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 Mat. Sb., 1994, Volume 185, Number 9, Pages 3–28 (Mi msb923)

Orders of moduli of continuity of operators of almost best approximation

P. V. Al'brecht

Abstract: Let $X$ be a normed linear space, $Y\subset X$ a finite-dimensional subspace, $\varepsilon>0$. A multiplicative $\varepsilon$-selection $M\colon K\to Y$, where $K\subset X$, is a single-valued mapping such that
$$\forall x\in K\qquad \|Mx-x\|\leqslant\inf\{\|x-y\|:y\in Y\}\cdot(1+\varepsilon).$$

It is proved in the paper that when $X=L^p(T,\Sigma,\mu)$, $1<p<\infty$, for any $Y\subset X$ and $\varepsilon>0$ there exists an $\varepsilon$-selection $M\colon K\to Y$ such that
$$\forall x_1,x_2\in K\qquad \|Mx_1-Mx_2\|\leqslant c(n,p)(1+\varepsilon^{-|1/2-1/p|})\|x_1-x_2\|,$$
where the estimate is order-sharp in the space $L^p[0,1]$. It is also established that the Lipschitz constant for the $\varepsilon$-selection is of proximate order $1/\varepsilon$ in the spaces $L^1[0,1]$ and $C[0,1]$.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 83:1, 1–22

Bibliographic databases:

UDC: 517.5
MSC: 41A35, 41A50, 41A65
Received: 02.10.1992 and 21.12.1993

Citation: P. V. Al'brecht, “Orders of moduli of continuity of operators of almost best approximation”, Mat. Sb., 185:9 (1994), 3–28; Russian Acad. Sci. Sb. Math., 83:1 (1995), 1–22

Citation in format AMSBIB
\Bibitem{Alb94} \by P.~V.~Al'brecht \paper Orders of moduli of continuity of operators of almost best approximation \jour Mat. Sb. \yr 1994 \vol 185 \issue 9 \pages 3--28 \mathnet{http://mi.mathnet.ru/msb923} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1305754} \zmath{https://zbmath.org/?q=an:0841.41030} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 83 \issue 1 \pages 1--22 \crossref{https://doi.org/10.1070/SM1995v083n01ABEH003578} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TQ10000001} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Marinov, “The Lipschitz constants of the metric $\varepsilon$-projection operator in spaces with given modules of convexity and smoothness”, Izv. Math., 62:2 (1998), 313–318
2. C. S. Rjutin, “Uniform Continuity of Generalized Rational Approximations”, Math. Notes, 71:2 (2002), 236–244
3. E. D. Livshits, “Stability of the operator of $\varepsilon$-projection to the set of splines in $C[0,1]$”, Izv. Math., 67:1 (2003), 91–119
4. P. Shvartsman, “Barycentric selectors and a Steiner-type point of a convex body in a Banach space”, Journal of Functional Analysis, 210:1 (2004), 1
5. P. A. Borodin, “The Linearity Coefficient of the Metric Projection onto a Chebyshev Subspace”, Math. Notes, 85:1 (2009), 168–175
6. A. R. Alimov, I. G. Tsar'kov, “Connectedness and other geometric properties of suns and Chebyshev sets”, J. Math. Sci., 217:6 (2016), 683–730
7. P. A. Borodin, Yu. Yu. Druzhinin, K. V. Chesnokova, “Finite-Dimensional Subspaces of $L_p$ with Lipschitz Metric Projection”, Math. Notes, 102:4 (2017), 465–474
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