Complex Korovkin Theory via Inequalities: A Quantitative Approach

Abstract

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