Polynomial approximation in spaces of analytic functions

Constructing approximations of complicated functions using simpler ones is a problem of fundamental importance in many fields of sciences and engineering. Such approximations are used, for example, to simplify calculations and make computer algorithms faster and more efficient. This is particularly important for real-time systems (flight control systems, medical devices, smartphones, sensors, etc.) where numerical calculations need to be performed as quickly as possible. Polynomials play a crucial role in constructing such approximations. They are used across many fields due to their versatility in simplifying complex functions and providing accurate estimations. In many instances, however, building explicit polynomial approximation schemes remains an open problem. In this project, the investigator and his colleagues study new methods to construct polynomial approximations for functions belonging to well-known spaces of functions. The research advances our knowledge of how to efficiently build such approximations and also connects the problem to other fields in mathematics such as matrix analysis and operator theory. The project also supports education by training one PhD student to become a new expert in the field. The PI will also engage in undergraduate and high school student mentoring and outreach activities. Function approximation constitutes a significant branch of analysis, involving the approximation of general functions by various families of simpler ones. This concept holds far-reaching applications across diverse mathematical disciplines and scientific domains. This project tackles challenging questions in complex analysis, focusing on polynomial approximation in spaces of analytic functions. While its history is extensive, a comprehensive understanding of polynomial approximation in many function spaces has been attained only recently, while others remain very active areas of research. The first part of the project explores constructive polynomial approximation schemes in weighted Dirichlet and in de Branges--Rovnyak spaces by re-framing the approximation problem as concrete matrix and operator theory problems. The second part of the project focuses on determining general conditions guaranteeing well-known approximation schemes (Cesaro, Abel, etc.) converge in general Banach holomorphic function spaces. The classical approximation property (AP) of Banach spaces plays a crucial role in this problem. Of particular interest is how strengthened versions of the AP can be used to characterize the existence of specific approximation schemes. Properties of the kernel of reproducing kernel Hilbert spaces of functions also have the potential to reveal when specific approximation schemes are valid. The techniques developed in the project are applicable to several fundamental spaces of analytic functions and therefore contribute to the broad problem of understanding constructive polynomial approximation in function spaces. 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.