Mathematical Sciences: Lifting Properties of Hankel Forms and Applications

该项目继续进行数学研究，重点是通常所说的“升力”的性质和应用。通过这一点，人们理解了这样一种框架，在该框架中，应用于某一集合的数学运算可以被证明是应用于更大集合的另一数学运算的限制。在许多重要的情况下，提升澄清了原始操作的性质，并允许人们预测在受限环境中不明显的其他特征。提升还可以表明，不同的理论是一个更大整体的一部分。这个项目的工作涉及双线性形式的提升性质，称为Hankel形式。最近的工作建立了Krein在提升形式的积分表示中的特征函数展开方法与正算子的Sz.Nagy-FOIAs提升理论之间的联系。这种研究方法将希尔伯特变换连续作用的勒贝格空间和与散射理论相关的膨胀和提升性质结合在一起。激励这项研究的一个基本结果是n-条件提升定理，它导致了一个攻击乘积空间中经典内插问题的机会，在乘积空间中，一元有界解析函数理论的主要工具是不可用的。第二组问题集中在矩阵的带扩张问题和相关的控制问题上。这项工作旨在推广Gohberg的一维结果，即当存在规定的负傅立叶系数的有界函数时。这项工作所涉及的问题将是多参数情况，它可以完全表述为提升理论中的一个问题。其他的应用还包括n-环面上的Hankel算子和Hilbert变换、辛空间上的Hilbert变换、Heisenberg群上的Hankel算子和非平稳随机过程理论。

This project continues mathematical research focusing on the properties and applications of what is generally referred to as 'lifting.' By this one understands a framework in which a mathematical operation, applied to a certain set, can be shown to be the restriction of another mathematical operation applied to a larger set. In many important cases, the lifting clarifies properties of the original operation and allows one to predict other characteristics which were not evident in the restricted environment. Lifting can also show how disparate theories are parts of a larger whole. Work on this project concerns lifting properties of bilinear forms known as Hankel forms. Recent work established links between Krein's method of eigenfunction expansions in the integral representations of lifted forms to the Sz.Nagy-Foias lifting theory of positive operators. The approach to the research brings together the Lebesgue spaces in which the Hilbert transform acts continuously and dilation and lifting properties associated with scattering theory. A fundamental result motivating this investigation is the n-conditional lifting theorem which leads to an opportunity to attack the classical interpolation problems in product spaces, where the main tools of the theory of bounded analytic functions in one variable are just not available. A second group of problems centers on the band extension problem for matrices and related problems in control. This work seeks to extend the one-dimensional results of Gohberg which tell when bounded functions with prescribed negative Fourier coefficients exist. The question addressed in this work will be the multiparametric case, which can be phrased entirely as a problem in lifting theory. Other applications will be made to Hankel operators and the Hilbert transform on the n-torus, on the symplectic space, on the Heisenberg group and to the theory of non-stationary stochastic processes.