Quantitative multivariate complex Korovkin theory

Abstract

Let K be a compact convex subset of Ck, k≥ 2 , 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. On the way to prove our main results we establish a complex multivariate Taylor’s formula and results on complex multivariate modulus of continuity. We consider related approximations under convexity and we finish with important applications.

Publication Title

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas

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