Harmonic analysis and spaces of analytic functions in several variables

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Operator theory, harmonic analysis, and analytic function theory of one variable form a collection of interwoven fields that were created to solve some of the most important problems in pure and applied mathematics. For example, Norbert Wiener initiated the study of time series for the purpose of tackling engineering "prediction" problems in World War II. Time series is a term for a sequence of "random" data, such as stock market prices or the positions of a jet being tracked. Harmonic analysis, which can loosely be described as a set of techniques for breaking up a "signal," such as a sound, into fundamental "tones," provided a natural set of tools for solving time series prediction problems. Operator theory and analytic function theory in one variable have had similar success in related engineering "control" problems involving feedback, for example, designing a thermostat or an automatic pilot. These problems all involve "one-dimensional" data in a certain sense, and operator theory, harmonic analysis, and analytic function theory bring an impressive toolbox of related techniques to bear on them. Multivariate versions of these theories exist to address multidimensional problems (e.g., image or video processing), but the connections between the different theories are weaker and less well understood. It is the goal of this project to expand some known and particularly strong interactions of these fundamental subjects to build a more robust toolbox for addressing fundamentally multidimensional problems.The specific topics this project will develop further are (1) characterizations of when a positive trigonometric polynomial can be factored as a single square, (2) weak factorization of Hardy spaces on the polydisk, and (3) operator theoretic models for bounded analytic functions. Success in topic (1) will deepen our understanding of multivariate moment problems, multivariable polynomials, sums of squares decompositions and determinantal representations, and multivariate operator theory related to von Neumann's inequality. Success in topic (2) will bridge a gap between harmonic analysis and function theory in several variables and could lead to applications in Dirichlet series. This project will use recent innovations in item (3) to study spaces of entire functions in several variables and their connections with Fourier analysis, which could have profound implications in old and difficult completeness problems in Fourier analysis.