In mathematics, a fusion frame of a vector space is a natural extension of a frame. Modules dongweili,jinsong leng andtingzhuhuang abstract. New characterizations of fusion bases and riesz fusion. Introduction the concept of hilbert space frames was. It is an additive construct of several, potentially overlapping frames. Upon improving and extending the concept of redundancy of frames, we introduce the notion of redundancy of fusion frames, which is concerned with the properties of lower and upper redundancies.
Construction of fusion frame systems in finite dimensional. In this paper, we survey the notion of weaving hilbert space fusion frames. Through this paper, h and k are hilbert spaces and hi. Fusion frames and g frames were considered recently as generalizations of frames in hilbert spaces, in this paper we generalize some of the known results in frame theory to fusion frames and g frames. Hilbert cmodules are generalization of hilbert spaces by allowing the inner product to take values in a calgebra rather than in the field of complex numbers. Fusion frames have been applied to various fields where distributed or parallel processing is required. Fusion s unique and patented, factory produced preinsulated external wall system stif panel, releases the unmatched efficiency of light gauge steel lgs as a structural building system. Redundancy of fusion frames in hilbert spaces core. In this paper, we present a new identity for gframes which was. Section 2 contains the necessary background about fusion frames, k frames. Huangandyangjournalofinequalitiesandapplications20202020. The authors have provided a complete charac terization of triples m,n,m for which tight fusion frames exist, where m.
Fusionriesz frame riesz frame of subspace whose all subsets are fusion frame. Bayer d, rahimi a, multipliers for continuous frames in hilbert spaces. Next we introduce a new notion of a riesz fusion bases in a hilbert space. This cited by count includes citations to the following articles in scholar. On the equality of fusion frames1 yao xiyan 2, gao guibao and mai ali dept. The positive real constants r1 and r2, respectively, are called lower and upper frame bounds for the frame xnn. It is essential for constructing fusion frame systems to get their local frames. Finally, we specialize the results to fusion frames. Fusionriesz frame in hilbert space 343 proposition 3. Moreover, we give some new results on fusion frames and weaving fusion frames under operator perturbations. Besides, we study the equality of fusion frames and. A subset cof a vector space xis said to be convex if for all x,y. Fusion frames and g frames in hilbert spaces are generalizations of frames, and frames were extended to banach spaces. Some properties of k frames in hilbert spaces springerlink.
In addition, we study the relationship between redundancy of frames fusion frames and dual frames dual fusion frames. From her discussions there are many differences between k frames and ordinary frames, so in this paper we further discuss the interchangeability of two bessel sequences with respect to a k frame. Fusion frames are widely studied recently due to its many kinds of applications. In this paper, we establish some new identities and inequalities for fusion frames with a scalar in hilbert c. Some results about operator perturbation of fusion frames in. The notion of frame has some generalizations such as frames of subspaces, fusion frames and g frames. New characterizations of fusion frames frames of subspaces. Fusion frames are generalizations of frames in hilbert spaces which were introduced by casazza et al. In this article we introduce fusion frames, g frames, banach g frames in banach spaces and we show that they share many useful.
The ones marked may be different from the article in the profile. The motivation for this concept comes from the event that a signal can not be acquired by a single sensor alone, rather the partial components of the signal must be collected via a network of sensors, and the partial signal representations are then fused. A frame is a redundant set of vectors in a hilbert space with the property that provide usually nonunique representations of vectors in terms of the frame elements. Kframes were recently introduced by gavruta in hilbert spaces to study atomic systems with respect to a bounded linear operator. I be a redundant sequence of subspaces of a hilbert space each. On the duality of cfusion frames in hilbert spaces core. Properties of jfusion frames in krein space mafiadoc. Hilbert space, emerged from hilberts e orts to generalize the concept of euclidean space to an in nite dimensional space 7. On some new inequalities for fusion frames in hilbert spaces dongweili andjinsong leng abstract. Research article construction of fusion frame systems in. Aug 28, 2015 upon improving and extending the concept of redundancy of frames, we introduce the notion of redundancy of fusion frames, which is concerned with the properties of lower and upper redundancies. From her discussions there are many differences between kframes and ordinary frames, so in this paper we further discuss the interchangeability of two bessel sequences with respect to a kframe, where k is a bounded linear operator with closed range.
Section 2 contains the necessary background about fusion frames, k frames and weaving frames in separable hilbert spaces to make the paper selfcontained. Kg fusion frames are generalizations of fusion frames, k fusion frames and g fusion frames. It will be shown that generally the dual of c fusion frames may not be defined. The purpose of introducing fusion frame or frame of subspace is to first construct local components and then build a global frame from these. We first investigate the construction of a fusion frame system in a finitedimensional hilbert space when its fusion frame operator matrix is given and provides a corresponding algorithm. These properties are achieved by considering the minimum and maximum values of the redundancy function which is defined from the unit sphere of the hilbert space into the positive real. Problem about frames and frames for hilbert cmodules are more complicated than those for hilbert spaces. Finally we generalize a perturbation result in frame theory to gframes in hilbert spaces. Then, we discuss the redundancy of kg fusion frames. A new notion in frame theory has been introduced recently that called woven frames.
China abstract in this paper, we discuss the property of fusion frames and fusion frames operators for hilbert spaces by utilizing the method of operator theory. Also, fusion frames are an important extension of frames, used in many areas especially for wireless sensor networks. K fusion frames are generalizations of fusion frames in frame theory. Chapter 3 hilbert spaces there are really three types of hilbert spaces over c. The second step, we discuss the relationships between the operator perturbation of fusion frames and the operator perturbation of fusion frame sequences. Vector spaces the notion of a vector space should be known from linear algebra. A relay fusion frame for a vector space equipped with an inner product can also process signals, but the distance between transmitters and receivers is not required. Several perturbation results on k fusion frames are formulated and analyzed. In 1991, grochenig 11 extended the notion of frames from hilbert spaces to the banach spaces and introduced banach frames. On some new inequalities for fusion frames in hilbert spaces. In this paper, we introduce the generalized fusion frames or gfusion frames for hilbert spaces and give characterizations of these frames from. It is also considered a generalization of riesz frame since it shares some important properties of riesz frame.
We find that the operator perturbation of fusion frames is equivalent to the operator perturbation of fusion frame sequences under some conditions see theorem 3. Our results are more general than those previously obtained by balan et al. Pdf construction of fusion frame systems in finite. Research article construction of fusion frame systems in finite dimensional hilbert spaces jinsonglengandtingzhuhuang school of mathematical sciences, university of electronic science and technology of china, chengdu, china. In section 3, we present necessary and su cient conditions for weaving k fusion frames in separable hilbert spaces. Construction of kgfusion frames and their duals in hilbert spaces. Request pdf fusionriesz frame in hilbert space fusionriesz frame riesz frame of subspace whose all subsets are fusion frame sequences with the same bounds is a special fusion frame. Aug 27, 2019 a relay fusion frame for a vector space equipped with an inner product can also process signals, but the distance between transmitters and receivers is not required. In this paper we investigate a new notion of bases in hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in hilbert spaces. Recently fusion frame was considered as a generalization of frame in hilbert spaces.
Specially, we introduce new construction of subfusion frames and derive new results. In this paper we focus on the erasure of a fusion frame. For instance, in a coding transmission process, the encoded and quantized data must be put in numbers of packets. In this paper, we introduce and study frame of operators in quaternionic hilbert spaces as a generalization of g frames which in turn generalized various notions like pseduo frames, bounded quasiprojectors and frame of subspaces fusion frames in separable quaternionic hilbert spaces.
In 39 we defined fusion frame in krein spaces in a more geometric setting motivated by the work of giribet et al. Both of these facts follow from the assertion that an orthonormal or riesz basis has a biorthogonal sequence. In this paper, we establish several new inequalities for fusion frames with a scalar in hilbert spaces. Introduction frames for hilbert spaces have been introduced in 1952 by du. K gfusion frames in hilbert spaces journal of inequalities and. The matrix representations of its local frame operators and inverse frame operators are naturally obtained. We show that the constructed local frame of a subspace with dimension can inherit some properties from the corresponding frame of hilbert space f such as parseval and harmony.
In this article, we show that every fusion frame in a separable hilbert space hcorresponds to a frame in the hilbert bcm. Relay fusion frame system is a device that can receive and transmit signals over a long distance. Ji issaidtobearelayfusionframe,orsimplyr fusion frame,ifthereexistconstants0 hilbert spaces. We also generalize a perturbation result in frame theory to g frames in hilbert spaces. The inequality 1 is called the frame inequality for the frame xnn. Let v be a set endowed with two operations, the operation addition, denoted by. This makes the study of the frames for hilbert cmodules important and interesting. The theory of hilbert space that hilbert and others developed has not only greatly enriched the world of mathematics but has proven extremely useful in the development of scienti c theories, particularly quantum. Gfusion frames which are obtained from the combination of gframes and fusion frames were recently introduced for hilbert spaces.
Pdf frame of operators in quaternionic hilbert spaces. In this article, we show that every fusion frame in a separable hilbert space hcorresponds to a frame in the hilbert bcmmodule hm. An introduction 156 1 the hilbert space l2 156 2 hilbert spaces 161 2. Bierens pennsylvania state university june 24, 2007 1. In this paper we are interested to investigate the properties of j fusion frame in krein space frame theory. Also generalized frames are important frames in the hilbert space of bounded linear operators.
Throughout the paper, we suppose h is a separable hilbert space, ia. In this paper, we introduce fusion frames and g frames in hilbert cmodules and we show that they share many useful properties with their corresponding notions in hilbert space. Erasures and equalities for fusion frames in hilbert spaces. A frame is a redundant set of vectors in a hilbert space h with the property that. These properties are achieved by considering the minimum and maximum values of the redundancy function which is defined from the unit sphere of the hilbert space into the positive real numbers. Improving and extending the concept of dual for frames, fusion frames and continuous frames, the notion of dual for continuous fusion frames in hilbert spaces will be studied. When one or more packets are scrambled, lost, or delayed, fusion frames can enhance the robustness to the packet erasures. There are many di erences between standard frames and k frames, so we study weaving k fusion frames in hilbert spaces. Fusion riesz frame riesz frame of subspace whose all subsets are fusion frame sequences with the same bounds is a special fusion frame.
Moreover, we indicate some results about excess of fusion frames. We obtain new g frames by considering g frames for its components. On some equalities and inequalities of fusion frame in. K frames were recently introduced by gavruta in hilbert spaces to study atomic systems with respect to a bounded linear operator. Throughout this paper h is a separable hilbert space, and glh denotes the set of all bounded linear operators which have bounded inverses. This progression allows us to do things such as measure similarity between two images by. Fusion light gauge steel super structure solutions. C algebra of coe cients and therefore, similar procedures used for frames in hilbert spaces are applied to deal with frames in hilbert c modules. Fusion frames and g frames are also studied in 1, 3, 20, 22. Fusion frames and gframes in hilbert spaces are generalizations of frames, and frames were extended to banach spaces. This article characterizes various kinds of property of k fusion frames. Fusion frame riesz fusion basis besselian fusion frame fredholm operator abstract in this paper we investigate the connection between fusion frames and obtain a relation between indexes of the synthesis operators of a besselian fusion frame and associated frame to it. Pdf a new notion in frame theory has been introduced recently that called woven frames. Such frames play an important role in gabor and wavelet analysis.
Theorem 2 a sequence fx kgin a hilbert space his a riesz basis for hif and only if fx kg satis es the frame condition and is. N in h is a tight frame if it is there exist r1, r2 satisfying inequality 1 with r1 r2 and a parseval frame if it is a tight frame with r1 r2 1. We then provide an easily implementable algorithm for computing such 2sparse fusion frames. Keywords frames fusion frames hilbert spaces selfadjoint operators signal processing sparsity. We state the relationship between redundancy of local frames and fusion frames in a particular case. Casazza and kutyniok 2,3,4 introduced fusion frames frames of subspaces and fusion frames systems in hilbert spaces to study the relation between a frame and its local components. Fusion frames and gframes in hilbert cmodules semantic.
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