RUI: Orthonormal Fourier Bases and Iterated Function Systems

研究人员研究了实线上和高维上与仿射迭代函数系(IFSS)相关的Hilbert空间的傅立叶基。在过去的二十年里，许多工作都集中在迭代函数系统的吸引子是具有非整数Hausdorff维的分形体的情况下，而研究者们研究的是具有重叠的迭代函数系统。在一维情形下，与这些吸引子相关的测度是鄂尔多斯的自相似卷积测度，一般是不变的Hutchinson测度。关于非有限重叠情形下与Hutchinson测度相关的正交基的研究几乎一无所知，事实上，研究者和P.Jorgensen以前的工作已经表明，重叠情形下的测度具有与非重叠情形下不同的性质。提出的关于IFSS的工作是调和分析的一个令人兴奋的分支，它与分形学、小波和随机游动有关；它还应用于数论、动力学和组合几何。调查人员在他们的工作中借鉴了所有这些领域的方法。这项提案中的工作将涉及与爱荷华大学的P.Jorgensen进一步合作。迭代函数系统(IFSS)体现了递归和自相似的共同主题。研究人员将研究IFSS的性质及其相关几何图形。事实上，这项工作的一个中心主题是几何和分形谱分析之间的相互作用。从几何角度和从频谱角度研究分形学之间存在着天然的张力；用于频谱分析的传统时频方法是线性的，研究人员研究的系统是非线性的。了解非线性现象的需要是由许多现实生活应用程序推动的。IFS在信号处理和通信中反复出现。例如，JPEG2000压缩标准背后的关键小波就是一种特殊的迭代函数系统。迭代函数系统的应用远远超出了通信-迭代函数系统被用于数据压缩、量子计算、模式识别、DNA计算和大量其他领域。此外，由于其广泛的适用性，这项提案中的工作将吸引本科生进入数学研究。研究人员将与格林内尔学院的本科生一起进行研究，P.Jorgensen和调查人员发表的工作已经被纳入爱荷华大学本科生的研究项目，作为研究生教育和教授联盟(AGEP Alliance)和哈特兰数学伙伴关系的一部分。

ABSTRACTThe investigators study Fourier bases of Hilbert spaces associated with affine iterated function systems (IFSs) on the real line and in higher dimensions. While much work in the last two decades has focused on the case in which the attractor of the IFS is a fractal with non-integral Hausdorff dimension, the investigators study IFSs with overlap. In the one-dimensional setting, the measures associated with these attractors are Erdos's self-similar convolution measures; in general the measures are invariant Hutchinson measures. Almost nothing is known about orthonormal bases associated with Hutchinson measures in the case of non-finite overlap, and in fact previous work of the investigators and P. Jorgensen has shown that the measures in the overlap cases have different properties from the non-overlap measures. The proposed work on IFSs is an exciting branch of harmonic analysis which connects to fractals, wavelets, and random walks; it also has applications to number theory, dynamics, and combinatorial geometry. The investigators draw on methods from all these fields in their work. The work in this proposal will involve further collaborations with P. Jorgensen at the University of Iowa. Iterated function systems (IFSs) embody the common themes of recursion and self-similarity. The investigators will study properties of IFSs and their associated geometry. In fact, a central theme in this work is the interplay between geometry and spectral analysis on fractals. There is a natural tension between studying fractals from a geometric point of view and from a spectral point of view; traditional time-frequency methods used in spectral analysis are linear, and the systems the investigators study are non-linear. The need to understand non-linear phenomena is motivated by a host of real-life applications. IFSs occur repeatedly in signal processing and communications. For example, wavelets, the key behind the JPEG 2000 compression standard, are a particular example of an IFS. Applications of iterated function systems extend far beyond communications---IFSs are used in data compression, quantum computing, pattern recognition, DNA computations, and a vast array of other fields. Furthermore, due to its wide applicability, work in this proposal will draw undergraduate students into mathematical research. The investigators will conduct research with undergraduates at Grinnell College, and published work by P. Jorgensen and the investigators has already been incorporated into research projects for undergraduates at the University of Iowa, as part of the Alliances for Graduate Education and the Professoriate (AGEP Alliance) and the Heartland Mathematics Partnership.