Approximation Theory and Complex Dynamics

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This project involves the study of approximation theory in the setting of complex functions, with applications to complex dynamics. Approximation theory seeks to understand the extent to which the behavior of a general function can be effectively modeled by that of functions drawn from a more restricted class. Efficient approximation of functions is of relevance for numerical calculation. Since the only calculations that can be carried out numerically are the elementary operations of addition, subtraction, multiplication, and division, in practical terms it is of importance to understand when the values of general functions are well approximated by the values of either polynomial or rational functions. In many situations, the values of the approximant resemble those of the general function only for a sampling of input values. What can be said about values of the approximant for other choices of input? This is the main question studied in this project, with the following application in mind: when a general function is iterated to produce a dynamical system, to what extent does the dynamical behavior of an approximant resemble the dynamical behavior of the original function? The project will also contribute to the development of human resources through educational outreach at the high school level as well as mentoring and training at the undergraduate and graduate levels, and will facilitate the interaction of different fields of mathematics through the organization of conferences and seminars.The Principal Investigator will study the approximation of analytic functions in one complex variable by polynomials, rational functions, and transcendental entire functions. Quasiconformal mappings will be a major tool in this study. The quasiconformal approach to this particular subject is largely unexplored, and affords control over geometric properties of the approximants such as location of critical points and critical values. Such control can be used to understand the intricate dynamics recently proven to exist for transcendental entire functions. Another anticipated application lies in an improved understanding of the geometries of lemniscates (level sets of polynomials or rational functions) which relate to the emerging field of pattern recognition as a tool to distinguish between different planar shapes. The PI also will investigate whether a better understanding of the geometry of polynomial or rational approximants yields new insight on the numerical implementation of root-finding algorithms which rely essentially on such approximants. Interactions between the different fields alluded to above (approximation theory, geometric function theory, complex dynamics, and numerical analysis) will be fostered via the organization of conferences, meetings, and seminars.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.