Research in Analysis

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This project investigates several questions in mathematical analysis. The topics under study are related to important areas of application, including the scattering process in electromagnetic and acoustic wave propagation, an important and ubiquitous physical phenomenon. Several of the questions concern the circle of ideas surrounding the Steklov conjecture in the theory of orthogonal polynomials. The project also studies other questions in approximation theory and complex analysis, many of which are of central interest in probability and mathematical physics. A primary goal of this project is to study how the size of an orthogonal polynomial depends on the properties of the measure of orthogonality. To improve the upper estimates, the project aims to develop methods based on techniques related to singular integral operator estimates in weighted spaces. Another goal of this project is to obtain the spectral characterization of all measures on the real line for which the logarithmic integral converges. This condition, known as the Szego condition, plays a key role in the theory of Gaussian stochastic processes with continuous time. The project also aims to develop spectral theory based on the theory of multiple orthogonal polynomials. In addition, the project will explore wave propagation in multidimensional electromagnetic and acoustic scattering, investigating the minimal assumptions on coefficients in elliptic operators that guarantee existence of wave operators and presence of absolutely continuous spectral type. The questions to be addressed are central for analysis; the results are expected to have impact in probability, mathematical physics, and other areas.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.