Extensions of Bernstein's Lethargy Theorem

TL;DR

This paper surveys recent extensions of Bernstein's Lethargy Theorem, highlighting a significant reduction in approximation intervals and revealing a novel link to bounded linear operators between Banach spaces.

Contribution

It introduces two new extensions of Bernstein's Lethargy Theorem, one improving approximation bounds and another establishing a connection to operator theory.

Findings

One extension halves the interval for best approximation.

Another extension connects the theorem to bounded linear operators.

The paper provides a comprehensive survey of recent developments.

Abstract

In this paper, we examine the aptly-named "Lethargy Theorem" of Bernstein and survey its recent extensions. We show that one of these extensions shrinks the interval for best approximation by half while the other gives a surprising connection to the space of bounded linear operators between two Banach spaces.