Mathematical Sciences: Lifting Properties of Hankel Forms and Applications

该项目继续数学研究，重点是 通常被称为“纳米材料”的材料的性质和应用 “举起。“通过这一点，人们可以理解一个框架， 数学运算，适用于某一特定的集合，可以被证明， 是另一个数学运算的限制， 更大的设置。 在许多重要的情况下，解除制裁澄清了 原始操作的属性，并允许预测 其他特征在受限制的 环境 举也可以显示出理论是多么的不同 一个更大的整体的一部分。 这个项目的工作涉及提升性质的双线性 形式称为汉克尔形式。 最近的工作建立了联系 Krein的本征函数展开法与 提升形式的积分表示 正算子的提升理论 的途径 研究汇集了勒贝格空间，其中希尔伯特 变换连续作用，伸缩和提升性质 与散射理论有关。 一个基本的结果 激发这项研究的是n-条件提升 定理，导致一个机会，攻击经典的 插值问题的产品空间，其中的主要工具， 一元有界解析函数的理论 不可用. 第二组问题集中在乐队上 矩阵的延拓问题及控制中的相关问题。 这项工作旨在推广Gohberg的一维结果 它告诉我们，当有界函数具有规定的负 傅立叶系数存在。 这项工作所解决的问题是， 将是多参数的情况下，这可以完全措辞 提升理论中的一个问题 其他申请将在 到Hankel算子和Hilbert变换的n-环面上， 辛空间，海森堡群和理论 非平稳随机过程

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.