共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
Fundamental domains of pairs of lattices were used by Han and Wang to construct multivariate Gabor frames for separable lattices.
We build upon their results to obtain Gabor frames with smooth and compactly supported window functions. Our results are applicable,
for example, if certain pairs of lattices with equal density allow for a common compact and star-shaped fundamental domain. 相似文献
3.
We consider the construction of tight Gabor frames (h, a=1,
b=1) from Gabor systems (g, a=1, b=1) with g a window
having few zeros in the Zak transform domain via the operation
h=Z
-1(Zg/|Zg|), where Z is the standard Zak transform. We
consider this operation with g the Gaussian, the hyperbolic
secant, and for g belonging to a class of positive, even,
unimodal, rapidly decaying windows of which the two-sided
exponential is a typical example. All these windows g have the
property that Zg has a single zero, viz. at
(1/2,\1/2), in the unit square [0,1)2. The Gaussian
and hyperbolic secant yield a frame for any a,b > 0 with ab < 1,
and we show that so does the two-sided exponential. For these
three windows it holds that S
a
-1/2
g
h as a
1,
where S
a
is the frame operator corresponding to the Gabor frame
(g,a,a). It turns out that the hs corresponding to gs of
the above type look and behave quite similarly when scaling
parameters are set appropriately. We give a particular detailed
analysis of the h corresponding to the two-sided exponential. We
give several representations of this h, and we show that
, and is continuous and
differentiable everywhere except at the half-integers, etc., and
we pay particular attention to the cases that the time constant of
the two-sided exponential g tends to
. We also consider
the cases that the time constants of the Gaussian and of the
hyperbolic secant tend to 0 or to
. It so turns out that
h thus obtained changes from the box function
into its Fourier transform
when the time constant changes from 0 to
相似文献
4.
《数学的实践与认识》2017,(21)
引入多个加细函数生成的最小能量多重仿射框架的概念.运用矩阵理论和时频分析方法,给出对应于最小能量多重仿射框架的多重加细函数所满足的条件.得到最小能量多重仿射紧框架的特征与构造方法.对原多重加细函数和最小能量多小波仿射框架做正交变换,得到新的多重加细函数和对应的最小能量多重仿射框架. 相似文献
5.
Jean-Pierre Gabardo 《Acta Appl Math》2009,107(1-3):49-73
We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L 2(?) when the translations and modulations of the window are associated with certain non-separable lattices in ?2 which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ?2n with the property that the short-time Fourier transform defines an isometric embedding from L 2(? n ) to L μ 2 (?2n ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point. 相似文献
6.
A.J.E.M. Janssen Peter L. Søndergaard 《Journal of Fourier Analysis and Applications》2007,13(2):211-241
In this article we investigate the computational aspects of some recently proposed iterative methods for approximating the
canonical tight and canonical dual window of a Gabor frame (g, a, b). The iterations start with the window g while the iteration
steps comprise the window g, the k-th iterand γk, the frame operators S and Sk corresponding to (g, a, b) and (γk, a, b), respectively, and a number of scalars. The structure of the iteration step of the method is determined by the envisaged
convergence order m of the method. We consider two strategies for scaling the terms in the iteration step: Norm scaling, where
in each step the windows are normalized, and initial scaling where we only scale in the very beginning. Norm scaling leads
to fast, but conditionally convergent methods, while initial scaling leads to unconditionally convergent methods, but with
possibly suboptimal convergence constants. The iterations, initially formulated for time-continuous Gabor systems, are considered
and tested in a discrete setting in which one passes to the appropriately sampled-and-periodized windows and frame operators.
Furthermore, they are compared with respect to accuracy and efficiency with other methods to approximate canonical windows
associated with Gabor frames. 相似文献
7.
Mathematical Notes - Tangent-valued forms, tangent and cotangent vectors of the first and the second order are considered. For an affine connection, second-order tangent-valued (vertical and... 相似文献
8.
Given g∈L2(R n ), we consider irregular wavelet for the form\(\left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j \) > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L2(R n ) are given. For a class of functions g∈L22(R n ) we prove that certain growth conditions on {λ j } will frames, and that some other types of sequences exclude the frame property. We also give a sufficient condition for a Gabor system\(\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} \)to be a frame. 相似文献
9.
Given g { l\fracn2 g( lj x - kb ) }jezjezn ,where lj \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L
2(R
n
) are given. For a class of functions g{ ezrib( j,x ) g( x - lk ) }jezn ,kez\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame. 相似文献
10.
Journal of Fourier Analysis and Applications - We generalize three main concepts of Gabor analysis for lattices to the setting of model sets: fundamental identity of Gabor analysis, Janssen’s... 相似文献
11.
Ole Christensen Baiqiao Deng Christopher Heil 《Applied and Computational Harmonic Analysis》1999,7(3):292
A Gabor system is a set of time-frequency shifts S(g, Λ) ={e2 π ibxg(x − a)}(a, b) Λ of a function g L2(Rd). We prove that if a finite union of Gabor systems k = 1rS(gk, Λk) forms a frame for L2(Rd) then the lower and upper Beurling densities of Λ = k = 1r Λk satisfy D−(Λ) ≥ 1 and D + (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection k = 1r{gk(x − a)}a Γk of pure translates can form a frame for L2(Rd). 相似文献
12.
In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic. 相似文献
13.
Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames for improving the orientation selectivity of tensor product real-valued wavelet bases/frames. In this paper, we studied the general theory of discrete Gabor frames for finite signals, and constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation. Besides high orientation selectivity, the proposed multi-scale discrete 2D Gabor frames also allow us to simultaneously exploit sparsity prior of cartoon image regions in spatial domain and the sparsity prior of textural image regions in local frequency domain. Using a composite sparse image model, we showed the advantages of the proposed discrete Gabor frames over the existing wavelet frames in several image recovery experiments. 相似文献
14.
15.
Karlheinz Gröchenig 《Journal of Fourier Analysis and Applications》2014,20(4):865-895
This is a survey about the theory of Gabor frames. We review the structural results about Gabor frames over a lattice and then discuss the few known results about the fine structure of Gabor frames. We add a new result about the relation between properties of the window and properties of the frame set and conclude with a vision of how a more complete theory of the fine structure might look like. 相似文献
16.
The aim of this paper is to investigate the symmetry properties of tight frames,
with a view to constructing tight frames of orthogonal polynomials
in several variables which share the symmetries of the weight function,
and other similar applications.
This is achieved by using representation theory to
give methods for constructing tight frames as orbits of groups of unitary
transformations acting on a given finite-dimensional Hilbert space. Along the way, we show
that a tight frame is determined by its Gram matrix and discuss how the symmetries of a
tight frame are related to its Gram matrix. We also give a complete classification of
those tight frames which arise as orbits of an abelian group of symmetries. 相似文献
17.
Frames have become standard tools in signal processing due to their robustness against transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they do exist. However, it is known that an ETF of d + 1 vectors in a d dimensional space always exists. This article gives an explicit construction of ETFs of d + 1 vectors in a d dimensional space. This construction works for both real and complex cases and is simpler than existing methods. The absence of ETFs of arbitrary sizes in a given space leads to generalizations of ETFs. One way to do this to consider tight frames where the set of (acute) angles between pairs of vectors has k distinct values. This article presents a construction of tight frames such that for a given value of k, the angles between pairs of vectors take at most k distinct values. These tight frames can be related to regular graphs and association schemes. 相似文献
18.
19.
20.
Jakob Lemvig Kamilla Haahr Nielsen 《Journal of Fourier Analysis and Applications》2016,22(6):1440-1451
The frame set conjecture for B-splines \(B_n\), \(n \ge 2\), states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form \(ab=r\), where r is a rational number smaller than one and a and b denote the sampling and modulation rates, respectively, has infinitely many pieces, located around \(b=2,3,\dots \), not belonging to the frame set of the nth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines \(B_n\), \(n \ge 2\). 相似文献