Geodesic
Differentiable operators of nearly best approximation
Izvestiya. Mathematics , Tome 63 (1999) no. 4, pp. 631-647.

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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 $L^p[0,1]$. 
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     author = {P. V. Al'brecht},
     title = {Differentiable operators of nearly best approximation},
     journal = {Izvestiya. Mathematics },
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     number = {4},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a0/}
}
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P. V. Al'brecht. Differentiable operators of nearly best approximation. Izvestiya. Mathematics , Tome 63 (1999) no. 4, pp. 631-647. http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a0/

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