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The construction of Parseval fusion frames is highly desirable in a wide range of signal processing applications. In this paper, we study the problem of modifying the weights of a fusion frame in order to generate a Parseval fusion frame. To this end, we extend the notion of the scalability to the fusion frame setting. We then proceed to characterize scalable fusion Riesz bases and $1$-excess fusion frames. Furthermore, we provide the necessary and sufficient conditions for the scalability of certain $k$-excess fusion frames, $k\geq 2$. Finally, we present several pertinent examples to confirm the obtained results., Comment: 26 pages

In this paper, we explore the interplay between topological structures and phase retrieval in the context of projective Hilbert spaces. This work provides not only a deeper understanding and a new classification of the phase retrieval property in Hilbert spaces but also a way for further investigations into the topological underpinnings of quantum states.

Computing the excess as a method of measuring the redundancy of frames was recently introduced to address certain issues in frame theory. In this paper, the concept of excess for fusion frames is studied. Then, several explicit methods are provided to compute the excess of fusion frames and their $Q$-duals. In particular, some upper bounds for the excess of $Q$-dual fusion frames are established. It turns out that, unlike ordinary frames, for every $n \in \Bbb{N}$ we can provide a fusion frame with its $Q$-dual whose the difference of their excess is $n$. Furthermore, the connection between the excess of fusion frames and their orthogonal complement is completely characterized. Finally, several examples are exhibited to confirm the obtained results., Comment: 25 pages

Recovering a signal up to a unimodular constant from the magnitudes of linear measurements has been popular and well studied in recent years. However, numerous unsolved problems regarding phase retrieval still exist. Given a phase retrieval frame, may the family of phase retrieval dual frames be classified? And is such a family dense in the set of dual frames? Can we present the equivalent conditions for a family of vectors to do weak phase retrieval in complex Hilbert space case? What is the connection between phase, weak phase and norm retrieval? In this context, we aim to deal with these open problems concerning phase retrieval dual frames, weak phase retrieval frames, and specially investigate equivalent conditions for identifying these features. We provide some characterizations of alternate dual frames of a phase retrieval frame which yield phase retrieval in finite dimensional Hilbert spaces. Moreover, for some classes of frames, we show that the family of phase retrieval dual frames is open and dense in the set of dual frames. Then, we study weak phase retrieval problem. Among other things, we obtain some equivalent conditions on a family of vectors to do phase retrieval in terms of weak phase retrieval.

The purpose of this work is to examine the structure of optimal dual fusion frames and get more exibility in the use of dual fusion frames for erasures of subspaces. We deal with optimal dual fusion frames with respect to different definitions of duality and compare the advantages of these approaches. In addition, we introduce a new concept so called partial optimal dual which involves less time and computation for detecting optimal dual for erasures in known locations. Then we study the relationship between local and global optimal duals by partial optimal duals which leads to some applicable results. In the sense that, we obtain an overcomplete frame and a family of associated optimal duals by a given Riesz fusion basis. We present some examples to exhibit the effect of error rate when dual fusion frames are applied in reconstruction., Comment: 19 pages

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. In this paper, we study the notion of excess for woven frames and prove that any two frames in a separable Hilbert space that are woven have the same excess. We also show that every frame with a large class of duals is woven provided that its redundant elements have small enough norm. Also, we try to transfer the woven property from frames to their duals and vise versa. Finally, we look at which perturbations of dual frames preserve the woven property, moreover it is shown that under some conditions the canonical Pareseval frame of two woven frames are also woven., Comment: This article is containing 16 pages and it is published in journal of Bulletin of the Malaysian

For finding the numerical solution of operator equations in many applications a decomposition in subspaces is needed. Therefore, it is necessary to extend the known method of matrix representation to the utilization of fusion frames. In this paper we investigate this representation of operators on a Hilbert space $\Hil$ with Bessel fusion sequences, fusion frame and Riesz decompositions. We will give the basic definitions. We will show some structural results and give some examples. Furthermore, in the case of Riesz decompositions, we prove that those functions are isomorphisms. Also, we want to find the pseudo-inverse and the inverse (if there exists) of such matrix representations. We are going to apply this idea to the Schatten $p$-class operators. Finally, we show that tensors of fusion frame are frames in the space of Hilbert-Schmidt operators.

Weaving Hilbert space frames have been introduced recently by Bemrose et al. to deal with some problems in distributed signal processing. In this paper, we survey this topic from the viewpoint of the duality principle, so we obtain new properties in weaving frame theory related to dual frames. Specifically, we give some sufficient conditions under which a frame with its canonical dual, alternate duals or approximate duals constitute some concrete pairs of woven frames. Moreover, we survey the stability of weaving frames under perturbations and different operators.

In this paper, we introduce the concept of $K$-fusion frames and propose the duality for such frames. The relation between the local frames of $K$-fusion frames with their dual is studied. The elements from the range of a bounded linear operator $K$ can be reconstructed by $K$-frames. Also, we establish $K$-fusion frame multipliers and investigate reconstruction of the range of $K$ by them.

K-frames are strongly tools for the reconstruction elements from the range of a bounded linear operator K on a separable Hilbert space H. In this paper, we study some properties of K-frames and introduce the K-frame multipliers. We also focus to represent elements from the range of K by K-frame multipliers.

The Gram matrix is defined for Bessel sequences by combining synthesis with subsequent analysis operators. If different sequences are used and an operator U is inserted we reach so called U-cross Gram matrices. This can be seen as reinterpretation of the matrix representation of operators using frames. In this paper we investigate some necessary or sufficient conditions for Schatten p-class properties and the invertibility of U-cross Gram matrices. In particular, we show that under mild conditions the pseudo-inverse of a U-cross Gram matrix can always be represented as a U-cross Gram matrix with dual frames of the given ones. We link some properties of U-cross Gram matrices to approximate duals. Finally, we state several stability results. More precisely, it is shown that the invertibility of U-cross Gram matrices is preserved under small perturbations., Comment: 29 pages

A new notion in frame theory, so called weaving frames has been recently introduced to deal with some problems in signal processing and wireless sensor networks. Also, fusion frames are an important extension of frames, used in many areas especially for wireless sensor networks. In this paper, we survey the notion of weaving Hilbert space fusion frames. This concept can be had potential applications in wireless sensor networks which require distributed processing using different fusion frames. Indeed, we present several approaches for identifying and constructing of weaving fusion frames in terms of local frames, bounded operators in Hilbert spaces and also dual fusion frames. To this end, we present some conditions under which a fusion frame with its duals constitute some pair of woven fusion frames. As a result, we show that Riesz fusion bases are woven with all of their duals. Finally, we obtain some new results on fusion frames and weaving fusion frames under operator perturbations.

For applications like the numerical solution of physical equations a discretization scheme for operators is necessary. Recently frames have been used for such an operator representation. In this paper, we apply fusion frames for this task. We interpret the operator representation using fusion frames as a generalization of fusion Gram matrices. We present the basic definition of $U$-fusion cross Gram matrices of operators for a bounded operator $U$. We give sufficient conditions for their (pseudo-)invertibility and present explicit formulas for the inverse. In particular, we characterize fusion Riesz bases and fusion orthonormal bases by such matrices. Finally, we look at which perturbations of fusion Bessel sequences preserve the invertibility of the fusion Gram matrix of operators., Comment: 19 pages

K-frames, a new generalization of frames, were recently considered by L. Gavruta in connection with atomic systems and some problems arising in sampling theory. Also, fusion frames are an important generalization of frames, applied in a variety of applications. In the present paper, we introduce the notion of K-fusion frames in Hilbert spaces and obtain several approaches for identifying of $K$-fusion frames. The main purpose is to reconstruct the elements from the range of the bounded operator $K$ on a Hilbert space H by using a family of closed subspaces in H. This work will be useful in some problems in sampling theory which are processed by fusion frames. For this end, we present some descriptions for duality of K-fusion frames and also resolution of the operator K to provide simple and concrete constructions of duals of K-fusion frames. Finally, we survey the robustness of K-fusion frames under some perturbations.

Fusion frames are valuable generalizations of discrete frames. Most concepts of fusion frames are shared by discrete frames. However, the dual setting is so complicated. In particular, unlike discrete frames, two fusion frames are not dual of each other in general. In this paper, we investigate the structure of the duals of fusion frames and discuss the relation between the duals of fusion frames with their associated discrete frames., Comment: 12 pages

In this paper we extend the notion of approximate dual to fusion frames and present some approaches to obtain dual and approximate alternate dual fusion frames. Also, we study the stability of dual and approximate alternate dual fusion frames., Comment: 14 pages

Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ be the semidirect product of $H$ and $K$ with respect to $\tau$. In this article we define the Zak transform $\mathcal{Z}_L$ on $L^2(G_\tau)$ with respect to a $\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\mathrm{SL}(2,\mathbb{Z})\ltimes_\tau\mathbb{R}^2$ and also the Weyl-Heisenberg groups.