Result: Differentiable operators of nearly best approximation

Title:
Differentiable operators of nearly best approximation
Authors:
P V Al'brecht
Source:
Izvestiya: Mathematics. 63:631-647
Publisher Information:
Steklov Mathematical Institute, 1999.
Publication Year:
1999
Subject Terms:
Best approximation, Chebyshev systems, epsilon-selection, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), normed linear space, Approximation by arbitrary linear expressions, subspace, 0101 mathematics, best approximation, 01 natural sciences
Document Type:
Academic journal Article
File Description:
application/xml
ISSN:
1468-4810
1064-5632
DOI:
10.1070/im1999v063n04abeh000249
Access URL:
https://zbmath.org/1517670
https://doi.org/10.1070/im1999v063n04abeh000249
http://ui.adsabs.harvard.edu/abs/1999IzMat..63..631A/abstract
Accession Number:
edsair.doi.dedup.....11f6d87d5a51293f5461b0fbb7baeddc
Database:
OpenAIRE

Further Information

Let \(X\) be a normed linear space, and let \(Y\subset X\) be its finite-dimensional subspace. \textit{V. I. Berdyshev} [Math. USSR, Izv. 16, 431-456 (1981; Zbl 0468.90084)] proved that for any \(Y\) there exists a Lipschitzian multiplicative \(\varepsilon\)-selection \(\varphi:X\to Y\) (i.e., \(\|x-\varphi(x)\|\leq \text{dist}(x,Y)(1+\varepsilon)\)). The author proves that in this setting there exists a multiplicative \(\varepsilon\)-selection such that its smoothness will coincide with that of norm in~\(X\). It is also shown that it is impossible to find an \(\varepsilon\)-selection of greater smoothness in \(L^[0,1]\).