Weighted Inequalities via Dyadic Operators and A Learning Theory Approach to Compressive Sensing

Dissertation Defense
Thursday, March 30, 2017 - 11:00
1 hour (actually 50 minutes)
Skiles 202
Georgia Institute of Technology
This thesis explores topics from two distinct fields of mathematics. The first part addresses a theme in abstract harmonic analysis, while the focus of the second part is a topic in compressive sensing. The first part of this dissertation explores the application of dominating operators in harmonic analysis by sparse operators. We make use of pointwise sparse dominations weighted inequalities for Calder\'on-Zygmund operators, Hardy-Littlewood maximal operator, and their fractional analogues. Dominating bilinear forms by sparse forms allows us to derive weighted inequalities for oscillatory integral operators (polynomially modulated CZOs) and random discrete Hilbert transforms. The later is defined on sets of initegers with asymptotic density zero, making these weighted inequalitites particulalry attractive. We also discuss a characterization of a certain weighted BMO space by commutators of multiplication operators with fractional integral operators. Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems.  It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of slogā”(n/s) -- n is ambient dimension and s is sparsity threshhold.The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix.  A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing.  Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere.  We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.