RUI: Orthonormal Fourier Bases and Iterated Function Systems

RUI：正交傅立叶基和迭代函数系统

• 批准号: 0701164
• 负责人: Karen Shuman
• 金额: --
• 依托单位: Grinnell College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-15 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701164&HistoricalAwards=false
• 关键词: RUI; Orthonormal; Fourier; Bases; Iterated

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.

本文研究了在真实的直线上和高维空间中与仿射迭代函数系统（IFS）相关联的Hilbert空间的傅立叶基。虽然在过去的二十年里，许多工作都集中在IFS的吸引子是具有非整数Hausdorff维数的分形的情况下，研究人员研究IFS的重叠。在一维环境中，与这些吸引子相关的测度是鄂尔多斯自相似卷积测度;一般来说，这些测度是不变的哈钦森测度。几乎没有什么是已知的正交基与哈钦森措施在非有限重叠的情况下，事实上，以前的工作的调查人员和P. Jorgensen已经表明，措施在重叠的情况下，有不同的性质，从非重叠措施。关于IFS的拟议工作是调和分析的一个令人兴奋的分支，它与分形、小波和随机游走有关;它也应用于数论、动力学和组合几何。调查人员在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州大学的P. Jorgensen的进一步合作。迭代函数系统（IFS）体现了递归和自相似的共同主题。研究人员将研究IFS的特性及其相关的几何形状。事实上，这项工作的一个中心主题是分形几何和光谱分析之间的相互作用。从几何的角度和从频谱的角度研究分形之间有一种天然的张力;传统的时频方法用于频谱分析是线性的，而研究人员研究的系统是非线性的。理解非线性现象的需要是由许多实际应用激发的。IFS在信号处理和通信中反复出现。例如，小波，JPEG 2000压缩标准背后的关键，是IFS的一个特定示例。迭代函数系统的应用远远超出了通信-IFS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，这项建议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和研究人员发表的工作已经被纳入爱荷华州大学本科生的研究项目，作为研究生教育和教授联盟（AGEP联盟）和中心地带数学伙伴关系的一部分。