本文主要研究不同场合下的预正交自适应Fourier 分解(pre-orthogonaladaptive Fourier decomposition, POAFD) 理论及其应用问题.

首先，在一个Hilbert 空间中通过内积核定义的线性算子对应一个自然的再生核Hilbert 空间结构，称为H-HK 结构. 这个结构本身内蕴一个基方法, 可以用于解答线性算子若干最基本的问题, 包括确定或刻画值域空间、解算子方程以及解Moore-Penrose 伪- (广义-) 逆算子问题. 本文将建立H-HK 结构下的POAFD算法. 在这个方法之下导出上述3 个问题的解的稀疏表示. 在逐次跟踪匹配的优化方法论中, POAFD 的优选原理保证了它在算法上的可行性. 所提供的方法可有效地应用于实际问题, 包括信号与图像重构、常微分方程、偏微分方程和优化问题的数值解等.

其次, 我们知道，POAFD 是一种确定性信号的稀疏表示. 特别是在Hardy空间中，通过对Takenaka-Malmquist 系统参数的自适应选择而建立的自适应分解理论（AFD），是POAFD 的特殊情况. POAFD 作为AFD 的多元推广，是在Hilbert 空间具有再生核的背景下建立的. 目前，它们已经被推广到随机信号的情形中. 这里我们研究两种类型的随机信号. 一种是可表示为带有误差项的确定性信号（如白噪声）；另一种是服从一定分布的几类随机信号的混合. 我们将证明单位圆盘上随机复Hardy 空间中n -最佳Szeg¨o 核逼近的存在性. 它是经典Hardy空间对应结果在随机信号上的推广. 这一结果还可进一步推广到随机多圆盘的情况，在图像处理等问题中具有潜在的应用价值. 此外，我们还给出了相应算法.

再次，我们研究了利用振幅测量值来重构单位圆盘上Hardy 空间函数的问题，也就是相位恢复问题. 采用的方法是，通过Nevanlinna 分解定理将其转化为计算相应的内、外函数. 即采用机械求积法计算Hilbert 变换，得到外函数，再用找最小值的方法求出Blashcke 乘积的零点，从而在假设奇异内函数部分为1 的前提下，给出内函数的表达式. 针对具体算法我们还做了几个说明性实验.

最后，我们提出了一些有潜在应用价值的后续课题，可以在未来的研究中作进一步考虑.

This study mainly concerns the pre-orthogonal adaptive Fourier decomposition(POAFD) theory and its applications in different contexts.

First, a linear operator defined by an inner product kernel in a Hilbert space correspondsto a natural structure of reproducing kernel Hilbert space, called H-HK formulation.The H-HK formulation itself contains a basis method, which can straightforwardlysolve three basic type problems, namely, the image function identification,inverse problem, and Moore-Penrose pseudo-inverse problem. After a summary on theclassical basis method we introduce the POAFD, a “non-basis”method in the H-HKformulation. The maximal selection principle of POAFD makes itself be the best amongall the existing matching pursuit methods in the one-step-optimal-selection category. It,therefore, can be used to give fast converging sparse numerical solutions of approximation,ordinary and partial differential equations, and optimization problems.

Secondly, it’s known that POAFD is a kind of sparse representation of deterministicsignals. Especially in Hardy space, adaptive Fourier decomposition (AFD), whichis a special case of POAFD, is established through adaptive selections of the parametersdefining a Takenaka-Malmquist system in one-complex variable. POAFD whichis the multivariate generalization of AFD, is established with the context Hilbert spacepossessing a unique reproducing kernel. Recently, they have been extended to randomsignals. In this study we work on two types of random signals. One is those expressibleas the sum of a deterministic signal with an error term such as a white noise; andthe other is, in general, as mixture of several classes of random signals obeying certaindistributive law. We first prove the existence of the n-best approximation in terms ofthe parameterized Szeg? kernels in the stochastic complex Hardy space of the unit disc.It is a generalization to random signals of the corresponding result for the Hardy spaceof the disc, and has applications in signal analysis. The result may be generalized to therandomized polydisc case with potential applications in image processing. A practicalalgorithm for the approximation is proposed.

Thirdly, we focus on the study of reconstructing a function f in the Hardy spaceof the unit disc from amplitude measurements. It’s known as the problem of phase retrieval.We transform it into solving the corresponding outer and inner function throughthe Nevanlinna factorization Theorem. The outer function will be established basedon the mechanical quadrature method, while we determine the zero points of Blashckeproduct by finding out the minimum value, thereby computing the inner function underthe assumption that the singular inner function part is trivial. Hereafter the concretealgorithm and illustrative experiments follow.

Last but not least, we present some potential studies that can be further consideredin the future.