Nowhere Weak Differentiability of the Pettis Integral

For an arbitrary infinite-dimensional Banach space \X, we construct examples of strongly-measurable \X-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 \X, from which it follows that our examples are reasonably sharp. This is an expanded version of a previously posted paper with the same name

Explicit constructions of RIP matrices and related problems

We give a new explicit construction of $n\times N$ matrices satisfying the Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This overcomes the natural barrier k=O(n^{1/2}) 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 k-th moments are uniformly small for 1\le k\le N (Turan's power sum problem), which improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le (\log N)^{4+o(1)}. This latter construction produces elementary explicit examples of n by N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log N)^{5/2+o(1)}.Comment: v3. Minor correction