Research in Analysis

这个项目研究了数学分析中的几个问题。正在研究的课题涉及重要的应用领域，包括电磁和声波传播中的散射过程，这是一种重要的和普遍存在的物理现象。其中几个问题涉及到正交多项式理论中围绕Steklov猜想的思想圈。该项目还研究了近似理论和复数分析中的其他问题，其中许多问题是概率和数学物理的核心兴趣。这个项目的一个主要目标是研究正交多项式的大小如何取决于正交性度量的性质。为了改进上限估计，该项目旨在开发基于加权空间中奇异积分算子估计相关技术的方法。这个项目的另一个目标是获得对数积分收敛的实线上所有测量的谱特征。这一条件被称为Szego条件，在连续时间高斯随机过程理论中起着关键作用。该项目还旨在发展基于多重正交多项式理论的频谱理论。此外，该项目还将探索多维电磁和声波散射中的波传播，研究确保波算符存在和绝对连续谱类型存在的椭圆算子系数的最小假设。需要解决的问题是分析的中心；结果预计将在概率、数学物理和其他领域产生影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.