子空间标架理论的若干问题及应用

去噪是信号处理的关键问题之一，标架与伪标架在信号去噪中有重要应用；多址技术一直都是无线通信的关键技术之一，超标架在多址技术方面有重要应用。到目前为止，全空间(平方可积函数空间)标架理论的研究成果丰硕，约化子空间标架理论的研究取得了一定进展，子空间超标架的研究未见报道。约化子空间是Hardy空间与全空间的推广，实际问题中有大量约化子空间中的信号。研究表明，约化子空间标架理论与子空间超标架理论不是全空间标架理论的平凡推广，如子空间超Gabor标架Ron-Shen对偶原则不成立等。本项目拟研究一般约化子空间中Parseval标架小波的分类、标架小波乘子、伪标架小波，向量值子空间中超小波与超Gabor标架，并尝试其在信号处理、无线通信等方面的应用。其研究成果将进一步丰富和完善小波与Gabor标架的理论、拓宽标架理论的数学应用，为信号去噪提供理论借鉴和启示，为设计高效的多址通信方式提供一个新的视角。

Noise suppression is crucial for signal processing, in which frames and pseudoframes have important applications. Multiplexing has been being one of key techniques in wireless communications, in which superframes play a significant role. Up to now, the whole space (the Hilbert space of square integrable functions) frame theory has seen its great achievements, reducing subspace frame theory has seen some progresses, and the results on subspace superframe theory have not been reported.The notion of reducing subspace is a generalization of Hardy space and the whole space, and many signals in applications belong to a reducing space. Our study has shown that reducing subspace frame theory and subspace superframe theory are not trivial generalizations of the whole space frame theory. For example, Ron-Shen dual principle fails to hold for subsapce superframes. This project focuses on the classification of Parseval frame wavelets, frame wavelet multipliers, pseudoframe wavelets in the setting of reducing subspaces, vector-valued subspace superwavelet frames and super Gabor frames, and their applications in signal processing and wireless communications. Research for this project will further enrich the theory of wavelet and Gabor frames, broaden the mathematical applications of frame theory, provide a theoretical reference and inspiration for signal denoising, and provide a new perspective for the design of efficient multiplex communication.