TY - JOUR

T1 - Subspaces and orthogonal decompositions generated by bounded orthogonal systems

AU - Guédon, Olivier

AU - Mendelson, Shahar

AU - Pajor, Alain

AU - Tomczak-Jaegermann, Nicole

PY - 2007/5

Y1 - 2007/5

N2 - We investigate properties of subspaces of L 2 spanned by subsets of a finite orthonormal system bounded in the L ∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L 1 and the L 2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L 2 n, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin's splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L 2 and the L p norms are close (again, up to a logarithmic factor).

AB - We investigate properties of subspaces of L 2 spanned by subsets of a finite orthonormal system bounded in the L ∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L 1 and the L 2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L 2 n, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin's splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L 2 and the L p norms are close (again, up to a logarithmic factor).

KW - Bounded orthogonal systems

KW - Empirical processes

KW - Generic chaining

KW - Orthogonal decompositions

UR - http://www.scopus.com/inward/record.url?scp=34247578706&partnerID=8YFLogxK

U2 - 10.1007/s11117-006-2059-1

DO - 10.1007/s11117-006-2059-1

M3 - Article

SN - 1385-1292

VL - 11

SP - 269

EP - 283

JO - Positivity

JF - Positivity

IS - 2

ER -