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Optimal linear approximation and isometric extensions

Alexander Kushpel

Abstract

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.

Optimal linear approximation and isometric extensions

Abstract

Let be a Banach space with the unit ball and be a convex origin-symmetric compact in . Let be an isometric extension of . It is well-known that linear widths may decrease in order when compared with and absolute widths are realized in the space which is the Banach space of bounded functions on the unit ball of the conjugate space . We show that it is sufficient to use just -dimensional extensions of to attain absolute linear widths. This unexpected fact significantly reduces the space . 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.
Paper Structure (3 sections, 2 theorems, 79 equations)

This paper contains 3 sections, 2 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preabsolute linear widths
  3. Examples and application

Key Result

Theorem 1

Let $A\subset X$ be a convex origin symmetric compact, $\mathrm{diam}\left( A,X\right) <\infty$, $\phi _{k}\in X^{\ast }$ be such that and $c_{k}:B\left( X^{\ast }\right) \rightarrow \mathbb{R}$, $1\leq k\leq n$ be the functionals of the best approximation of $\phi \in B\left( X^{\ast }\right)$ by $\mathrm{lin}\left\{ \phi _{k}\text{, }1\leq k\leq n\right\}$ in $\left\Vert \cdot \right\Vert _{\lef

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof