Mathematical Sciences: Lifting Properties of Invariant Kernels and Applications

本研究将集中在应用的理论， 广义Toeplitz核的分析和 概率论，特别是加权范数不等式， 瞬间的问题 这项工作的一个方向是关于 随机过程 广义平稳性概念 这种过程源于奇异积分理论 通过考虑广义Toeplitz核。 的 具有协方差核的过程的特征， 是这样给出的使用提升性能的内核。 的 经典平稳过程是那些具有协方差 Toeplitz或不变的核作为特殊情况出现。 目前的工作计划在处理内核， 在单参数变换群下不变， 处理二维插值问题的目标是 提供了经典Helson-Szego的二维版本 预测理论的定理 这项工作的第二个应用将寻求在 解析函数类缺项系数的研究。 该理论的原型是在佩利的工作， 在哈代空间中具有缺项系数的函数是 平方可积 . 从Hankel形式导出一个对偶结果， Toeplitz形式的镜像。 期待新成果 它可以转化为加权范数不等式， 傅立叶变换 除了预测理论，这项工作还涉及到思想 在量子理论、散射结构和系统理论中， 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.