Mathematical Sciences: Wavelets and Time-Frequency Analysis

小行星9401340 该奖项支持一系列谐波分析相关领域的数学研究计划，专注于时频分析和小波理论，并具有信号处理的潜在应用。 该计划有三个主要主题。 第一个问题涉及的分析和重建信号的非均匀样本的Gabor变换。 采样集的不均匀性需要引入新的技术，其结果根据相干态的性质来制定。 问题，如是否一个给定的集合的时间频率偏移的一个固定的窗口函数是完整的，是最小的，形成一个框架或形成一个Riesz的基础将采取行动。 第二条调查线研究Weyl算子的奇异值。 这些算子与具有良好局部化性质的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。 小波和膨胀方程的工作继续早期的工作，开发了重要的应用程序的一个旧的概念，联合频谱半径的存在性和光滑性特征的解决方案，膨胀方程。 本文主要研究非正则解的性质。 这些包括非compatible支持的或分布式的解决方案。 将考虑与随机矩阵的连接。 调和分析结合了这些数学元素，最好地体现了综合的思想。 一种是试图将复杂的问题分解为基本的组成部分。 然后分析这些组件的基本特性。 最后，通过组件的重组来重构解决方案。 傅立叶级数和傅立叶变换是在这种情况下使用的工具的例子;一个离散，另一个表示连续分解。最近小波理论增加了新的维度，一些更经典的方法，谐波分析。

9401340 Heil 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 window function 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; one dis crete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.