解析再生核希尔伯特空间的自适应傅里叶分解理论及其相关应用

In the recent ten years, the so-called adaptive Fourier decomposition has been developed, which provides an adaptive function approximation based on the Takenaka-Malquist system in the Hardy space, and it has significance both in theory and practice. The research topics in this project contain several important but unsolved problems in the current development of the theory of adaptive Fourier decomposition, and we will solve them in this project. More specifically, we will continue to investigate the theory of adaptive Fourier decomposition in reproducing kernel Hilbert spaces based on the methods in complex and harmonic analysis: 1. establish the adaptive Fourier decomposition of Hardy and Bergman spaces in various domains in the several complex variables setting; 2. prove the Paley-Wiener type theorems corresponding to Clifford-Fourier transform, and then establish the adaptive Fourier decomposition in the related reproducing kernel Hilbert spaces in the Clifford algebra setting. In practice, we will consider the the algorithm of adaptive Fourier decomposition in the cases on unbounded domains for the purpose of developing the related algorithm in general reproducing kernel Hilbert spaces (e.g. Paley-Wiener space, Hardy space and Bergman space).

自适应Fourier分解理论是最近十年发展的一种基于自适应Takenaka-Malquist系统的函数逼近，它同时具有理论和应用意义。本项目的研究课题将解决的都是当前自适应Fourier分解理论在理论和应用发展中重要但一直未被解决的问题。更具体地，我们将利用复分析和调和分析的方法进一步发展自适应Fourier分解理论：1. 建立多复变量设定下不同区域（如管域，拟凸域等）Hardy空间和Bergman空间的自适应Fourier分解；2. 在Clifford代数设定下给出关于Clifford-Fourier变换的Paley-Wiener型定理，从而建立相关的再生核Hilbert空间的自适应Fourier分解。在应用上，我们将考虑在无界区域的自适应Fourier分解算法，从而在更多的再生核Hilbert空间（如Paley-Wiener空间，Hardy空间和Bergman空间等）实现相关算法。

自适应Fourier分解是近十多年来发展起来的一种新的函数分解方法，其同时具有理论与应用意义。本项目的目的是将其推广到多复变分析与Clifford分析中的解析再生核希尔伯特空间中，此外，在算法上本项目给出了一种可实现的上半平面的Hardy空间的自适应Fourier分解算法。