Denka Kutzarova

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).

Remarks about Schlumprecht space

Let S denote the Schlumprecht space. We prove that 1. l∞ is finitely disjointly respresentable in S; 2. S contains an l1-spreading model; 3. for any sequence (nk) of natural numbers, S is isomorphic to the space ( ∑∞ k=1⊕ lnk ∞ ) S . Let (ei) ∞ i=1 be the standard basis of the linear space c00, the set of all finitely supported sequences. For x = ∞ ∑ i=1 aiei ∈ c00, suppx denotes the set {i ∈ N : ai 6= 0}. A subset E of N is said to be an interval if there exist a, b such that E = {c ∈ N : a < c < b}. For finite subsets E,F of N, E < F means maxE < minF or E is an empty set. For x = ∞ ∑ i=1 aiei and a subset E of N, Ex denotes the vector Ex = ∑ i∈E aiei. Let f : [1,∞) → [1,∞) be the function defined by f(x) = log2(x+1). The Schlumprecht space S = (S, ‖ · ‖) is the completion of c00 with respect to the norm ‖ · ‖ which satisfies the following implicit equation: (1) ‖x‖ = max { ‖x‖∞, sup E1

On drop property for convex sets

Let (X, ‖ · ‖) be a real Banach space. Let C be a closed convex set in X. By a drop D(x, C) determined by a point x ∈ X, x / ∈ C, we shall mean the convex hull of the set {x} ∪ C. We say that C has the drop property if C 6= X and if for every nonvoid closed set A disjoint with C, there exists a point a ∈ A such that D(a, C) ∩ A = {a}. For a given C a sequence {xn} in X will be called a stream if xn+1 ∈ D(xn, C) \ C (cf. [6]). When the set A has a positive distance from C, a variety of “Drop theorems” has been obtained in [1, 2, 3] and [8]. If C is the closed unit ball and has the drop property then we say that the norm ‖ · ‖ has the drop property [9]. Norms with the drop property have been investigated in papers [4, 6] and [9]. The drop property for closed bounded sets has been considered in [5]. There was proved that a bounded closed convex symmetric set having the drop property is compact or has a nonempty interior. We shall prove this theorem without assumptions on boundness and symmetry of sets under consideration. The Kuratowski measure of noncompactness of a set G in a Banach space X is the infimum α(G) of those ε > 0 for which there is a covering of G by a finite number of sets of diameter less than ε. For a closed convex set C denote by F (C) the set of all linear continuous functionals f ∈ X, f 6= 0, which are bounded above on C. For f ∈ F (C) and δ > 0 put

Asymptotic geometry of Banach spaces and uniform quotient maps

Recently, Lima and Randrianarivony pointed out the role of the property

Breaking the k 2 barrier for explicit RIP matrices

(\beta)

Renorming spaces with greedy bases

of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of

On nearly uniformly convex sets

(\beta)

Some more weak Hilbert spaces

of the domain space. We also provide conditions under which this comparison can be improved.

Non-equivalent greedy and almost greedy bases in lp

We give a new explicit construction of n x N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε>0, large k and k2-ε ≤ N ≤ k2+ε, we construct RIP matrices of order k with n=O(k2-ε). This overcomes the natural barrier n >> k2 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.

A separable space with no Schauder decomposition

We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given e 0 , so that the basis becomes ( 1 + e ) -democratic, and hence ( 2 + e ) -greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is ( 1 + e ) -greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in L p 0 , 1 , 1 < p < ∞ , and in dyadic Hardy space H 1 , as well as the unit vector basis of Tsirelson space.