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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 4, Pages 3–18 (Mi izv249)  

Differentiable operators of nearly best approximation

P. V. Al'brecht

Moscow Aviation Institute

Abstract: Let $X$ be a normed linear space, let $Y\subset X$ be a finite-dimensional subspace, and let $\varepsilon>0$. We define a multiplicative $\varepsilon$-selection $M\colon X\to Y$ to be a map such that
$$ \forall x\in X \qquad \|Mx-x\|\leqslant \inf\{\|x-y\|\colon y\in Y\}(1+\varepsilon). $$

We prove that there is an $\varepsilon$-selection $M$ whose smoothness coincides with that of the norm in $X$. We show that, generally speaking, it is impossible to find an $\varepsilon$-selection of greater smoothness in $L^p[0,1]$.

DOI: https://doi.org/10.4213/im249

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English version:
Izvestiya: Mathematics, 1999, 63:4, 631–647

Bibliographic databases:

MSC: 90C48, 90C25, 54C60, 46E30, 41A30, 41A25
Received: 09.01.1998

Citation: P. V. Al'brecht, “Differentiable operators of nearly best approximation”, Izv. RAN. Ser. Mat., 63:4 (1999), 3–18; Izv. Math., 63:4 (1999), 631–647

Citation in format AMSBIB
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