Some convergence problems for weak norms

Author:
I. K. Daugavet

Translated by:
A. Plotkin

Original publication:
Algebra i Analiz, tom **15** (2003), nomer 4.

Journal:
St. Petersburg Math. J. **15** (2004), 575-585

MSC (2000):
Primary 46N40

DOI:
https://doi.org/10.1090/S1061-0022-04-00823-4

Published electronically:
July 6, 2004

MathSciNet review:
2068983

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a normed space compactly embedded in a space , let be a sequence of finite-dimensional subspaces of the dual space , and let

If the sequence is asymptotically dense in , then , where denotes the operator that embeds in . In particular, if is a sequence of finite-dimensional projections in , and the sequence is asymptotically dense in , then . The norm is estimated in terms of the best approximation of the elements of the unit ball in (this ball is compact in ) by elements of . Usually, for projection methods of solving functional equations, the metric in which the convergence should be studied is dictated by the general convergence theorems (we mean, e.g., the energy metric for the Ritz method). The above arguments make it possible to establish a faster convergence of projection methods in weaker metrics. Some results of this type are obtained in the paper for the Ritz and Galerkin methods and for the method of moments.

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Additional Information

**I. K. Daugavet**

Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, St. Petersburg 198504, Russia

DOI:
https://doi.org/10.1090/S1061-0022-04-00823-4

Keywords:
Superconvergence,
projection,
projection methods,
Ritz and Galerkin methods,
method of moments

Received by editor(s):
December 18, 2002

Published electronically:
July 6, 2004

Article copyright:
© Copyright 2004
American Mathematical Society