JournalsrmiVol. 24, No. 3pp. 1075–1095

Majorizing measures and proportional subsets of bounded orthonormal systems

  • Olivier Guédon

    Université Paris-Est, Marne-la-Vallée, France
  • Shahar Mendelson

    Technion - Israel Institute of Technology, Haifa, Israel
  • Alain Pajor

    Université Paris-Est, Marne-la-Vallée France
  • Nicole Tomczak-Jaegermann

    University of Alberta, Edmonton, Canada
Majorizing measures and proportional subsets of bounded orthonormal systems cover
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Abstract

In this article we prove that for any orthonormal system (φj)j=1nL2(\varphi_j)_{j=1}^n \subset L_2 that is bounded in LL_{\infty}, and any 1<k<n1 < k < n, there exists a subset II of cardinality greater than nkn-k such that on span{φi}iI\mathrm{span}\{\varphi_i\}_{i \in I}, the L1L_1 norm and the L2L_2 norm are equivalent up to a factor μ(logμ)5/2\mu (\log \mu)^{5/2}, where μ=n/klogk\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.

Cite this article

Olivier Guédon, Shahar Mendelson, Alain Pajor, Nicole Tomczak-Jaegermann, Majorizing measures and proportional subsets of bounded orthonormal systems. Rev. Mat. Iberoam. 24 (2008), no. 3, pp. 1075–1095

DOI 10.4171/RMI/567