Frames generated by unitary systems.

这个建议涉及处理系统的现代傅立叶分析的各个方面（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，我们主要感兴趣的情况下的Gabor系统，小波系统和加窗指数。当用于定义对应的酉算子的参数是正则的，例如形成离散群时，存在处理这些系统的公知技术。然而，当参数是不规则的，这些技术通常会崩溃，有许多未解决的问题，关于相应的不规则系统，我想追求。 与此相关，我们还建议研究谱对和谱测度的理论，这可以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。自从他们的发现，他们发挥了基础作用，在纯数学和应用数学，更特别是在理论的偏微分方程。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值，它们在给定区间上的平方可积函数空间中是正交和完备的。人们可以用任意的（可测的）集合E来代替这个区间，并询问是否存在一族复指数，它们在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，所有这些集似乎“瓦”的真实的行的翻译，在这个意义上说，真实的行可以涵盖无限数量的翻译的一套E不重叠（至套零措施）。这导致Fuglede制定他现在著名的猜想在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，这个猜想现在已经被证明在3维或更高维的两个方向上都是错误的，尽管低维问题仍然是开放的。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允许指数的正交基。C.最近的结果- K. Lai和我认为，至少在某些情况下，测度是谱的这一事实与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在更多的细节，希望更多地阐明了迷人的性能谱集和频谱措施有关Fuglede的猜想。

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.