Frames generated by unitary systems.

这一建议涉及现代傅立叶分析的各个方面，涉及系统*(例如，正交基。Riesz底座、框架...)由作用于希尔伯特空间的单个或有限多个元素的一族(通常是一组)酉运算符生成。*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运算符的参数是规则的，例如，形成一个离散的群时，存在着处理这些系统的众所周知的技术。然而，当参数是不规则的时，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。**与这些相关的，我们还建议研究谱对和谱*测量的理论，这可以被看作是傅立叶级数概念在区间上的自然推广。傅里叶级数是由傅里叶在寻找热方程解的合适表达式的过程中首先发现的。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基础作用，尤其是在偏微分方程式理论中。傅立叶级数背后的主要数学成分是复指数函数，对于用来定义它们的参数的适当选择，它们在给定区间上的平方可积函数的*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，并询问是否存在一族复指数族*，它构成了E上相应的平方可积函数空间的一个完备的正交系。如果是这样的话，E称为谱集。构造不是区间的这种集合的例子并不困难，B.Fuglede注意到，所有*这样的集合似乎都通过平移来‘平铺’真实的直线，在这个意义上，真实的直线可以*被集合E的不重叠的无限数量的平移(直到*个零度量的集合)所覆盖。这使得Fuglede在任意维欧氏空间中提出了他现在著名的猜想，指出集合E允许*复指数的正交基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在*维3或更高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的理论。*如果相应的平方可积函数空间*允许指数的正交基，则概率测度是谱的。Lai C.-K.Lai和我最近的结果表明，至少在某些情况下，度量*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。我想更详细地研究这些类型的问题，希望*更多地阐明谱集和谱度量*与Fuglede猜想有关的迷人性质。

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 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.