Mathematical Sciences: Research in Fourier Analysis

小行星9307781 这个项目将在谐波分析的一般框架内开发几个主题。 一个是加权不等式工作的延续。 这些不等式将函数的范数与其变换的范数联系起来;即通过偏微分算子或傅立叶变换。 范数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。 特别强调的是卷积算子的双权估计的推导. 还将就抽样问题开展工作。 有一个广泛的理论，处理定期抽样，但只有分散的结果，目前可用于不定期抽样-这往往是唯一可行的手段收集信息。 这项研究继续努力扩大最近的不规则采样结果，通过使用逆运算符的系数或通过改变代表函数。 一个具体的目标是构建一个真正的多维版本的现有理论。 小波理论的其他工作也将继续进行。 我们的目标是Riesz产品的发展，从套正交镜像滤波器的小波包的建设和建设的L-1理论的多分辨率分析。 调和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已被证明是成功的，在产生更精细的能力，分析当地的时间和频率特性。 新的发展对信号分析、数据压缩和采样理论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. ***