Widths and rigidity

TL;DR

This paper investigates the rigidity of orthonormal systems in various function spaces, establishing conditions under which they cannot be well approximated by low-dimensional subspaces, with implications for understanding function approximation limits.

Contribution

It provides new sufficient conditions for rigidity in different $L_p$ spaces, including probabilistic independence implying rigidity in $L_1$ and $L_0$, and demonstrates positive approximation results for specific function systems.

Findings

Independence implies rigidity in $L_1$ and $L_0$.

$S_{p'}$-systems are $L_p$-rigid for $1<p<2$.

First $N$ trigonometric functions can be approximated by very-low-dimensional spaces in $L_0$.

Abstract

We consider Kolmogorov widths of finite sets of functions. Any orthonormal system of $N$ functions is rigid in $L_2$, i.e. it cannot be well approximated by linear subspaces of dimension essentially smaller than $N$. This is not true for weaker metrics: it is known that in every $L_p$, $p<2$, the first $N$ Walsh functions can be $o(1)$-approximated by a linear space of dimension $o(N)$. We give some sufficient conditions for rigidity. We prove that independence of functions (in the probabilistic meaning) implies rigidity in $L_1$ and even in $L_0$ -- the metric that corresponds to convergence in measure. In the case of $L_p$, $1<p<2$, the condition is weaker: any $S_{p'}$-system is $L_p$-rigid. Also we obtain some positive results, e.g. that first $N$ trigonometric functions can be approximated by very-low-dimensional spaces in $L_0$, and by subspaces generated by $o(N)$ harmonics in $L_p$, $p<1$.