Mathematical Sciences: Wavelets Frames and Discrete Group Representations

DMS-9500269 PI: Johnston Johnston将使用离散群的酉表示理论来统一连续和离散小波变换理论。研究了实仿射群和实Heisenberg群的离散可数子群的酉表示的平方可积性。这将给Hilbert空间中离散小波集的已知结果提供新的框架结构和群论证明。谐波分析结合了最能体现综合思想的数学元素。人们试图把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性。最后，通过组件的重组重构解。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，另一个是连续的。最近，振荡积分理论为一些更经典的谐波分析方法增加了新的维度。

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.