Generalized Picard singular integrals
In this article, we introduce and study a new type of Picard singular integral operators on Rn constructed by means of the concept of the nonisotropic β-distance and the q-exponential functions. The central role here is played by the concept of nonisotropic β-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 β-distance with respect to the uniform norm. Then we give the definition of β-Lebesque points depending on nonisotropic β-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. © 2008 Elsevier Ltd. All rights reserved.
Computers and Mathematics with Applications
Anastassiou, G., & Aral, A. (2009). Generalized Picard singular integrals. Computers and Mathematics with Applications, 57 (5), 821-830. https://doi.org/10.1016/j.camwa.2008.09.026