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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=674739
• 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.

这个建议涉及现代傅立叶分析处理系统 * 的各个方面（例如正交基。Riesz基础、框架...）由作用于希尔伯特空间的单个或多个元素的酉算子族（通常是一个群）* 生成。* 更具体地说，我们主要对Gabor系统，* 小波系统和加窗指数的情况感兴趣。当用于定义相应的酉算子的参数是正则的，例如形成一个离散群时，有一些众所周知的技术来处理这些系统。然而，当参数不规则时，这些技术通常会失效 *，并且关于相应的不规则系统 * 还有许多未解决的问题，我想继续研究。与此相关，我们还建议研究谱对和谱 * 测度的理论，它们可以被看作是区间上傅立叶级数 * 概念的自然推广。傅立叶级数首先由傅立叶在寻找热方程解的合适表达式的过程中发现。自从他们被发现以来，他们在纯数学和应用数学中，特别是在偏微分方程理论中，都发挥了重要的作用。傅立叶级数背后的主要数学成分是复指数函数，对于用于定义它们的参数的适当选择的 * 值，它们在给定区间上的平方可积函数的 * 空间中是正交和完备的。我们可以用任意的（可测的）集合E来代替这个区间 *，并询问是否存在一个复指数族 *，它在E上的 * 可积函数的平方空间中形成一个完备的正交系。如果是这种情况，则E称为谱集。构造这样的集合的例子并不困难，它们不是区间和B。Fuglede注意到，所有 * 这样的集似乎“瓷砖”的真实的行的翻译，在这个意义上说，真实的行可以 * 覆盖无限数量的翻译的一套E不重叠（达 * 套零措施）。这导致Fuglede制定他现在著名的猜想在一个 * 欧几里德空间的任意维度指出，一组E承认一个正交的基础上 * 复指数当且仅当它瓷砖欧几里德空间的翻译。* 不幸的是，这个猜想现在已被证明在 * 维度3或更高维度的两个方向上都是错误的，尽管低维问题仍然悬而未决。 我们建议研究与谱集密切相关的谱测度理论。一个概率测度是谱的，如果平方可积函数的相应空间 * 容许指数的正交基。C.最近的结果- K. Lai和我建议 *，至少在某些情况下，在测度 * 是谱的这一事实和卷积性质之间存在关系，这可以被视为平铺性质的某种推广。我想调查这些类型的问题，更详细的希望 *，以阐明更多的迷人性质的谱集和谱措施 * 有关Fuglede的猜想。