2000/11/28 by R. Vershynin, Vershynin, R.
Mathematics · #43A46 #43A46) #46B09 (60G50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.AP #math.FA #math.PR #msc:43A46 #msc:46B09
paper · pdf · doi:10.48550/arxiv.math/0011232
arxiv created 2000/11/28 · arxiv updated 2009/11/30
This paper addresses the problem of improving properties of a linear operator u in l2n by restricting it onto coordinate subspaces. We discuss how to reduce the norm of u by a random coordinate restriction, how to approximate u by a random operator with small "coordinate" rank, how to find coordinate subspaces where u is an isomorphism. The first problem in this list provides a probabilistic extension of a suppression theorem of Kashin and Tzafriri, the second one is a new look at a result of Rudelson on the random vectors in the isotropic position, the last one is the recent generalization of the Bourgain-Tzafriri's invertibility principle. The main point is that all the results are independent of n, the situation is instead controlled by the Hilbert-Schmidt norm of u. As an application, we provide an almost optimal solution to the problem of harmonic density in harmonic analysis, and a solution to the reconstruction problem for communication networks which deliver data with random losses.