Majorizing measures and proportional subsets of bounded orthonormal systems

TL;DR

This paper demonstrates that within any bounded orthonormal system, large subsets exist where the $L_1$ and $L_2$ norms are equivalent up to a logarithmic factor, using majorizing measures and empirical process estimates.

Contribution

It introduces a novel approach employing majorizing measures to identify large subsets of bounded orthonormal systems with norm equivalence properties.

Findings

Existence of large subsets with norm equivalence in bounded orthonormal systems

New estimate of empirical process supremum using majorizing measures

Quantitative bounds involving logarithmic factors

Abstract

In this article we prove that for any orthonormal system $(\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\spa\{\vphi_i\}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.