Mathematical Sciences: Wavelets Frames and Discrete Group Representations

数学科学：小波框架和离散群表示

• 批准号: 9500269
• 负责人: Carolyn Johnston
• 金额: $ 5.26万
• 依托单位: Florida Atlantic University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500269&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Wavelets; Frames; Discrete

DMS-9500269 PI: Johnston Johnston will use the unitary representation theory of discrete groups to unify the continuous and discrete wavelet transform theories. The square-integrability of unitary representations of certain discretized countable subgroups of the real affine group and the real Heisenberg group will be investigated. This will give new frame constructions as well as group-theoretic proofs for well-known results about discrete wavelet sets in Hilbert spaces. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the theory of oscillatory integrals added new dimensions to some of the more classical approaches to harmonic analysis.

DMS-9500269 PI：Johnston Johnston将使用离散群的么正表示理论来统一连续和离散小波变换理论。我们将研究实仿射群和实Heisenberg群的某些离散化的可数子群的酉表示的平方可积性。这将为Hilbert空间中关于离散小波集的著名结果提供新的框架结构和群论证明。调和分析结合了这些数学元素，最好地例证了综合的思想。一种是试图将复杂的问题分解成基本的组成部分。然后分析这些组件的基本特征。最后，通过组件的重新组合来重建解决方案。傅里叶级数和傅里叶变换是在这种情况下使用的工具的例子；一个是离散的，另一个代表连续分解。最近，振荡积分理论为一些更经典的调和分析方法增加了新的维度。