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 -