Mathematical Sciences: Research in Fourier Analysis

数学科学：傅立叶分析研究

• 批准号: 9002420
• 负责人: John Benedetto
• 金额: $ 11.4万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-06-01 至 1994-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002420&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Fourier; Analysis

This project continues work on modern aspects of Fourier analysis related to wavelet theory, weighted inequalities for the Fourier transform and restriction theorems for the transform. Weighted Fourier transform inequalities are motivated by some of the central issues of signal processing. In linear system theory, weights correspond to various filters in energy concentration problems. Another area of application, prediction theory, weighted Lebesgue spaces arise for weights corresponding to power spectra of stationary stochastic processes. The object of the research is to obtain bounds on the q-th power norm of the weighted Fourier transform in terms of the weighted p-th power norm of the signal (original function). The program seeks best possible estimates. This search gives rise to uncertainty principle inequalities; these provide natural constraints for determining the effectiveness of spectrum estimation techniques used in signal processing. The methods of wavelet theory play a basic role in this activity. Initial efforts will focus on the special case of quadratic (weighted) norms. Inequalities may be obtained when information concerning the (integrals of) decreasing rearrangements of the weights can be computed. The same condition obtains without rearranging the weights when the inequalities are valid. What needs to be worked out is what happens between the necessary and sufficient conditions. Techniques derived from balayage combined with atomic decompositions of underlying tent spaces will be employed in efforts to clarify the different conditions.

该项目继续在与小波理论相关的傅立叶分析的现代方面进行工作，对转换的傅立叶变换和限制定理的加权不平等。 加权傅立叶变换不平等是由信号处理的一些主要问题激发的。 在线性系统理论中，权重对应于能量浓度问题中的各种过滤器。 对应于固定随机过程的功率谱的权重出现了另一个应用领域，预测理论，加权lebesgue空间。 研究的目的是根据信号的加权p-th功率标准（原始函数）获得加权傅立叶变换的Q-the功率规范的界限。 该计划寻求最好的估计。 这种搜索引起了不确定性原则的不平等。这些提供了自然限制，以确定信号处理中使用的频谱估计技术的有效性。 小波理论的方法在这项活动中起着基本作用。 最初的努力将集中在二次（加权）规范的特殊情况下。 当可以计算有关减少权重重排的（积分）的信息时，可能会获得不平等。 当不等式有效时，相同的条件在不重新安排重量的情况下获得。 需要解决的是必要条件和充分条件之间会发生什么。 将采用源自巴莱奇（Balayage）的技术与基本帐篷空间的原子分解相结合，以阐明不同的条件。