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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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
Published electronically: July 6, 2004
MathSciNet review: 2068983
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Abstract: Let $U$ be a normed space compactly embedded in a space $V$, let $\{U_n^*\}$ be a sequence of finite-dimensional subspaces of the dual space $U^*$, and let \[ U^{(n)}=\{ u\in U \mid \chi (u)=0, \chi \in U_n^*\}. \] If the sequence $\{U_n^*\}$ is asymptotically dense in $U^*$, then $\|I_n\|\to 0$, where $I_n$ denotes the operator that embeds $U^{(n)}$ in $V$. In particular, if $\{P_n\}$ is a sequence of finite-dimensional projections in $U$, and the sequence $\{{\mathcal R}(P_n^*)\}$ is asymptotically dense in $U^*$, then $\|u-P_nu\|_V/\|u-P_nu\|_U\to 0$. The norm $\|I_n\|$ is estimated in terms of the best approximation of the elements of the unit ball in $V^*$ (this ball is compact in $U^*$) by elements of $U_n^*$. 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

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