半空间时频分析的理论与方法研究及相关问题

构造具有良好时频局部化性质、优美结构及计算方面快捷有效的标架是函数空间标架研究与时频分析追求的目标。小波与Gabor标架可以做到这一点。过去三十年来，我们见证了直线上平方可积函数空间L2(R)上小波与Gabor标架丰硕的理论成果和巨大的应用潜力，然而半直线上平方可积函数空间L2(R+)上具有优美结构的标架的研究鲜见报道（R+表示正实数集）。这是因为R按加法作成群，而R+却不能。本项目拟研究L2(R+)上的伸缩调制系（MD系）时频分析理论，其中调制因子是伸缩周期的（非加法周期）。注意到：L2(R+)在Fourier变换下不封闭，且不容许小波与Gabor标架；Hardy空间中小波系的Fourier变换不具有MD系形式；局部紧群上的Gabor分析不适用于L2(R+)。因此，本项目研究的问题不是平凡的，其研究成果将进一步丰富时频分析的理论与方法，为标架理论的数学与工程应用提供新的理论借鉴。

To construct frames with good time-frequency localized properties, beautiful structure and efficient computational properties is a goal in the function space frame theory and time-frequency analysis. Wavelet and Gabor frames are among them. The last thirty years have seen great achievements in the theory and applications of the wavelet and Gabor analysis on the square integral function space on the real line, but have seen little progress of frames with beautiful structure on the square integral function space on the half real line. It is because the real line is a group under addition, while the half real line is not. This project intends to study the time-frequency analysis of dilation-and-modulation systems in L2(R+), where the modulation factors are dilation periodic instead of addition periodic. Observe that L2(R+) is not closed under Fourier transform, and does not admit wavelet and Gabor frames, that the Fourier transform version of wavelet systems in the Hardy space is not of MD systems, and that Gabor analysis on local compact does not apply to L2(R+). Therefore, the problems of this project are nontrivial. This research will further enrich the theory and methods of time-frequency analysis, and provide a new theoretical reference for the mathematical and engineering application of the frame theory.