On concentrators and related approximation constants
Abstract
Pippenger ([Pippenger, 1977]) showed the existence of $(6m,4m,3m,6)$concentrator for each positive integer $m$ using a probabilistic method. We generalize his approach and prove existence of $(6m,4m,3m,5.05)$concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from $44.5$ (established in [Kalton, Roberts, 1983]) to $39$. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in $\mathbb{R}^d$ by linear functions, and improve the estimate of this Whitney constant from $802$ (proved in [Brudnyi, Kalton, 2000]) to $73$.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1404.2161
 Bibcode:
 2014arXiv1404.2161B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Combinatorics;
 Primary 41A63 (46A10). Secondary 05D40;
 05C35
 EPrint:
 J. Math. Anal. Appl. 402 (2013), no. 1, 234241