High order approximation theory for Banach space valued functions

Authors

  • George A. Anastassiou University of Memphis, USA

DOI:

https://doi.org/10.33993/jnaat462-1112

Keywords:

Banach space valued differentiable functions, positive linear operator, convexity, modulus of continuity, rate of convergence
Abstract views: 332

Abstract

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued di§erentiable functions to the unit operator. 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 whose we study their approximation properties. We derive pointwise and uniform estimates which imply the approximation of these operators to the unit assuming di§erentiability of functions. At the end we study the special case where the high order derivative of the on hand function fulÖlls a convexity condition resulting into sharper estimates.

MR3724631

Downloads

Download data is not yet available.

References

G.A. Anastassiou, Moments in Probability and Approximation Theory, Pitman Research Notes in Math., 287, Longman Sci. & Tech., Harlow, U.K., 1993.

G.A. Anastassiou, Lattice homomorphism - Korovkin type inequalities for vector valued functions, Hokkaido Mathematical Journal, Vol. 26 (1997), pp. 337-364, https://doi.org/10.14492/hokmj/1351257969 DOI: https://doi.org/10.14492/hokmj/1351257969

T. Nishishiraho, Bernstein-type approximation processes for vector-valued functions, Acta Math. Hungar. 84 (1999), no. 1-2, pp. 149-158. DOI: https://doi.org/10.1023/A:1006611205884

R. Paltanea, Vector variants of some approximation theorems of Korovkin and of Sendov and Popov. Constructive theory of functions, pp. 366-373, DARBA, Sofia, 2003.

R. Paltanea, Approximation of functions in Banach spaces using linear and positive operators, in Proceedings of RoGer seminar 2004, Mediamira Science Publisher, Cluj-Napoca, 2004, pp. 5-20.

T. Popoviciu, Sur l’approximation des fonctions convexes d’ordre superieur, Mathematica, 10 (1935), pp. 49–54, Cluj, Romania, https://ictp.acad.ro/approximation-higher-order-convex-functions-i/

G.E. Shilov, Elementary Functional Analysis, Dover Publications, Inc., New York, 1996

Downloads

Published

2017-11-08

How to Cite

Anastassiou, G. A. (2017). High order approximation theory for Banach space valued functions. J. Numer. Anal. Approx. Theory, 46(2), 113–130. https://doi.org/10.33993/jnaat462-1112

Issue

Section

Articles