Optimal linear approximation and isometric extensions
Let $X$ be a Banach space with the unit ball $B(X)$ and $A\subset X$ be a convex origin-symmetric compact in $X$. Let $\mathrm{j}:X\rightarrow \widetilde{X}$ be an isometric extension of $X$. It is well-known that linear widths $λ_{n}\left( \mathrm{j}\left( A\right) \text{,}% \widetilde{X}\right) $ may decrease in order when compared with $λ_{n}\left( A\text{,}X\right) $ and absolute widths $Λ\left( A,% \widehat{X}\right) =\inf_{\mathrm{j}}\left( \mathrm{j}\left( A\right) ,% \widetilde{X}\right) $ are realized in the space $\widehat{X}$ which is the Banach space of bounded functions $f:B\left( X^{\ast }\right) \rightarrow \mathbb{R}$ on the unit ball $B\left( X^{\ast }\right) $ of the conjugate space $X^{\ast }$. We show that it is sufficient to use just $n$-dimensional extensions of $X$ to attain absolute linear widths. This unexpected fact significantly reduces the space $\ \widehat{X}$. This allows us to introduce the notion of preabsolute widths. We give the respective optimal extensions explicitly and establish order estimates for preabsolute widths of a wide range of sets of smooth functions considered in \cite{C11}. In particular, in the case of super-small and super-high smoothness considered in \cite{C11} the orders of preabsolute linear widths coincide with the orders of absolute linear widths. In the intermediate cases of finite and infinite smoothness the respective orders are different.