Two‐dimensional adaptive Fourier decomposition
Two‐dimensional adaptive Fourier decomposition
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DOI：
10.1002/mma.3649
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发表时间：
2014-06
影响因子：
2.9
通讯作者：
Tao Qian
中科院分区：
文献类型：
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作者：
Tao Qian
一维自适应傅立叶分解（英语：One‐dimensional adaptive Fourier decomposition，缩写为1‐D AFD或AFD）是一种将物理上可实现的信号自适应表示为上下文的参数化Szegö核和高阶Szegö核的线性组合。本文基于多元复哈代空间理论研究了多维AFD。我们继续使用两种方法，一种使用Product‐TM系统;另一种使用Product‐Szegö字典。使用乘积TM系统方法，我们证明了在每一个参数对的选择，可以达到最大能量，因此，我们证明了收敛性。利用乘积-Szegö字典方法，我们证明了纯贪婪算法是适用的。接下来，我们介绍一种新的贪婪算法，称为预正交贪婪算法（P-OGA）。我们证明了它的收敛性和收敛速度估计，也允许弱型版本的P-OGA。最后，通过对P-OGA收敛速度的估计，证明了该算法优于正交贪婪算法（OGA）.在最后一部分中，我们深入分析了P-OGA，并介绍了P-OGA-诱导的完整词典（缩写为完整词典）的概念。证明了P-OGA在完备字典下适用于2-torus上的哈代H2空间.版权所有© 2016约翰威利父子有限公司.
One‐dimensional adaptive Fourier decomposition, abbreviated as 1‐D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher‐order Szegö kernels of the context. In the present paper, we study multi‐dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product‐TM Systems; and the other uses Product‐Szegö Dictionaries. With the Product‐TM Systems approach, we prove that at each selection of a pair of parameters, the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product‐Szegö dictionary approach, we show that pure greedy algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre‐orthogonal Greedy Algorithm (P‐OGA). We prove its convergence and convergence rate estimation, allowing a weak‐type version of P‐OGA as well. The convergence rate estimation of the proposed P‐OGA evidences its advantage over orthogonal greedy algorithm (OGA). In the last part, we analyze P‐OGA in depth and introduce the concept P‐OGA‐Induced Complete Dictionary, abbreviated as Complete Dictionary. We show that with the Complete Dictionary P‐OGA is applicable to the Hardy H2 space on 2‐torus. Copyright © 2016 John Wiley & Sons, Ltd.