Mathematical Sciences: Research in Fourier Analysis

数学科学：傅里叶分析研究

• 批准号: 9307781
• 负责人: John Benedetto
• 金额: $ 4万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9307781&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Fourier; Analysis

9307781 Benedetto This project will develop several themes within the general framework of harmonic analysis. One is a continuation of work on weighted inequalities. These are inequalities relating norms of functions with the norms of their transforms; i.e. through a partial differential operator or the Fourier transform. The norms are computed against weighted measures and their comparisons provide fundamental information about the range of the transformation, that is, about the set of solutions. Particular emphasis will be placed on the derivation of two- weight estimates of convolution operators. Work will also be done on sampling problems. There is an extensive theory which treats regular sampling but only scattered results are currently available for irregular sampling - which is often the only feasible means for gathering information. This research continues efforts to expand on recent irregular sampling results by using either coefficients from inverse operators or by varying the representing functions. A specific goal is that of constructing a genuine multidimensional version of existing theory. Additional work on wavelet theory will also be pursued. The object is the development of Riesz products from sets of quadrature mirror filters for the construction of wavelet packets and the construction of an L-1 theory of multiresolution analysis. Harmonic analysis is a branch of mathematics which seeks to analyze complex phenomena through processes of decomposition and synthesis. The classical models for such analysis can be found in Fourier series and Fourier integral representations. Recent innovations have led to wavelet theory whic h has proved successful in yielding much a refined capability in analyzing local time and frequency characteristics. New developments have had a major impact on studies related to signal analysis, data compression and sampling theory. ***

9307781 Benedetto这个项目将在谐波分析的一般框架内开发几个主题。 一种是对加权不平等的延续。 这些是功能规范与其转换规范有关的不平等。即通过部分差分运算符或傅立叶变换。 规范是根据加权度量计算的，其比较提供了有关转换范围（即有关解决方案集）的基本信息。 特别重点将放在卷积操作员的两次权重估计的推导上。 还将在抽样问题上完成工作。 有一个广泛的理论可以治疗常规采样，但目前仅可用于不规则抽样的结果 - 这通常是收集信息的唯一可行手段。 这项研究继续努力通过使用逆操作员或改变表示函数的系数来扩展最近的不规则采样结果。 一个具体的目标是构建现有理论的真正多维版本。 还将进行有关小波理论的其他工作。 该对象是从一组正交镜过滤器中开发Riesz产品，用于构建小波数据包，以及构建L-1多分辨率分析理论。 谐波分析是数学的一个分支，旨在通过分解和合成过程来分析复杂现象。 这种分析的经典模型可以在傅立叶序列和傅立叶积分表示中找到。 最近的创新导致了小波理论，事实证明，在分析当地时间和频率特征方面已经成功地产生了许多精致的能力。 新的发展对与信号分析，数据压缩和采样理论相关的研究产生了重大影响。 ***