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，标准正交基）。Riesz基，坐标系，…)由一族（通常是一群）作用于希尔伯特空间的单个或有限多个元素的幺正算子生成。更具体地说，我们主要对Gabor系统，小波系统和窗口指数感兴趣。当用于定义相应酉算子的参数是正则的，例如形成一个离散群时，已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有许多未解决的问题，这是我想要追求的。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里叶级数概念的自然推广。傅里叶级数是傅里叶在寻找热方程解的合适表达式时首次发现的。自从它们被发现以来，它们在纯数学和应用数学中都发挥了重要作用，尤其是在偏微分方程理论中。傅里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，复指数函数在给定区间上的平方可积函数空间中是正交的和完全的。我们可以用一个任意的（可测的）集合E来代替这个区间，并问是否存在复指数族，它们在E上的可积函数的相应的平方空间中形成一个完整的正交系统。如果是这样，E就被称为谱集。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，所有这样的集合似乎都通过平移来“平铺”实线，在这个意义上，实线可以被集合E的无限数量的平移所覆盖，这些平移不重叠（直到零测度的集合）。这使得Fuglede在任意维的欧几里得空间中提出了他现在著名的猜想，即当且仅当一个集合E通过平移来覆盖欧几里得空间时，它承认复指数的正交基。不幸的是，这个猜想现在已经被证明在3维或更高维度的方向上都是错误的，尽管低维的问题仍然是开放的。我们建议研究与谱集密切相关的谱测度理论。如果平方可积函数的相应空间允许指数的正交基，则概率测度是谱测度。c - k。Lai和我认为，至少在某些情况下，一个测度是频谱的这一事实与卷积性质之间存在着某种关系，卷积性质可以被看作是平铺性质的某种推广。我想更详细地研究这类问题，希望能更多地阐明与Fuglede猜想有关的光谱集和光谱测度的迷人性质。

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 techniquesto deal with these systems when the parameters used to define the correspondingunitary operators are regular, forming a discrete group for example.However, when the parameters are irregular, these techniques generally break downand there are many unsolved problems regarding the corresponding irregular systemswhich 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 chosenvalues 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 intervalby an arbitrary (measurable) set E and ask if there exists a family of complex exponentialswhich 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 toconstruct examples of such setswhich are not intervals and B. Fuglede noticed that allsuch 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 indimension 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 hopeto shed more light to the fascinating properties of spectral sets and spectral measures in relation to Fuglede's conjecture.