Revista Matemática Iberoamericana
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Published online: 2008-12-31
Majorizing measures and proportional subsets of bounded orthonormal systems
Olivier Guédon[1], Shahar Mendelson[2], Alain Pajor[3] and Nicole Tomczak-Jaegermann[4] (1) Université Paris-Est, Marne-la-Vallée, France(2) Technion - Israel Institute of Technology, Haifa, Israel
(3) Université Paris-Est, Marne-la-Vallée France
(4) University of Alberta, Edmonton, Canada
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.
Keywords: Empirical process, majorizing measure, orthonormal system
Guédon Olivier, Mendelson Shahar, Pajor Alain, Tomczak-Jaegermann Nicole: Majorizing measures and proportional subsets of bounded orthonormal systems. Rev. Mat. Iberoam. 24 (2008), 1075-1095. doi: 10.4171/RMI/567