Mathematical Sciences: Lifting Properties of Invariant Kernels and Applications

这项研究将集中于广义Toeplitz核理论在分析和概率论问题中的应用，特别是在加权范数不等式和矩问题上的应用。这项工作的一个方向是关于随机过程的平稳性。通过考虑广义Toeplitz核，从奇异积分理论出发，提出了这类过程的广义平稳性概念。利用协方差核的提升性质，给出了具有协方差核的过程的特征。经典平稳过程是那些具有Toeplitz或不变协方差核的过程，它们作为特例出现。本工作计划处理在单参数变换群下不变的核和处理二维内插问题，目的是提供经典预测理论的Helson-Szego定理的二维版本。这项工作的第二个应用将被用于研究解析函数族的缺位系数。该理论的原型是Paley的工作，他断言Hardy空间中具有缺项系数的函数是平方可积的。一个对偶结果来自汉克尔形式，它实际上是Toeplitz形式的镜像。期望得到新的结果，这些结果可以转化为傅立叶变换的加权范数不等式。除了预测理论之外，这项工作还涉及到量子理论、散射结构和系统理论中的思想，以及工程中的矩问题。

This research will focus on applications of the theory of generalized Toeplitz kernels to problems in analysis and probability theory, especially to weighted norm inequalities and moment problems. One direction this work takes concerns stationarity of stochastic processes. A generalized notion of stationarity for such processes arises from the theory of singular integrals through the consideration of generalized Toeplitz kernels. The characterization of the processes having covariance kernels that are such was given using lifting properties of the kernels. The classical stationary processes being those that have covariance kernels that are Toeplitz or invariant occur as special cases. Present work is planned in dealing with kernels that are invariant under one-parameter groups of transformations and the treatment of bidimensional interpolation problems with a goal of providing a bidimensional version of the classical Helson-Szego theorem of prediction theory. A second application of this work will be sought in the study of lacunary coefficients of classes of analytic functions. The prototype of the theory is in the work of Paley which asserts that functions in Hardy space with lacunary coefficients are square integrable . A dual result derives from Hankel forms which are effectively the mirror images of Toeplitz forms. New results are expected which can be translated into weighted norm inequalities for Fourier transforms. In addition to prediction theory this work relates to ideas in quantum theory, scattering structures and systems theory, as well as moment problems in engineering.