Mathematical Sciences: Wavelets Frames and Discrete Group Representations

DMS-9500269 PI：约翰斯顿 约翰斯顿将利用离散群的酉表示理论来统一连续和离散小波变换理论。 研究了真实的仿射群和真实的海森堡群的某些离散可数子群的酉表示的平方可积性。 这将给出新的框架结构，以及著名的结果的群论证明离散小波集在希尔伯特空间。 调和分析结合了这些数学元素，最好地体现了综合的思想。 一种是试图将复杂的问题分解为基本的组成部分。 然后分析这些组件的基本特性。 最后，通过组件的重组来重构解决方案。 傅立叶级数和傅立叶变换是在这种情况下使用的工具的例子;一个离散，另一个表示连续分解。最近的振荡积分理论增加了新的层面，一些更经典的方法，调和分析。

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.