Frames generated by unitary systems.

由单一系统生成的框架。

• 批准号: RGPIN-2014-05935
• 负责人: Gabardo, JeanPierre
• 金额: $ 0.8万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611932
• 关键词: Frames; generated; unitary; systems.

This proposal involves various aspects of modern Fourier analysis dealing with systems (e.g. orthonormal bases. Riesz bases, frames,...) generated by a family (usually a group) of unitary operators acting on a single or finitely many elements of a Hilbert space. More specifically, we are mainly interested in the case of Gabor systems, wavelet systems and windowed exponentials. There exist well-known techniques to deal with these systems when the parameters used to define the corresponding unitary operators are regular, forming a discrete group for example. However, when the parameters are irregular, these techniques generally break down and there are many unsolved problems regarding the corresponding irregular systems which I would like to pursue. Related to these, we also propose to work on the theory of spectral pairs and spectral measures, which can be seen as natural generalizations of the concept of Fourier series on an interval. Fourier series where first discovered by Fourier in the process of finding a suitable expression for the solutions of the heat equation. Since their discovery,they have played a fundamental role in both pure and applied mathematics, more particularly in the theory of partial differential equations. The main mathematical ingredients behind Fourier series are the complex exponential functions which turm out, for suitably chosen values of the parameters used to define them, to be orthogonal and complete in the space of square-integrable functions on the given interval. One can replace this interval by an arbitrary (measurable) set E and ask if there exists a family of complex exponentials which forms a complete orthogonal system for the corresponding space of square in integrable functions on E. If this is the case, E is called a spectral set. It is not difficult to construct examples of such setswhich are not intervals and B. Fuglede noticed that all such sets seem to ''tile" the real line by translations, in the sense that the real line could be covered by an infinite number of translates of the set E which do not overlap (up to sets of zero measure). This lead Fuglede to formulate his now famous conjecture in an Euclidean space of arbitrary dimension stating that a set E admits an orthogonal basis of complex exponential if and only if it tiles the Euclidean space by translation. Unfortunately, this conjecture has now been shown to be false in both direction in dimension 3 or higher, although the lower-dimensional problems are still open. We propose to work on the theory of spectral measures which are closely related to spectral sets. A probability measure is spectral if the corresponding space of square-integrable functions admits an orthogonal basisof exponentials. Recent results by C.-K. Lai and myself suggest that, at least in some situations,there is a relationship between the fact that a measure is spectral and a convolution property, which can be seen as some generalization of a tiling property. I would like to investigate these type of problems in more details with the hope to shed more light to the fascinating properties of spectral sets and spectral measures in relation to Fuglede's conjecture.

这一建议涉及现代傅立叶分析处理系统的各个方面 (e.g.标准正交基Riesz基础、框架...）由一个家庭（通常是一个团体）产生的 作用在希尔伯特空间的单个或多个元素上的酉算子。 更具体地说，我们主要对Gabor系统的情况感兴趣， 小波系统和加窗指数。有一些众所周知的技术 处理这些系统时所用的参数要定义相应的 酉算子是正则的，例如形成离散群。 然而，当参数不规则时，这些技术通常失效 相应的非正则系统还有许多问题没有解决 我想继续调查 与此相关，我们还提出了谱对理论和谱 措施，这可以被看作是自然推广的概念傅立叶级数 在一个间隔。傅立叶级数是傅立叶在寻找 热方程解的合适表达式。自从他们被发现以来，他们 在纯数学和应用数学中发挥了重要作用， 在偏微分方程理论中。背后的主要数学成分 傅立叶级数是复指数函数，对于适当选择的 用于定义它们的参数的值，在 给定区间上的平方可积函数空间。人们可以把这个间隔 由任意（可测）集合E，并询问是否存在一个复指数族 它形成了一个完整的正交系统的相应空间的平方， E上的可积函数如果是这种情况，则E称为谱集。就不难 构造不是区间和B的集合的例子。Fuglede注意到，所有 这样的集合似乎通过平移“平铺”真实的线，在这个意义上，真实的线可以 被集合E的不重叠的无限数量的平移覆盖（直到 零的测量）。这导致Fuglede制定了他现在著名的猜想， 任意维数的欧几里得空间，说明一个集合E有一个正交基， 复指数当且仅当它通过平移平铺欧几里得空间。 不幸的是，这个猜想现在已经被证明是错误的，在两个方向， 3维或更高，尽管低维问题仍然是开放的。我们提出 研究与谱集密切相关的谱测度理论。 一个概率测度是谱的，如果相应的平方可积函数空间 admits承认an orthogonal正交basis基of exponentials指数. C.最近的结果- K.赖和我建议 至少在某些情况下， 是频谱和卷积属性，可以看作是平铺的某种推广 财产我想调查这些类型的问题在更多的细节与希望 为了更好地阐明谱集和谱测度的迷人特性， 与Fuglede猜想的关系