Mathematical Sciences: Lifting Properties of Invariant Kernels and Applications

本文将重点研究广义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注随机过程的平稳性。这类过程的广义平稳性概念是通过考虑广义Toeplitz核从奇异积分理论中产生的。具有这样的协方差核的过程的表征是利用核的提升性质给出的。经典的平稳过程是那些协方差核是Toeplitz或不变量的过程，作为特殊情况出现。目前的工作计划是处理在单参数变换群下不变的核和处理二维插值问题，目的是提供经典预测理论的Helson-Szego定理的二维版本。本工作的第二个应用将在分析函数类的空白系数的研究中寻求。该理论的原型是在Paley的著作中，他断言Hardy空间中具有空系数的函数是平方可积的。汉克尔形式是Toeplitz形式的镜像，它产生了双重结果。新的结果有望转化为傅里叶变换的加权范数不等式。除了预测理论外，这项工作还涉及量子理论、散射结构和系统理论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.