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.

这个项目继续研究与小波理论有关的傅里叶分析的现代方面，傅里叶变换的加权不等式和变换的限制定理。加权傅里叶变换不等式是由信号处理的一些中心问题引起的。在线性系统理论中，权重对应于能量集中问题中的各种过滤器。另一个应用领域，预测理论，产生了与平稳随机过程的功率谱相对应的权重的加权勒贝格空间。研究的目的是根据信号(原始函数)的加权p次方模求出加权傅里叶变换的q次方范数的界。该计划寻求可能的最佳估计。这种搜索产生了不确定原理不等式，这为确定在信号处理中使用的谱估计技术的有效性提供了自然的约束。小波理论的方法在这一活动中起着基础性的作用。最初的努力将集中在二次(加权)范数的特殊情况上。当可以计算关于权重递减重排(积分)的信息时，可以得到不等式。当不等式有效时，无需重新排列权重即可得到相同的条件。需要弄清楚的是在必要条件和充分条件之间发生了什么。在澄清不同条件的工作中，将采用从压载中衍生出来的技术，并结合底层帐篷空间的原子分解。