Multivariate and abstract approximation theory for Banach space valued functions

• Autorzy: Anastassiou G. A.

Identyfikatory

DOI

10.1515/dema-2017-0020

• Języki publikacji: EN

Abstrakty

EN

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

Słowa kluczowe

EN

continuous Banach space valued functions; Banach space valued Fréchet differentiable functions; positive linear operator; convexity; modulus of continuity; rate of convergence

PL

przestrzeń Banacha; operator liniowy; wypukłość; moduł ciągłości; zbieżność; konwergencja

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2017
• Tom: Vol. 50, nr 1
• Strony: 208--222
• Opis fizyczny: Bibliogr. 4 poz.

Twórcy

autor

Anastassiou G. A.

• Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, U.S.A.

Bibliografia

• [1] Anastassiou G. A., Moments in probability and approximation theory, Pitman Research Notes in Math., Longman Sci. & Tech., Harlow, U.K., 1993, 287
• [2] Anastassiou G. A., Lattice homomorphism - Korovkin type inequalities for vector valued functions, Hokkaido Mathematical Journal, 1997, 26, 337-364
• [3] Cartan H., Differential calculus, Hermann, Paris, 1971
• [4] Rall L. B., Computational solution of nonlinear operator equations, John Wiley & Sons, New York, 1969

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).