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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
5/2009

Error estimates for two-dimensional Cross Approximation

Jan Schneider

Abstract

We deal with the approximation of a given function $f$ on $[0,1]^2$ by special bilinear forms $\sum_{i=1}^k g_i\otimes h_i$ via the so-called cross approximation. In particular we are interested in estimating the error function $f-\sum_{i=1}^k g_i\otimes h_i$ of the corresponding algorithm in the maximum norm. Because of the large amount of papers available that deal with similar matrix algorithms in applied situations without giving satisfactory error estimates, we concentrate on the theoretical issues of the problem in the language of functions. We connect it with related results from other areas of Analysis in an historical survey and give a lot of references. Our main result is the connection of the error of our algorithm with the error of best approximation by arbitrary bilinear forms.

Received:
Jan 9, 2009
Published:
Jan 14, 2009
MSC Codes:
65G99, 68W25, 41A50
Keywords:
Cross Approximation, error estimates, best approximation, bilinear forms

Related publications

inJournal
2010 Repository Open Access
Jan Schneider

Error estimates for two-dimensional cross approximation

In: Journal of approximation theory, 162 (2010) 9, pp. 1685-1700