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的更精确的估计。第一部分是对谐波分析中的不确定性原理进行了新的推导。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。量子力学中著名的海森堡测不准原理是调和分析中更一般的测不准原理的例子之一。特别是，研究者将研究具有特定类型的密度集上支持的傅里叶变换的函数，以获得新版本的不确定性原理，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整数格上的函数及其周期化之间的关系。调和分析中经常使用周期作为傅里叶级数和傅里叶积分之间的联系。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气工程、信号处理和晶体学等应用科学中的重要工具。 在这个建议中，研究者研究了傅立叶变换的各种性质。傅立叶变换伊萨种主要的数学工具，广泛应用于信号处理、计算机科学与电子工程、晶体学和断层扫描等领域，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，这将导致更深入地了解傅里叶变换在各种限制下的行为。谐波分析中的不确定性原理的发展将导致工具来确定多少信息足以恢复信号和数据。傅立叶变换的研究不仅在数学物理，特别是偏微分方程等理论领域，而且在图像处理、电气工程和数值方法等应用科学领域都将提供新的技术。

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.