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首页> 外文期刊>Mathematics of computation >DUAL GRAMIAN ANALYSIS: DUALITY PRINCIPLE AND UNITARY EXTENSION PRINCIPLE
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DUAL GRAMIAN ANALYSIS: DUALITY PRINCIPLE AND UNITARY EXTENSION PRINCIPLE

机译：双克分析：二元原理和统一扩展原理

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摘要

Dual Gramian analysis is one of the fundamental tools developed in a series of papers by Amos Ron and Zouwei Shen for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shift-invariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames and the unitary extension principle for wavelet frames. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in a paper by Ron and Shen, e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.
机译：双克里姆人分析是Amos Ron和Zouwei Shen用于学习框架的一系列论文中开发的基本工具之一。使用双克百万分析，帧运算符可以表示为由（广义）换档不变系统的发电机的傅立叶变换组成的矩阵系列，这允许我们在其发电机方面表征帧和紧帧的大多数属性。这种表征应用于上述纸张到两个广泛使用的帧系统，即Gabor和小波框架系统。在许多结果中，我们在此提到了对Gabor帧的二元原理的发现以及小波框架的单一扩展原理。本文旨在建立一般的Hilbert空间中的帧的双克拉姆分析，并随后使用由其元素产生的双克克矩阵的给定系统的帧质属性。因此，对于希尔伯特空间中的帧，例如，可以获得许多有趣的结果，例如，在帧元素和二元原理方面的帧界限的估计。此外，这种新的表征在罗恩和沉的纸上提供了新的见解，例如，罗恩和沉，例如，统一扩展原理与二元原理之间的联系在弱道中。这种连接的一个应用是简化来自给定可再原体掩模的多变量紧密小波框架的构造。与需要从给定行完成具有三角多项式条目的单一矩阵的现有方法相反，我们的方法通过将其转换为恒定的矩阵完成问题来大大简化了紧密的小波框架结构。为了说明其简单性，所提出的施工方案用于构造来自具有某些所需特性的箱花键的多变量紧密小波框架的少数示例，例如紧凑的支撑，对称或抗对称。

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