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    Generalized Picard singular integrals
    (Pergamon-Elsevier Science Ltd, 2009) Anastassiou, George A.; Aral, Ali
    in this article, we introduce and study a new type of Picard singular integral operators on R-n constructed by means of the concept of the nonisotropic beta-distance and the q-exponential functions. The central role here is played by the concept of nonisotropic beta-distance, which allows one to improve and generalize the results given for classical Picard and q-Picard singular integral operators. in order to obtain the rate of convergence we introduce a new type of modulus of continuity depending on the nonisotropic beta-distance with respect to the uniform norm. Then we give the definition of beta-Lebesque points depending on nonisotropic beta-distance and a pointwise approximation result shown at these points. Furthermore, we study the global smoothness preservation property of these new type Picard singular integral operators and prove a sharp inequality. (C) 2008 Elsevier Ltd. All rights reserved.
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    Öğe
    ON GAUSS-WEIERSTRASS TYPE INTEGRAL OPERATORS
    (Sciendo, 2010) Anastassiou, George A.; Aral, Ali
    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