共查询到20条相似文献,搜索用时 812 毫秒
1.
Akram Aldroubi David Larson Wai-Shing Tang Eric Weber 《Transactions of the American Mathematical Society》2004,356(12):4767-4786
We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.
2.
A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those
Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using
the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames.
We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction
operator contains no copy of c
0. In particular, a Schauder frame of a Banach space with no copy of c
0 is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c
0. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the
kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames
and near-Riesz bases. 相似文献
3.
Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Given a spectrum for a desired fusion frame operator and dimensions for subspaces, one existing method for creating unit-weight fusion frames with these properties is the flexible and elementary procedure known as spectral tetris. Despite the extensive literature on fusion frames, until now there has been no construction of fusion frames with prescribed weights. In this paper we use spectral tetris to construct more general, arbitrarily weighted fusion frames. Moreover, we provide necessary and sufficient conditions for when a desired fusion frame can be constructed via spectral tetris. 相似文献
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5.
本文研究了可分的Hilbert空间H中带符号广义框架,利用算子理论方法,给出了H中一族向量{hm}m∈M是一个带符号广义框架当且仅当带符号广义框架的框架算子的正部S 和负部S-是有界线性算子,讨论了H中带符号广义框架的框架算子S的可逆性,并且得到了H中每个向量f关于带符号广义框架{hm}m∈M和其对偶带符号广义框架{~hm}m∈M的表示式. 相似文献
6.
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
7.
Fusion-Riesz frame (Riesz frame of subspace) whose all subsets are fusion frame sequences with the same bounds is a special fusion frame. It is also considered a generalization of Riesz frame since it shares some important properties of Riesz frame. In this paper, we show a part of these properties of fusion-Riesz frame and the new results about the stabilities of fusion-Riesz frames under operator perturbation (simple named operator perturbation of fusion-Riesz frames). Moreover, we also compare the operator perturbation of fusion-Riesz frame with that of fusion frame, fusion-Riesz basis (also called Riesz decomposition or Riesz fusion basis) and exact fusion frame. 相似文献
8.
Peter G. Casazza 《Journal of Fourier Analysis and Applications》1998,4(6):727-732
We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. We
next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination
of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple
of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis for H. Finally, every
frame can be written as a (multiple of a) average of two orthonormal bases for a larger Hilbert space.
Acknowledgements and Notes. This research was supported by NSF DMS 9701234. Part of this research was conducted while the author was a visitor at the
“Workshop on Linear Analysis and Probability”, Texas A&M University. 相似文献
9.
本文利用引入的KS性质,刻划了那些其Scott拓扑可由开滤子生成的分配备格,该结果也是对[1]中一公开问题的一种解答.本文的刻划定理对于判定分配备格的Scott拓扑是否与Scott开滤子拓扑一致具有较强的可操作性,应用该刻划定理给出大量非连续格,其Scott拓扑具有开滤子基. 相似文献
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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
14.
Yoo Young Koo 《Linear and Multilinear Algebra》2013,61(7):856-870
Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In particular, we find a necessary and sufficient condition for the sum of two strongly disjoint (orthogonal) frame sequences (in the same Hilbert space) to be a frame sequence, and thereby show that the sum of two strongly disjoint frame sequences may not be a frame sequence. We also show that the closedness of the sum of the synthesis operators of two frame sequences and that of the sum of the frame operators of the same frame sequences are not related. Other observations are also included. 相似文献
15.
Robert Calderbank Peter G. Casazza Andreas Heinecke Gitta Kutyniok Ali Pezeshki 《Advances in Computational Mathematics》2011,35(1):1-31
Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data
processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame
for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal
into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications,
and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this
paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a ‘uniform basis’
over all subspaces, thereby enabling low-complexity fusion frame decompositions. We study the existence and construction of
sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice
to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has
a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization
of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then
introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method
and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion
frame. We proceed by establishing existence conditions for 2-sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable
algorithm for computing such 2-sparse fusion frames. 相似文献
16.
Johan Schubert 《International Journal of Approximate Reasoning》2012,53(2):176-189
We construct alternative frames of discernment from input belief functions. We assume that the core of each belief function is a subset of a so far unconstructed frame of discernment. The alternative frames are constructed as different cross products of unions of different cores. With the frames constructed the belief functions are combined for each alternative frame. The appropriateness of each frame is evaluated in two ways: (i) we measure the aggregated uncertainty (an entropy measure) of the combined belief functions for that frame to find if the belief functions are interacting in interesting ways, (ii) we measure the conflict in Dempster’s rule when combining the belief functions to make sure they do not exhibit too much internal conflict. A small frame typically yields a small aggregated uncertainty but a large conflict, and vice versa. The most appropriate frame of discernment is that which minimizes a probabilistic sum of the conflict and a normalized aggregated uncertainty of all combined belief functions for that frame of discernment. 相似文献
17.
In this paper, the sum of standard generalized flames of Hilbert W^*-module is studied intensively by using operator-theoretic-methods, and some conditions are given to assure that the sum of two or more standard generalized frames is a standard generalized frame. 相似文献
18.
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
19.
A. A. Zakharova 《Mathematical Notes》2008,83(1-2):190-200
In this paper, we introduce the notion of generalized frames and study their properties. Discrete and integral frames are special cases of generalized frames. We give criteria for generalized frames to be integral (discrete). We prove that any bounded operator A with a bounded inverse acting from a separable space H to L 2(Ω) (where Ω is a space with countably additive measure) can be regarded as an operator assigning to each element x ∈ H its coefficients in some generalized frame. 相似文献
20.
g-Besselian frames in Hilbert spaces 总被引:1,自引:0,他引:1
In this paper, we introduce the concept of a g-Besselian frame in a Hilbert space and discuss the relations between a g-Besselian frame and a Besselian frame. We also give some characterizations of g-Besselian frames. In the end of this paper, we discuss the stability of g-Besselian frames. Our results show that the relations and the characterizations between a g-Besselian frame and a Besselian frame are different from the corresponding results of g-frames and frames. 相似文献