Complex Korovkin theory

Let K be a compact convex subspace of C and C (K; C) the space of continuous functions from K into C. We consider bounded linear functionals from C (K; C) into C and bounded linear operators from C (K; C) into itself. We assume that these are bounded by companion real positive linear entities, respectively. We study quantitatively the rate of convergence of the approximation of these linearities to the corresponding unit elements. Our results are inequalities of Korovkin type involving the complex modulus of continuity and basic test functions

Summability on Mellin-type nonlinear integral operators

In this study, approximation properties of the Mellin-type nonlinear integral operators defined on multivariate functions are investigated. In order to get more general results than the classical aspects, we mainly use the summability methods defined by Bell. Considering the Haar measure with variation semi-norm in Tonelli's sense, we approach to the functions of bounded variation. Similar results are also obtained for uniformly continuous and bounded functions. Using suitable function classes we investigate the rate of convergence in the approximation. Finally, we give a non-trivial application verifying our approach

Quantitative Multivariate Complex Korovkin Approximation Theory

Let K be a compact convex subset of (Formula Presented), and C(K, C) be the space of continuous functions from K into C. We consider bounded linear operators from C(K, C) into itself. We assume that these are bounded by companion positive linear operators from C(K, R) into itself. We study quantitatively the rate of convergence of the approximation and high order approximation of these multivariate complex operators to the unit operators. Our results are inequalities of Korovkin type involving the multivariate complex modulus of continuity of the engaged function or its partial derivatives and basic test functions