Abstract
In this article we prove that for any orthonormal system $(\varphi_j)_{j=1}^n \subset L_2$ 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 $\mathrm{span}\{\varphi_i\}_{i \in I}$, 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.
Citation
Olivier Guédon . Shahar Mendelson . Alain Pajor . Nicole Tomczak-Jaegermann . "Majorizing measures and proportional subsets of bounded orthonormal systems." Rev. Mat. Iberoamericana 24 (3) 1075 - 1095, November, 2008.
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