Abstract
The aim of this note is to present an elementary functional-analytical proof of a Bernstein-type inequality, given by DeVore in 1973 for one-dimensional trigonometric polynomials. This proof allows extensions of the inequality to arbitrary total biorthogonal systems in Banach spaces. Furthermore, the general inequality may indeed be used to prove a uniform-boundedness principle with rates.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation. Springer-Verlag, Berlin, 1967.
R. DeVore, Multipliers of uniform convergence. Enseignement Math. 14 (1969), 175–188.
R. DeVore, An extension of Bernstein’s inequality, pp. 325–336 in G. G. Lorentz (ed.), Approximation Theory (Proc. Conf., Austin, Texas, 1973). Academic Press, New York, 1973.
W. Dickmeis and R. J. Nessel, A unified approach to certain counterexamples in approximation theory in connection with a uniform boundedness principle with rates. J. Approximation Theory 31 (1981), 161–174.
W. Dickmeis and R. J. Nessel, A uniform boundedness principle with rates and an application to linear processes. In: P. L. Butzer, E. Görlich, and B. Sz.-Nagy (eds.), Functional Analysis and Approximation (Proc. Conf., Oberwolfach, 1980, ISNM60), Birkhäuser Verlag, Basel, Stuttgart, and Boston, 1981 (in print).
W. Dickmeis and R. J. Nessel, On uniform boundedness principles and Banach-Steinhaus theorems with rates. Numer. Funct. Anal. Optim. 3 (1981), 19–52.
A. F. Timan, Theory of Approximation of Functions. Pergamon Press, New York, 1963.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer Basel AG
About this chapter
Cite this chapter
Dickmeis, W., Nessel, R.J. (1983). On an Inequality of Devore. In: Beckenbach, E.F., Walter, W. (eds) General Inequalities 3. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 64. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6290-5_30
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6290-5_30
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6292-9
Online ISBN: 978-3-0348-6290-5
eBook Packages: Springer Book Archive