Mathematical Sciences: Wavelets and Time-Frequency Analysis

9401859 Ramanathan该奖项支持谐波分析相关领域的数学研究项目，重点是时频分析和小波理论，具有信号处理的潜在应用。该计划有三个主要主题。第一个问题是分析和重建信号的非均匀样本的Gabor变换。采样集的非均匀性要求引入新的技术，用相干态的性质来表述结果。诸如固定窗口函数的给定时频移集合是否完整，是否最小，是否形成框架或是否形成Riesz基等问题将被采纳。第二部分研究Weyl算子的奇异值。这些算子与具有良好局域性的Gabor变换密切相关，即将函数的能量集中在给定空间区域的算子。这些算子的局部化性质可以从其奇异值的行为中推断出来。将对这些值的渐近衰减进行研究。关于小波和膨胀方程的工作延续了早期的工作，这些工作发展了联合谱半径的旧概念在膨胀方程解的存在性和光滑性表征中的重要应用。在这项工作中，将研究非正则解的性质。这些包括非紧凑支持或分布式解决方案。将考虑与随机矩阵的连接。谐波分析结合了最能体现综合思想的数学元素。人们试图把复杂的问题分解成基本的组成部分。然后分析这些组件的基本特性。最后，通过组件的重组重构解。傅里叶级数和傅里叶变换是在这种情况下使用的工具的例子；一个是离散的，另一个是连续的。最近，小波理论为一些更经典的谐波分析方法增加了新的维度。＊＊＊

9401859 Ramanathan This award supports a mathematical research program in a range of related areas in harmonic analysis, concentrating on time-frequency analysis and wavelet theory, with potential applications to signal processing. There are three main themes of the program. The first concerns the analysis and reconstruction of signals from nonuniform samples of their Gabor transforms. The nonuniformity of the sampling set requires the introduction of new techniques with results formulated in terms of the properties of coherent states. Questions such as whether a given collection of time-frequency shifts of a fixed windowfunction are complete, are minimal, form a frame or form a Riesz basis will be taken up. The second line of investigation studies singular values of Weyl operators. These operators are closely related to Gabor transform with good localizing properties, i.e. operators which concentrate the energy of a function in a given region of space. The localization properties of these operators can be inferred from the behavior of their singular values. Studies of the asymptotic decay of these values will be carried out. Work on wavelets and dilation equations continues earlier work which developed important applications of an older notion of joint spectral radius to the existence and smoothness characterizations of solutions to dilations equations. In this work, properties of non-canonical solutions will be studied. These include non-compactly supported or distributional solutions. Connections with random matrices will be considered. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; o ne discrete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis. ***