Mathematical Sciences: Research in Fourier Analysis

9307781 Benedetto这个项目将在谐波分析的总体框架内发展几个主题。一个是对加权不等式研究的延续。这些是函数的范数与其变换的范数之间的不等式；也就是通过偏微分算子或者傅里叶变换。规范是根据加权测度计算的，它们的比较提供了关于变换范围的基本信息，即关于解的集合。特别的重点将放在卷积算子的两权估计的推导上。我们还将研究抽样问题。有一个广泛的理论，它处理常规抽样，但目前只有分散的结果可用于不规则抽样-这往往是收集信息的唯一可行手段。本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。小波理论的进一步研究也将继续进行。目的是开发用于构造小波包和构造多分辨率分析的L-1理论的正交镜像滤波器集的Riesz产品。谐波分析是数学的一个分支，它试图通过分解和综合的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，它已被证明在分析局部时间和频率特性方面产生了非常精细的能力。新的发展对与信号分析、数据压缩和采样理论有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. ***