Frames generated by unitary systems.

这个建议涉及到处理系统的现代傅立叶分析的各个方面

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.