calculus is a generalization of many subjects, such as hypergeometric series, complex analysis, and particle physics. This monograph is an introduction to combining approximation theory and q-Calculus with applications, by using well- known operators. The presentation is systematic and the authors include a brief summary of the notations and basic definitions of q-calculus before delving into more advanced material. The many applications of q-calculus in the theory of approximation, especially on various operators, which includes convergence of operators to functions in real and complex domain? forms the gist of the book. This book is suitable for researchers and students in mathematics, physics and engineering, and for professionals who would enjoy exploring the host of mathematical techniques and ideas that are collected and discussed in the book. © Springer Science+Business Media New York 2013. All rights are reserved.Öğe Approximation of Some Classes of Functions by Landau Type Operators(Birkhauser, 2021) Agratini O.; Aral A.This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in Lp spaces and in weighted Lp spaces (1 ? p< ?). Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of continuous functions on convex compacts. The evaluation of the rate of convergence in one and multidimensional cases is performed by using adequate moduli of smoothness. © 2020, Springer Nature Switzerland AG.Öğe Approximation properties of Ibragimov-Gadjiev-Durrmeyer operators on Lp(R+)(Warsaw University, 2017) Ulusoy G.; Aral A.We deal with the approximation properties of a new class of positive linear Durrmeyer type operators which oer a reconstruction of integral type operators including well known Durrmeyer operators. This reconstruction allows us to investigate approximation properties of the Durrmeyer operators at the same time. It is rst shown that these operators are a positive approximation process in Lp R+. While we are showing this property of the operators we consider the Ditzian-Totik modulus of smoothness and corresponding Kfunctional. Then, weighted norm convergence, whose proof is based on Korovkin type theorem on Lp R+, is given. At the end of the paper we show several examples of classical sequences that can be obtained from the Ibragimov-Gadjiev-Durrmeyer operators. ' 2017 Glsm Ulusoy and Ali Aral,Öğe Generalized Szász Durrmeyer operators(2011) Aral A.; Gupta V.In this paper, we introduce and study a new sequence of positive linear operators acting on the spaces of continuous function on positive semi-axis. These operators are defined by means of the q-integral, which generalize the Szász Durrmeyer operators. We study their approximation property by presenting a direct approximation theorem on polynomial weighted spaces and the rate of convergence by means of weighted modulus of continuity. Also, we establish an asymptotic formula with respect to weighted norm. © 2011 Pleiades Publishing, Ltd.Öğe On Approximation Properties of Generalized Durrmeyer Operators(Springer New York LLC, 2016) Aral A.; Acar T.The concern of this paper is to introduce new generalized Durrmeyer-type operators from which classical operators can be obtained as a particular case, inspiring from the Ibragimov–Gadjiev operators (Gadjiev and Ibragimov, Soviet Math. Dokl. 11, 1092–1095, (1970) [8]).After the construction of newDurrmeyer operators is given, we obtain some pointwise convergence theorems and Voronovskaya-type asymptotic formula for new Durrmeyer-type operators. We establish a quantitative version of the Voronovskaya-type formula with the aid of the weighted modulus of continuity. Some special cases of new operators are presented as examples. © Springer Science+Business Media Singapore 2016.Öğe On gauss-weierstrass type integral operators(Walter de Gruyter GmbH, 2010) Anastassiou G.A.; Aral A.In this paper, we introduce a generalization of Gauss-Weierstrass operators based on q-integers using the q-integral and we call them q-Gauss-Weierstrass integral operators. For these operators, we obtain a convergence property in a weighted function space using Korovkin theory. Then we estimate the rate of convergence of these operators in terms of a weighted modulus of continuity. We also prove optimal global smoothness preservation property of these operators. © 2010 Warsaw University. All rights reserved.Öğe On generalized picard integral operators(Springer Singapore, 2018) Aral A.In the paper, we constructed a class of linear positive operators generalizing Picard integral operators which preserve the functions eµx and e2µx, µ > 0. We show that these operators are approximation processes in a suitable weighted spaces. The uniform weighted approximation order of constructed operators is given via exponential weighted modulus of smoothness.We also obtain their shape preserving properties considering exponential convexity. © Springer Nature Singapore Pte Ltd. 2018.Öğe Weighted approximation properties of generalized Picard operators(2011) Yilmaz B.; Aral A.; Başcanbaz-Tunca G.In this work, we continue the study of generalized Picard operator P?,? ([2]) depending on nonisotropic ?-distance, in the direction of weighted approximation process. For this purpose, we first define weighted n-dimensional Lp space by involving weight depending on nonisotropic distance. Then we introduce a new weighted ?-Lebesgue point depending on nonisotropic distance and study pointwise approximation of ?,? to the unit operator at these points. Also, we compare the order of convergence at the weighted ?-Lebesgue point with the order of convergence of the operators to the unit operator. Finally, we show that this type of convergence also occurs with respect to nonisotropic weighted norm. © 2011 EUDOXUS PRESS, LLCAll rights reserved.
