Stephen J. Dilworth

Explicit constructions of RIP matrices and related problems

We give a new explicit construction ofn×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some e > 0, largeN , and anyn satisfyingN1−e ≤ n ≤ N , we construct RIP matrices of order k ≥ n1/2+e and constant δ = n−e. This overcomes the natural barrier k = O(n1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1 ≤ k ≤ N (Turan’s power sum problem), which improves upon known explicit constructions when (logN)1+o(1) ≤ n ≤ (logN)4+o(1). This latter construction produces elementary explicit examples of n×N matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (logN )1+o(1) ≤ n ≤ (logN)5/2+o(1).

Complex convexity and the geometry of Banach spaces

The notion of PL -convexity was introduced in [4]. In the present article several results are proved which related PL -convexity to various aspects of the geometry of Banach spaces. The first section introduces the moduli of comples convexity and makes a comparison with the more familiar modulus of uniform convexity. It is shown that unconditional convergence of implies convergence of . In the next section the moduli and are shown to be related. The method of proof gives rise to a theorem about strict c -convexity of L p ( X ) and a result on the representability in L p ( X ).

Coefficient quantization for frames in Banach spaces

Let (ei) be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.

Equidistributed random variables in Lp, q

Abstract We consider linear combinations of independent identically distributed random variables in L p , q . In fact, we provide several norm inequalities for sums from a larger class of equidistributed random variables.

NOWHERE WEAK DIFFERENTIABILITY OF THE PETTIS INTEGRAL

For an arbitrary infinite-dimensional Banach space

Asymptotic geometry of Banach spaces and uniform quotient maps

\X

A general theory of almost convex functions

, we construct examples of strongly-measurable

On the geometry of the unit spheres of the Lorentz spaces L w ,1

\X

Breaking the k 2 barrier for explicit RIP matrices

-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentiability of the indefinite Pettis integral to the cotype of

The Fourier transform of order statistics with applications to Lorentz spaces

\X