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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdŕ (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Ličge, Belgium)

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Bernstein Theorems in an abstract setting
by
J. M. Almira
Departamento de Matemáticas. Universidad de Jaén
Coauthors: U. Luther (T.U. Chemnitz, Germany)

In 1912, S.N. Bernstein proved that for each null sequence {en} there exists a periodic function f such that its errors of best trigonometric polynomial approximation satisfy En(f)=en for all n. This result was proved by Timan for sequences of errors {E(f, Xn)}, where {Xn} is any chain of finite dimensional subspaces of a separable Banach space X. By the use of the Baire category theorem, Shapiro (see [2]) proved the existence of elements x in X such that E(x, Xn) =/= O(en) without assumptions on dim(Xn), and Albinus (see [1]) gave examples of metric spaces of measurable functions which admit a chain of finite dimensional subspaces {Xn} such that E(x, Xn) = O(en) for all f in X and a certain null sequence {en}. In this paper we will give new proofs of several of these results and we will also improve a few of them. For example, we will prove Bernstein's theorem in the Hilbert setting without assumptions on dim(Xn).

[1] Albinus, G., Remarks on a theorem of S.N. Bernstein, Studia Math. 38 (1970) 227-234.

[2] Shapiro, H., Some negative theorems of Approximation Theory, Michigan Math. J. 11 (1964) 211-217.

(T)

Date received: April 6, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caei-76.