Positivity 11 (2007), 269–283
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020269-15, published online April 6, 2007
Subspaces and Orthogonal Decompositions
Generated by Bounded Orthogonal Systems
Olivier Gu´edon, Shahar Mendelson
, Alain Pajor
and Nicole Tomczak-Jaegermann
Abstract. We investigate properties of subspaces of L
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
and the L
norms are close, up to a logarithmic factor. Considering
for example the Walsh system, we deduce the existence of two orthogonal
subspaces of L
, complementary to each other and each of dimension roughly
n/2, spanned by ±1 vectors (i.e. Kashin’s splitting) and in logarithmic dis-
tance to the Euclidean space. The same method applies for p>2, and, in
connection with the Λ
problem (solved by Bourgain), we study large sub-
sets of this orthonormal system on which the L
and the L
norms are close
(again, up to a logarithmic factor).
AMS Subject Classiﬁcation (2000). 46B07; 46B09; 42A05; 42A61; 94B75; 62G99.
Keywords. Empirical processes, generic chaining, bounded orthogonal sys-
tems, orthogonal decompositions.
In this note we consider a space L
of functions on a probability space and we
investigate properties of its subspaces spanned by a ﬁnite subset of an orthonor-
mal system which consists of functions bounded in L
. Typical examples of such
systems are the trigonometric and the Walsh systems.
The question we study is whether there is a subspace spanned by a large
subset of the orthonormal system on which the L
norms are close. The
two cases we focus on are when p>2andp = 1. Formally we address
Question 1. Let (ϕ
be an orthonormal system in L
Partially supported by an Australian Research Council Discovery grant.
This author holds the Canada Research Chair in Geometric Analysis.