Remark on weighted approximation properties of generalized Picard operators

Abstract

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.