Ron C. Blei

On trigonometric series associated with separable, translation invariant subspaces of ^{∞}()

G denotes a compact abelian group, and T denotes its dual. Our main result is that every non-Sidon set E C T contains a non-Sidon set F such that Lp(G) = (i}i ._, Cp. (G), where the F.s are finite, mutually disjoint, «d ur=i F.= F. 0. Introduction. Let G be a locally compact abelian group with Haar mea- sure dx, and let Y denote its dual with Haar measure dy. A(G) denotes the Gel- fand representation of the convolution algebra L (D = L (Y, dy). Aie) is a dense subspace of CJG), the Banach algebra of all continuous functions on G which vanish at infinity. A1(G) denotes the convolution algebra of all complex valued finite regular measures on G. If p £ M(G), then the Fourier-Stieltjes trans- form of p is the uniformly continuous function on F defined by

Rademacher chaos: tail estimates versus limit theorems

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.

METRIC ENTROPY OF HIGH DIMENSIONAL DISTRIBUTIONS

Let F d be the collection of all d-dimensional probability distribution functions on [0, 1] d , d > 2. The metric entropy of F d under the L 2 ([0,1] d ) norm is studied. The exact rate is obtained for d = 1, 2 and bounds are given for d > 3. Connections with small deviation probability for Brownian sheets under the sup-norm are established.

Multi-linear measure theory and multiple stochastic integration

SummaryStochastic integration in a Lebesgue Stieltjes sense is developed in a general context of multi-linear measure theory based on the Fundamental Grothendieck Inequality and Factorization Theorem.

Combinatorial dimension of fractional Cartesian products

The combinatorial dimension of a fractional Cartesian product is the optimal value of an associated linear programming problem

Fractional products of sets

For integers d > 1 and arbitrary 1 ⩽ α ⩽ d, α‐products in d‐fold Cartesian products are produced by probabilistic methods. Some explicit constructions are given by solutions of instances of the Turan problem for graphs.

The Grothendieck inequality revisited

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is the construction of a continuous map f from l2 (A) into L2 (O A, PA), where A is a set, OA = {-1,1}A, and PA is the uniform probability measure on OA.

A basic view of stochastic integration

Outlined is a bi-measure theoretic framework, including the fundamental Grothendieck inequality and factorization theorem of which self-contained proofs are presented. Stochastic integrators have a natural description in this framework.

Stochastic integrators indexed by a multi-dimensional parameter

SummaryMulti-parameter stochastic integrators are described and classified according to directions of integrability. Sets of directions are distinguished precisely by the combinatorial dimension of corresponding fractional Cartesian products. The main theorem establishes existence of stochastic processes which are integrators in prescribed directions but not others.

Fractional tensor products and P & M maps

Fractional P & M maps are constructed in a framework of harmonic analysis, linking fractional projective tensor algegras to algebras of absolutely convergent Fourier series.