This mathematics research project by Michael Lacey is in the general area of harmonic analysis. The Shannon Sampling Theorem asserts that a function in the Paley-Wiener class can be recovered by its values at the integers. This foundational fact has many quantifications, and extensions. Among the most delicate is to what extent the integers can be replaced by other discrete sets: For which other sets is sampling also well behaved. In the square-integrable case, this is just one instance of the profound two-weight inequality for the Hilbert transform. It asks for a real-variable characterization of those pairs of measures on the real line for which the Hilbert transform is bounded between the Hilbert spaces associated with the pair of measures (weights). Lacey has recently solved this problem, in complete generality. This and a family of related extensions will be under investigation by Lacey.
This mathematics research project by Michael Lacey is in the field of harmonic analysis, which has deep applications to other fields such as engineering, for instance to signal transmission. The hallmark of transmission of information is rapid and regular sampling of a signal, followed by a synthesis of the samples into, for instance, a voice conversation over a cell phone. The system is robust, with small perturbations of the sampling not affecting the synthesis, in part because the most obvious problems with the sampling are all accounted for. The abstraction of this sampling leads to fascinating mathematical problems: A dramatic oversampling of the signal increases the observed energy. How bad can the oversampling become, while preserving the energy of the signal? Just how degenerate of a sampling is permitted before the signal is lost? Lacey has provided a mathematical solution to the first problem, an advance that will open the door to a range of related questions. These techniques, disseminated across fields of science and engineering, will impact the underlying theories of sampling and information. Some of the problems in this project will be studied by Ph.D. students under Lacey's direction.
