Function Theory on Polydiscs

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This project concerns the study of complex functions of two or more variables with sufficiently nice structure. Such functions can be used to encode information about simple physical systems that accept input data and release output data, such as signal processors. They are also closely related to problems in robust control theory; here, the driving question is how to design systems that perform desired tasks while maintaining internal stability. The project's motivation is this: classic results by mathematicians such as Beurling and Herglotz show that nice functions of one variable can be represented using different and enlightening formulas. Different formulas are useful in different situations, and the collection serves as a robust toolbox for studying properties of complex functions and related objects. Recent results show that several essential one-variable formulas have two-variable analogues. The goals of this project include extending these known results, developing new formulas for functions of several variables, and investigating applications. While pursing this research, the principal investigator will continue to mentor students interested in science and mathematics through her teaching and outreach activities.The specific focus of this project is the study of holomorphic functions on the polydisc and multivariate analogues of classic representation formulas for such functions. The formulas of interest include transfer function realizations for Schur-Agler functions and weak factorizations of Hardy spaces. Specific topics of interest include canonical constructions, generalizations to other function spaces and domains, and applications. The study of transfer function realizations is closely related to both reproducing kernel Hilbert spaces and the "state space" methods of systems engineering. The proposed generalizations and refinements will use properties of related Hilbert spaces of functions and techniques that arise in the context of multidimensional input/state/output systems and scattering systems. The construction and study of weak Hardy space factorizations will require tools from multiparameter harmonic analysis, including atomic decompositions of product Hardy spaces and technical geometric results. Many properties of reproducing kernel Hilbert spaces can also be described with functions called generalized Bergman metrics. Within this context, the principal investigator plans a further study of functions on polydiscs, with an emphasis on boundary behavior. Such questions have applications to Cowen-Douglas operators, and the techniques suggest a method of generalizing results on polydiscs to classes of reproducing kernel Hilbert spaces.