Frames generated by unitary systems.

这一建议涉及现代傅里叶分析处理系统的各个方面。 (例如，正交基。Riesz底座、框架...)由一个家庭(通常是一个团体)生成 指作用于Hilbert空间的单个或有限多个元素上的酉算子。 更具体地说，我们主要对Gabor系统的案例感兴趣， 小波系和加窗指数。有一些众所周知的技术 处理这些系统时使用的参数定义了相应的 酉运算符是正则的，例如形成一个离散的群。 然而，当参数不规则时，这些技术通常会失效 而与之相对应的非规则系统也存在许多悬而未决的问题 这是我想要追求的。 与此相关，我们还提出了谱对和谱理论的研究。 度量，它可以看作是傅里叶级数概念的自然推广 每隔一段时间。傅里叶级数是由傅里叶在寻找的过程中首先发现的 热方程解的一个合适的表达式。自从他们被发现以来，他们 在纯数学和应用数学中都发挥了基础作用，更具体地说 在偏微分方程组理论中。背后的主要数学成分 傅里叶级数是复数的指数函数，可以适当地选择 用于定义它们的参数值在 在给定区间上平方可积函数的空间。人们可以替换这个间隔 通过任意(可测量的)集合E，并询问是否存在一族复指数族 它对相应的平方空间构成了一个完备的正交系。 E上的可积函数。如果是这样的话，E称为谱集。这并不难做到 构造不是区间的这种集合的例子，B.Fuglede注意到所有 从某种意义上说，这样的集合似乎是通过翻译将真实的线条“瓦片”起来 被集合E的不重叠的无限数量的平移所覆盖(最多 零度量组)。这导致Fuglede在一个 任意维的欧几里得空间，它表明集合E有如下正交基 复指数当且仅当它通过平移平铺欧几里得空间。 不幸的是，这一猜想现在已经被证明在两个方向上都是错误的 3维或更高维度，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度量理论。 概率测度是谱的如果对应的平方可积函数空间 允许指数的正交基。赖和我最近的研究结果表明 那就是，至少在某些情况下，一项措施 是光谱和卷积属性，这可以看作是分块的某种推广 财产。我想对这类问题进行更详细的调查，希望 更多地揭示了谱集和谱度量的迷人性质 与Fuglede猜想的关系。

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.