共查询到20条相似文献,搜索用时 390 毫秒
1.
A Weyl-Heisenberg frame (WH frame) for L2(ℝ) allows every square integrable function on the line to be decomposed into the infinite sum of linear combination of translated
and modulated versions of a fixed function. Some sufficient conditions for g ∈ L2(ℝ) to be a subspace Weyl-Heisenberg frame were given in a recent work [3] by Casazza and Christensen. Obviously every invariant
subspace (under translation and modulation) is cyclic if it has a subspace WH frame. In the present article we prove that
the cyclicity property is also sufficient for a subspace to admit a WH frame. We also investigate the dilation property for
subspace Weyl-Heisenberg frames and show that every normalized tight subspace WH frame can be dilated to a normalized tight
WH frame which is “maximal” In other words, every subspace WH frame is the compression of a WH frame which cannot be dilated
anymore within the L2(ℝ) space.
Communicated by Hans G. Feichtinger 相似文献
2.
Jerry Lopez 《Linear algebra and its applications》2010,432(1):471-1472
For any given frame (for the purpose of encoding) in a finite dimensional Hilbert space, we investigate its dual frames that are optimal for erasures (for the purpose of decoding). We show that in general the canonical dual is not necessarily optimal. Moreover, optimal dual frames are not necessarily unique. We present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem. As an application, we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight. 相似文献
3.
Richard Vale 《Linear algebra and its applications》2010,433(1):248-94
We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of Vale and Waldron (2005) [12] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames. 相似文献
4.
Gabor frames, unimodularity, and window decay 总被引:4,自引:0,他引:4
We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with
particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties
of g are inherited by its minimal dual γ0 and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying
(i.e., decay like |g(t)|≤Ce−α|t|
1/α for some 1/2≤α<1) window functions. Particularly, we find that g and γ0 have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e.,
its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent
to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically
associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent
the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi
representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on
results from the theory of entire functions, and on the theory of polynomial matrices. 相似文献
5.
Ole Christensen 《Journal of Fourier Analysis and Applications》2000,6(1):79-91
A frame in a Hilbert space
allows every element in
to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculations of
those coefficients and many other situations where frames occur, requires knowledge of the inverse frame operator. But usually
it is hard to invert the frame operator if the underlying Hilbert space is infinite dimensional. In the present paper we introduce
a method for approximation of the inverse frame operator using finite subsets of the frame. In particular this allows to approximate
the frame coefficients (even inl
2) using finite-dimensional linear algebra. We show that the general method simplifies in the important cases of Weil-Heisenberg
frames and wavelet frames. 相似文献
6.
Shannon Bishop Christopher Heil Yoo Young Koo Jae Kun Lim 《Linear algebra and its applications》2010,432(6):1501-1514
This paper determines the exact relationships that hold among the major Paley-Wiener perturbation theorems for frame sequences. It is shown that major properties of a frame sequence such as excess, deficit, and rank remain invariant under Paley-Wiener perturbations, but need not be preserved by compact perturbations. For localized frames, which are frames with additional structure, it is shown that the frame measure function is also preserved by Paley-Wiener perturbations. 相似文献
7.
We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in \(L^2(\mathbb {R}^k)\). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators. 相似文献
8.
We consider the notion of uncertainty for finite frames. Using a difference operator inspired by the Gauss-Hermite differential equation we obtain a time-frequency measure for finite frames. We then find the minimizer of the measure over all equal norm Parseval frames, dependent on the dimension of the space and the number of elements in the frame. Next we show that given a frame one can find the dual frame that minimizes this time-frequency measure, generalizing some work of Daubechies, Landau and Landau to the finite case and extending some recent work on Sobolev duals for finite frames. 相似文献
9.
In this paper, we study the feasibility and stability of recovering signals in finite-dimensional spaces from unordered partial frame coefficients. We prove that with an almost self-located robust frame, any signal except from a Lebesgue measure zero subset can be recovered from its unordered partial frame coefficients. However, the recovery is not necessarily stable with almost self-located robust frames. We propose a new class of frames, namely self-located robust frames, that ensures stable recovery for any input signal with unordered partial frame coefficients. In particular, the recovery is exact whenever the received unordered partial frame coefficients are noise-free. We also present some characterizations and constructions for (almost) self-located robust frames. Based on these characterizations and construction algorithms, we prove that any randomly generated frame is almost surely self-located robust. Moreover, frames generated with cube roots of different prime numbers are also self-located robust. 相似文献
10.
Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. In this paper we show a surprising result about the inverse of such operators, if any, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in ?2. We also introduce a class of frames (called pseudo-coherent frames), which includes Gabor frames and coherent frames, and investigate invertible pseudo-coherent frame multipliers, allowing a classification for frame-type operators for these frames. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case. 相似文献
11.
A dimension invariance property for finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation of duals in a very large dimensional space is now easily accessible by merely working in a space of some much smaller dimension. A number of uniform and non-uniform schemes are studied. To satisfy the support conditions, a method of finding valid alternate dual functions with small support via a known parametric dual frame formula is discussed. Oftentimes it is convenient to have truncated approximate duals that satisfy the support conditions. Stability studies of the dimension invariance principle via such approximate duals are also presented. 相似文献
12.
We construct non-tight frames in finite-dimensional spaces consisting of periodic functions. In order for these frames to
be useful in practice one needs to calculate a dual frame; while the canonical dual frame might be cumbersome to work with,
the setup presented here enables us to obtain explicit constructions of some particularly convenient oblique duals. We also
provide explicit oblique duals belonging to prescribed spaces different from the space where we obtain the expansion. In particular
this leads to oblique duals that are trigonometric polynomials. 相似文献
13.
Vincenza Del Prete 《Journal of Fourier Analysis and Applications》1999,5(6):545-562
We present a simple proof of Ron and Shen's frame bounds estimates for Gabor frames. The proof is based on the Heil and Walnut's representation of the frame operator and shows that it can be decomposed into a continuous family of infinite matrices. The estimates then follow from a simple application of Gershgorin's theorem to each matrix. Next, we show that, if the window function has exponential decay, also the dual function has some exponential decay. Then, we describe a numerical method to compute the dual function and give an estimate of the error. Finally, we consider the spline of order 2; we investigate numerically the region of the time-frequency plane where it generates a frame and we compute the dual function for some values of the parameters. 相似文献
14.
15.
The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example. 相似文献
16.
Demetrio Labate 《Journal of Geometric Analysis》2002,12(3):469-491
This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame
generators for shift-invariant subspaces of L2(ℝn). A number of applications of this general result are then obtained, among which are the characterization of tight frames
and dual frames for Gabor and wavelet systems. 相似文献
17.
We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity. We also exhibit a relationship between these equiangular tight frames, complex Seidel matrices, and highly regular, directed graphs. We construct examples of such frames with arbitrarily many vectors. 相似文献
18.
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
19.
In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space‐scale‐shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space‐scale‐shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space‐scale‐shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo‐spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results. 相似文献
20.
We show that a compactly supported tight framelet comes from an MRA if the intersection of all dyadic dilations of the space of negative dilates, which is defined as the shift-invariant space generated by the negative scales of a framelet, is trivial. We also construct examples of (non-tight) framelets, which are arbitrarily close to tight frame framelets, such that the corresponding space of negative dilates is equal to the entire space L 2ℝ.The first author was partially supported by the NSF grant DMS–0441817 相似文献