Differentiable operators of nearly best approximation

P. V. Al'brecht

Izvestiya. Mathematics , Tome 63 (1999) no. 4, pp. 631-647

Résumé

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 $L^p[0,1]$.