Mathematical Sciences: Research in Fourier Analysis

9307781贝内代托这个项目将在调和分析的一般框架内发展几个主题。其一是关于加权不等的工作的继续。这些是将函数的范数与其变换的范数联系起来的不等式，即通过偏微分算子或傅立叶变换。范数是根据加权度量计算的，它们的比较提供了关于变换范围的基本信息，也就是关于解的集合。将特别强调卷积算子的双权估计的推导。还将在抽样问题上开展工作。有一种广泛的理论来处理定期抽样，但目前只有零星的结果可用于非常规抽样--这往往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算子的系数或通过改变表示函数来扩展最近的不规则抽样结果。一个具体的目标是构建现有理论的真正多维版本。还将继续开展关于小波理论的其他工作。其目的是从用于构造小波包的多组正交镜像滤波器组发展Riesz积，并建立L-1多分辨分析理论。调和分析是数学的一个分支，它试图通过分解和合成过程来分析复杂的现象。这种分析的经典模型可以在傅里叶级数和傅里叶积分表示中找到。最近的创新导致了小波理论，事实证明，它在分析局部时间和频率特性方面取得了很大的成功。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。***

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. ***