Mathematical Sciences: Lifting Properties of Hankel Forms and Applications

该项目继续进行数学研究，重点关注通常被称为“提升”的性质和应用。通过这一点，人们理解了一个框架，在这个框架中，应用于某个特定集合的数学运算，可以被证明是应用于更大集合的另一个数学运算的限制。在许多重要的情况下，提升澄清了原始操作的特性，并允许人们预测在受限环境中不明显的其他特性。抬举还可以表明，不同的理论是一个更大整体的组成部分。这个项目的工作涉及提升双线性形式的性质，即汉高形式。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Sz之间的联系。正算子的Nagy-Foias提升理论。该研究方法将希尔伯特变换连续作用的勒贝格空间与散射理论相关的膨胀和提升特性结合在一起。推动本研究的一个基本结果是n条件提升定理，它导致有机会攻击积空间中的经典插值问题，其中单变量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能带扩展问题和控制中的相关问题上。这项工作旨在扩展Gohberg的一维结果，该结果告诉何时存在具有规定负傅立叶系数的有界函数。在这项工作中解决的问题将是多参数的情况下，这可以完全作为一个问题在提升理论。其他的应用将涉及到汉克尔算子和n环面上的希尔伯特变换、辛空间、海森堡群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.