Majorizing measures and proportional subsets of bounded orthonormal systems

Olivier Guédon; Shahar Mendelson; Alain Pajor; Nicole Tomczak-Jaegermann

doi:10.4171/RMI/567

Majorizing measures and proportional subsets of bounded orthonormal systems

Olivier Guédon^*, Shahar Mendelson, Alain Pajor, Nicole Tomczak-Jaegermann

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

22 Citations (Scopus)

Abstract

In this article we prove that for any orthonormal system (φ_j)_j=1ⁿ ⊂ L₂. that is bounded in L∞, and any 1 < k < n, there exists a subset I of cardinality greater than n - k such that on span{φ_i} _i∈I, the L₁ norm and the L₂ norm are equivalent up to a factor μ(log imu;)^5/2, where μ = √n/k √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.

Original language	English
Pages (from-to)	1075-1095
Number of pages	21
Journal	Revista Matematica Iberoamericana
Volume	24
Issue number	3
DOIs	https://doi.org/10.4171/RMI/567
Publication status	Published - 2008

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title = "Majorizing measures and proportional subsets of bounded orthonormal systems",

abstract = "In this article we prove that for any orthonormal system (φj)j=1n ⊂ L2. that is bounded in L∞, and any 1 < k < n, there exists a subset I of cardinality greater than n - k such that on span{φi} i∈I, the L1 norm and the L2 norm are equivalent up to a factor μ(log imu;)5/2, where μ = √n/k √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.",

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T1 - Majorizing measures and proportional subsets of bounded orthonormal systems

AU - Guédon, Olivier

AU - Mendelson, Shahar

AU - Pajor, Alain

AU - Tomczak-Jaegermann, Nicole

PY - 2008

Y1 - 2008

N2 - In this article we prove that for any orthonormal system (φj)j=1n ⊂ L2. that is bounded in L∞, and any 1 < k < n, there exists a subset I of cardinality greater than n - k such that on span{φi} i∈I, the L1 norm and the L2 norm are equivalent up to a factor μ(log imu;)5/2, where μ = √n/k √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.

AB - In this article we prove that for any orthonormal system (φj)j=1n ⊂ L2. that is bounded in L∞, and any 1 < k < n, there exists a subset I of cardinality greater than n - k such that on span{φi} i∈I, the L1 norm and the L2 norm are equivalent up to a factor μ(log imu;)5/2, where μ = √n/k √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.

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KW - Orthonormal system

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Majorizing measures and proportional subsets of bounded orthonormal systems

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