Estimates of Fourier Transforms and Applications

该项目的目标是研究傅里叶变换在各种限制下的行为，并获得傅里叶变换和周期化的更清晰的估计。这项研究将集中在两个主要目标上。一是获得谐波分析中不确定性原理的新版本。不确定性原理是一个陈述，它说一个函数和它的傅里叶变换不能同时集中在小集合上。著名的海森堡测不准原理在量子力学中是谐波分析中更普遍的测不准原理的一个例子。特别是，研究者将研究具有傅里叶变换的函数，该函数支持具有特定类型密度的集合，以获得不确定性原理的新版本，并将结果应用于偏微分方程和信号处理。第二个目标是研究高维积分上的函数及其周期化之间的关系。周期化是调和分析中常用的傅立叶级数和傅立叶积分之间的联系。它们出现在具有周期性结构的问题中，这就是为什么它们是电子工程、信号处理和晶体学等应用科学中的重要工具。在这个建议研究者研究傅里叶变换的各种性质。傅里叶变换是一种主要的数学工具，在信号处理、计算机科学与电子工程、晶体学和断层扫描等领域广泛用于表示、转换和恢复数字数据、信息和信号。作者建议进一步研究傅里叶变换的性质，这将导致对傅里叶变换在各种限制下的行为有更深的理解。谐波分析中的不确定性原理的发展将导致确定多少信息足以恢复信号和数据的工具。作为回报，傅里叶变换的研究不仅会在数学物理，特别是偏微分方程等理论领域，而且会在图像处理，电气工程和数值方法等应用科学领域提供新技术。

The goal of the project is to investigate the behavior ofFourier transforms under various restrictions and to obtainsharper estimates of Fourier transforms and Periodizations.The research will be focused on two main objectives. Thefirst one is to obtain new versions for the UncertaintyPrinciple in Harmonic Analysis. The Uncertainty Principleis a statement which says that a function and its Fouriertransform can not be simultaneously concentrated on smallsets. The famous Heisenberg Uncertainty Principle inQuantum Mechanics is one of the examples of the more general Uncertainty Principle in Harmonic Analysis. In particular,the investigator will study functions with Fouriertransforms supported on sets with certain type of densitiesto get new versions of the Uncertainty Principle and applythe results for Partial Differential Equations and SignalProcessing. The second objective is devoted to the relationbetween functions and their periodizations over integerlattices in higher dimensions. Periodizations are oftenused in Harmonic Analysis as a link between Fourier seriesand Fourier integrals. They arise in problems havingperiodic structure that is why they are an important toolin such applied sciences as Electrical Engineering, SignalProcessing and Crystallography. In this proposal the investigator studies variousproperties of Fourier transforms. The Fourier transform isa major mathematical tool extensively used to represent,convert and recover digital data, information and signalsin Signal Processing, Computer Science and ElectricalEngineering, Crystallography and Tomography. The authorproposes to further investigate properties of Fouriertransformations which should lead to a deeper understandingof the behavior of Fourier transforms under variousrestrictions. The development of the proposed UncertaintyPrinciple in Harmonic Analysis will lead to tools to determinehow much information is sufficient to recover signals anddata. In return, this study of Fourier transforms will givenew techniques not only in theoretical areas such asMathematical Physics, in particular, Partial DifferentialEquations but also in applied sciences such as ImageProcessing, Electrical Engineering and Numerical Methods.