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紧支撑Gabor框架的稳定性 总被引:1,自引:0,他引:1
1引言自Duffin和Schaeffer提出了框架理论以来,框架理论在数学和信息科学等领域占有非常重要的地位,在小波分析和时频分析中起着举足轻重的作用.现在Gabor框架系统被广泛地应用在信息论,量子力学,信号处理和图象处理等方面.Gabor框架的稳定性是应用中所需要的,同时也是人们关心的问题.但是,一直以来对Gabor框架的稳定性的研究主要集中在L~2(R)上,即使在L~2(R~d)上的研究也主要针对单项指标扰动进行了研究.本文借助于文[2]中的定理3和定理4,文[3]中的定理2和文[5]中的定理2.1,分别对L~2(R~d)上紧支撑Gabor框架的窗函数、平移指标、旋转指标以及多项混合扰动的稳定性进行了讨论. 相似文献
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主要讨论局部域上的Gabor紧框架.首先,建立局部域上Gabor系{xm(bx)g(x-u(n)a)}m.n∈p构成L~2(K)上紧框架的特征.其次,给出Gabor系{X_m(bx)g(x-u(n)a)}_(m,n∈p)成为L~2(K)上标准正交基的充要条件. 相似文献
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《数学的实践与认识》2016,(23)
仿射框架在信号处理中有实用性.运用算子理论与时频分析,将两个二重贝塞尔序列扩充为一对对偶二重仿射框架.再由已知的一对多重贝塞尔序列添加若干个函数使它们扩充为一对对偶多重仿射框架,得到了多重Gabor框架的特征不等式. 相似文献
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Gabor理论中的对偶原理(例如Ron-Shen对偶原理和Wexler-Raz双正交关系)在研究Gabor系统时起到了至关重要的作用. 对Banach空间中的任意序列, 该文定义了仅依赖两组 p-Riesz基的一个相关的序列(Riesz -对偶序列), 研究它与前一组序列相关的性质. 推广了P. G. Gasazza、G. Kutyniok和M. C. Lammers在可分Hilbert空间中框架的对偶原理的一些结果. 相似文献
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本文引入了Hilbert C*-模上的Gabor酉系统的概念,研究了它的基本性质,证明了膨胀定理,并给出某个酉系统的两个正规框架向量不相交的等价条件. 相似文献
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In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.
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IvanaCarrizo SergioFavier 《分析论及其应用》2003,19(3):238-254
In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically,we present the results obtained on frame stability when one disturbs the mother of wavelet frame, or the parameter of dilatation, and in Gabor frames when the generating function or the parameter of translation are perturbed. In all cases we work without demanding compactness of the support, neither on the generating function, nor on its Fourier transform. 相似文献
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Christopher Heil 《Journal of Fourier Analysis and Applications》2007,13(2):113-166
The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey
attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from
the one-dimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and
most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis
are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the
Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included. 相似文献
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Radu Balan Peter G. Casazza Christopher Heil Zeph Landau 《Journal of Fourier Analysis and Applications》2006,12(3):307-344
This work develops a quantitative framework for describing the overcompleteness of a large class of frames. A previous article
introduced notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via
a map a : I → G. This article shows that those abstract results yield an array of new implications for irregular Gabor frames.
Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their
meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the
relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally,
these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of
concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum
of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this
article, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with
most implications shown to be sharp. 相似文献
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Karlheinz Gröchenig 《Constructive Approximation》1993,9(2-3):283-297
We obtain irregular sampling theorems for the wavelet transform and the short-time Fourier transform. These sampling theorems yield irregular weighted frames for wavelets and Gabor functions with explicit estimates for the frame bounds. 相似文献
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Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating
linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators
that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal
Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation
by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations.
Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property
of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s
best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized
Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows. 相似文献
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Jean-Pierre Gabardo 《Journal of Functional Analysis》2009,256(3):635-672
We give a characterization for the weighted irregular Gabor tight frames or dual systems in L2(Rn) in terms of the distributional symplectic Fourier transform of a positive Borel measure on R2n naturally associated with the system and the short-time Fourier transform of the windows in the case where the window (or at least one of the windows in the case of dual systems) belongs to S(Rn). This result implies that, for certain classes of windows such as generalized Gaussians or “extreme-value” windows, the only weighted irregular Gabor tight frames (or even dual systems with both windows in the same class) that can be constructed with these windows are the trivial ones, corresponding to the measure μ=1 on R2n. Furthermore, we show that, if a such Gabor system admits a dual which is of Gabor type, then the Beurling density of the associated measure exists and is equal to one. 相似文献
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From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results. 相似文献
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This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels. 相似文献