共查询到20条相似文献,搜索用时 31 毫秒
1.
M.S. Asgari 《Journal of Mathematical Analysis and Applications》2005,308(2):541-553
In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {Wi}i∈I for a Hilbert space H, there exists a Hilbert space K⊇H and an orthonormal basis of subspaces {Ni}i∈I for K such that Wi=P(Ni), where P is the orthogonal projection of K onto H. We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula. 相似文献
2.
In this article, we study tensor product of Hilbert C*-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F, and we get more results.
For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K. 相似文献
3.
ON THE STABILITY OF FUSION FRAMES (FRAMES OF SUBSPACES) 总被引:1,自引:0,他引:1
Mohammad Sadegh Asgari 《数学物理学报(B辑英文版)》2011,31(4):1633-1642
A frame is an orthonormal basis-like collection of vectors in a Hilbert space, but need not be a basis or orthonormal. A fusion frame (frame of subspaces) is a frame-like collection of subspaces in a Hilbert space, thereby constructing a frame for the whole space by joining sequences of frames for subspaces. Moreover the notion of fusion frames provide a framework for applications and providing efficient and robust information processing algorithms.In this paper we study the conditions under which removing an element from a fusion frame, again we obtain another fusion frame. We give another proof of [5, Corollary 3.3(iii)] with extra information about the bounds. 相似文献
4.
Fusion frames and distributed processing 总被引:2,自引:0,他引:2
Peter G. Casazza Gitta Kutyniok Shidong Li 《Applied and Computational Harmonic Analysis》2008,25(1):114-132
Let {Wi}iI be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight vi, and let be the closed linear span of the Wis, a composite Hilbert space. {(Wi,vi)}iI is called a fusion frame provided it satisfies a certain property which controls the weighted overlaps of the subspaces. These systems contain conventional frames as a special case, however they reach far “beyond frame theory.” In case each subspace Wi is equipped with a spanning frame system {fij}jJi, we refer to {(Wi,vi,{fij}jJi)}iI as a fusion frame system. The focus of this article is on computational issues of fusion frame reconstructions, unique properties of fusion frames important for applications with particular focus on those superior to conventional frames, and on centralized reconstruction versus distributed reconstructions and their numerical differences. The weighted and distributed processing technique described in this article is not only a natural fit to distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. Another important component of this article is an extensive study of the robustness of fusion frame systems. 相似文献
5.
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
6.
<Emphasis Type="Italic">K</Emphasis>-fusion Frames and the Corresponding Generators for Unitary Systems
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Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces. By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator. 相似文献
7.
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
8.
Horst Herrlich 《Topology and its Applications》2011,158(17):2279-2286
Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:
- (1)
- For every family(Ai)i∈Iof sets there exists a family(Ti)i∈Isuch that for everyi∈I(Ai,Ti)is a compactT2space.
- (2)
- For every family(Ai)i∈Iof sets there exists a family(Ti)i∈Isuch that for everyi∈I(Ai,Ti)is a compact, scattered, T2space.
- (3)
- For every set X, every compactR1topology (itsT0-reflection isT2) on X can be enlarged to a compactT2topology.
- (a)
- (1) implies every infinite set can be split into two infinite sets.
- (b)
- (2) iff AC.
- (c)
- (3) and “there exists a free ultrafilter” iff AC.
9.
Alice Z.-Y. Chan Martin S. Copenhaver Sivaram K. Narayan Logan Stokols Allison Theobold 《Advances in Computational Mathematics》2016,42(3):721-756
A frame in an n-dimensional Hilbert space H n is a possibly redundant collection of vectors {f i } i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {f i } i∈I is said to be scalable if there exist nonnegative scalars {c i } i∈I such that {c i f i } i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or scalable subsets by using partially-ordered sets (posets). We define the factor poset of a frame {f i } i∈I to be a collection of subsets of I ordered by inclusion so that nonempty J?I is in the factor poset iff {f j } j∈J is a tight frame for H n . We study various properties of factor posets and address the inverse factor poset problem, which inquires when there exists a frame whose factor poset is some given poset P. We then turn our attention to scalable frames and present partial results regarding when a frame can be scaled to have a given factor poset; in doing so we present a bridge between erasure resilience (as studied via prime tight frames) and scalability. 相似文献
10.
Sums of Hilbert space frames 总被引:1,自引:0,他引:1
Sofian Obeidat Peter G. Casazza Janet C. Tremain 《Journal of Mathematical Analysis and Applications》2009,351(2):579-585
We give simple necessary and sufficient conditions on Bessel sequences {fi} and {gi} and operators L1, L2 on a Hilbert space H so that {L1fi+L2gi} is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames. 相似文献
11.
Let G be any graph and let {H i } i??I be a family of graphs such that $E\left( {H_i } \right) \cap E\left( {H_j } \right) = \not 0$ when i ?? j, ?? i??I E(H i ) = E(G) and $E\left( {H_i } \right) \ne \not 0$ for all i ?? I. In this paper we introduce the concept of {H i } i??I -super edge-magic decomposable graphs and {H i } i??I -super edge-magic labelings. We say that G is {H i } i??I -super edge-magic decomposable if there is a bijection ??: V(G) ?? {1,2,..., |V(G)|} such that for each i ?? I the subgraph H i meets the following two requirements: ??(V(H i )) = {1,2,..., |V(H i )|} and {??(a) +??(b): ab ?? E(H i )} is a set of consecutive integers. Such function ?? is called an {H i } i??I -super edge-magic labeling of G. We characterize the set of cycles C n which are {H 1, H 2}-super edge-magic decomposable when both, H 1 and H 2 are isomorphic to (n/2)K 2. New lines of research are also suggested. 相似文献
12.
13.
When A∈B(H) and B∈B(K) are given, we denote by MC the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form In this paper, for given A and B, the sets and ?C∈Inv(K,H)σl(MC) are determined, where σl(T),Bl(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H. 相似文献
14.
LetE be the Grassmann (or exterior) algebra of an infinite-dimensional vector space over a fieldK of characteristic 0. We compute the Hilbert series of the relatively free algebraF m(varE?K E)=K(x l,...,x m) /(T(E?K E)∩K〈x l,...,x m〉) of the variety of algebrasvarE?KE, whereT(E?K E) is the set of all polynomial identities forE?K EE from the free associative algebraK〈x 1,x2…〉. 相似文献
15.
In this paper, we study the relationship between frames for the super Hilbert space H⊕H and g-frames for H with respect to C2. We show that a g-frame associated with a frame for H⊕H remains a g-frame whenever any one of its elements is removed. Furthermore, we show that the excess of such a g-frame is at least dimH. 相似文献
16.
J.I. Giribet A. Maestripieri F. Martínez Pería 《Journal of Mathematical Analysis and Applications》2010,369(1):423-436
We present generalizations to Krein spaces of the abstract interpolation and smoothing problems proposed by Atteia in Hilbert spaces: given a Krein space K and Hilbert spaces H and E (bounded) surjective operators T:H→K and V:H→E, ρ>0 and a fixed z0∈E, we study the existence of solutions of the problems and . 相似文献
17.
Given a unimodal map f, let I=[c2,c1] denote the core and set E={(x0,x1,…)∈(I,f)|xi∈ω(c,f) for all i∈N}. It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of (I,f) is a proper subset of E and such that limk→∞Q(k)≠∞, where Q(k) is the kneading map.We use the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of (I,f) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of (I,f) is precisely E. Examples of this behavior are provided where limk→∞Q(k) does and does not equal infinity, and in the case where limk→∞Q(k)=∞, the collection of endpoints of (I,f) is always E. 相似文献
18.
Phil Howlett Konstantin Avrachenkov Charles Pearce Vladimir Ejov 《Journal of Mathematical Analysis and Applications》2009,353(1):68-2912
Let H and K be Hilbert spaces and for each z∈C let A(z)∈L(H,K) be a bounded but not necessarily compact linear map with A(z) analytic on a region |z|<a. If A(0) is singular we find conditions under which A−1(z) is well defined on some region 0<|z|<b by a convergent Laurent series with a finite order pole at the origin. We show that by changing to a standard Sobolev topology the method extends to closed unbounded linear operators and also that it can be used in Banach spaces where complementation of certain closed subspaces is possible. Our method is illustrated with several key examples.2 相似文献
19.
For a global field K and an elliptic curve Eη over K(T), Silverman's specialization theorem implies rank(Eη(K(T)))?rank(Et(K)) for all but finitely many t∈P1(K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family Ec,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ×) such that, under the parity conjecture, each Ec,d has elevated rank. 相似文献
20.
Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar, E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5-6) (2007) 479-484] it is proved that if S has a non-zero finite codimension in its completion, then Eq(S)≠E(S). In the present paper it is shown that if S has a total orthonormal system, then Eq(S)=E(S) implies completeness of S. In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a pre-Hilbert space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system. 相似文献