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Öğe The estimates of approximation by using a new type of weighted modulus of continuity(Pergamon-Elsevier Science Ltd, 2007) Gadjiev, A. D.; Aral, A.In this paper, we introduce a new type modulus of continuity for function f belonging to a particular weighted subspace of C (0, infinity) and show that it has some properties of ordinary modulus of continuity. We obtain some estimates of approximation of functions with respect to a suitable weighted norm via the new type moduli of continuity. Finally, we give some examples. (C) 2007 Elsevier Ltd. All rights reserved.Öğe Generalization of Szasz operators: quantitative estimate and bounded variation(Vasyl Stefanyk Precarpathian Natl Univ, 2021) Bozkurt, K.; Limmam, M. L.; Aral, A.Difference of exponential type Szasz and Szasz-Kantorovich operators is obtained. Similar es-timates are given for higher order mu-derivatives of the Szasz operators and the Szasz-Kantorovich type operators acting on the same order mu-derivative of the function. These differences are given in quantitative form using the first modulus of continuity. Convergence in variation of the operators in the space of functions with bounded variation with respect to the variation seminorm is obtained. The results propose a general framework covering the results provided by previous literature.Öğe I-convergence of positive linear operators on Lp weighted spaces(Eudoxus Press, Llc, 2008) Dirik, F.; Aral, A.; Demirci, K.In this paper, using the concept of I-convergence we prove a Korovkin type approximation by means of positive linear operators defined on the weighted space L-p,L-w(R). Also we state its n-dimensional analogue for the weighted space L-p,L-Omega(R-n). Also we display an example such that our method of convergence is stronger than the usual convergence in the weighted spaces L-p,L-w(R) and L-p,L-Omega(R-n).Öğe A Note on the Difference of Positive Operators and Numerical Aspects(SPRINGER BASEL AG, 2020) Aral, A.; Erbay, H.Recently, the differences between the two operators get the attention of scientists in approximation theory due to their ability to provide the approximation properties of the operator in the difference if the approximation properties of other operator in the difference are known. In other words, it gives us the ability to obtain a simultaneous approximation. On the other hand, the exponential-type operators possess better approximation properties than classical ones. Herein, the differences of the exponential-type Bernstein and Bernstein-Kantorovich operators and their differences between their higher order mu-derivatives applied to a function with the operators applied to the same order of mu-derivative of the function are considered. The estimates in the quantitative form are given in terms of the first modulus of continuity. Furthermore, quantitative estimates of the differences between Bernstein and Bernstein-Kantorovich operators as well as their Gruss-type difference are obtained. The numerical results obtained are in the direction of the theory, and some of them are presented.Öğe On behaviour of the Riesz and generalized Riesz potentials as order tends to zero(Element, 2007) Gadjiev, A. D.; Aral, A.; Aliev, Ilham A.In this paper, we present the Riesz potentials I-alpha and the generalized Riesz potentials I-v(alpha) as the families of positive linear operators, depending on parameter alpha > 0. We investigate their pointwise convergence and convergence in the norm as alpha -> 0. We investigate also the order of approximation of these families and show in particular that the order of approximation at the Lipschitz points is independent from Lipschitz degree.Öğe On Modified Mellin-Gauss-Weierstrass Convolution Operators(Springer Basel Ag, 2022) Aral, A.; Erbay, H.; Yilmaz, B.Mellin transform has various applications to real-life problems in function approximation, signal processing, and image recognition, thus, it has been the main ingredient of many studies in diverse fields. This study is devoted to Mellin operators and their variants to improve approximation accuracy and approximate ratio. Two Mellin type operators are reconstructed by using two sequences of functions to enable lower pointwise approximation error as well as higher pointwise convergence rate. Keeping the idea of Mellin convolution, these classes aim to be associated with functions defined on the semi-real axis, and the affine and quadratic functions pairs are fixed points. It has been shown, both theoretically and numerically, that operators can be used to approximate functions pointwise. Indeed the approximation accuracy can be adjusted by tuning the parameters. Moreover, weighted approximation, as well as Voronovskaya type results, are studied throughout the paper. The advantages of each operator over the other in terms of both approximation errors and convergence rates are presented.Öğe On Modified Mellin–Gauss–Weierstrass Convolution Operators(Birkhauser, 2022) Aral, A.; Erbay, H.; Yılmaz, B.Mellin transform has various applications to real-life problems in function approximation, signal processing, and image recognition, thus, it has been the main ingredient of many studies in diverse fields. This study is devoted to Mellin operators and their variants to improve approximation accuracy and approximate ratio. Two Mellin type operators are reconstructed by using two sequences of functions to enable lower pointwise approximation error as well as higher pointwise convergence rate. Keeping the idea of Mellin convolution, these classes aim to be associated with functions defined on the semi-real axis, and the affine and quadratic functions pairs are fixed points. It has been shown, both theoretically and numerically, that operators can be used to approximate functions pointwise. Indeed the approximation accuracy can be adjusted by tuning the parameters. Moreover, weighted approximation, as well as Voronovskaya type results, are studied throughout the paper. The advantages of each operator over the other in terms of both approximation errors and convergence rates are presented. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.Öğe On the Generalized Szasz-Mirakyan Operators(Springer Basel Ag, 2014) Aral, A.; Inoan, D.; Rasa, I.In this paper, we construct sequences of Szasz-Mirakyan operators which are based on a function.. This function not only characterizes the operators but also characterizes the Korovkin set {1, rho, rho(2)} in a weighted function space. We give theorems about convergence of these operators to the identity operator on weighted spaces which are constructed using the function rho and which are subspaces of the space of continuous functions on R+. We give quantitative type theorems in order to obtain the degree of weighted convergence with the help of a weighted modulus of continuity constructed using the function rho Further, we prove some shape-preserving properties of the operators such as the rho-convexity and the monotonicity. Our results generalize the corresponding ones for the classical Szasz operators.Öğe The Picard and Gauss-Weierstrass Singular Integrals in (p, q)-Calculus(MALAYSIAN MATHEMATICAL SCIENCES SOC, 2020) Aral, A.; Deniz, E.; Erbay, H.The vast development of the techniques in both the quantum calculus and the post-quantum calculus leads to a significant increase in activities in approximation theory due to applications in computational science and engineering. Herein, we introduce (p, q)-Picard and (p, q)-Gauss-Weierstrass integral operators in terms of the (p, q)-Gamma integral. We give a general formula for the monomials under both (p, q)-Picard and (p, q)-Gauss-Weierstrass operators as well as some special cases. We discuss the uniform convergence properties of them. We show that both operators have optimal global smoothness preservation property via usual modulus of continuity. Finally, we establish the rate of approximation using the weighted modulus of smoothness. Depending on the choices of parameters p and q in the integrals, we are able to obtain better error estimation than classical ones.Öğe Remark on weighted approximation properties of generalized Picard operators(Univ Miskolc Inst Math, 2013) Yilmaz, B.; Bascanbaz-Tunca, G.; Aral, A.In this article, we still hold the study of the generalized Picard operators P-lambda,P-beta (f; q) depending on nonisotropic beta-distance given in [3]. By continuing to deal with the nonisotropic weighted L-p,L-beta (R-n) space defined in [17], we introduce a new weighted L-p,L-beta modulus of continuity depending on the nonisotropic distance to obtain the weighted rate of convergence. We show that weighted convergence rate of P-lambda,P-beta (f; q) to f can be made better not only depending on the chosen q but also the choice of beta. We get that P-lambda,P-beta (f; q) satisfy the global smoothness preservation property via weighted L-p,L-beta modulus of continuity. Also we give direct approximation property of the generalized Picard operators P-lambda,P-beta (f; q) with respect to nonisotropic weighted norm.
