arXiv subjects
Publications and source records attributed to Fahimeh Arabyani Neyshaburi.
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.
This article explores the problem of modifying the subspaces of a fusion frame in order to construct a Parseval fusion frame. In this respect, the notion of scalability is extended to the fusion frame setting. Then, scalable fusion Riesz bases are characterized, and a concrete form for scalable 1-excess fusion frames is obtained. Furthermore, it is shown that 1-excess dual fusion frames of a fusion Riesz basis are not scalable. Finally, several examples are exhibited to confirm the acquired results.
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.
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.
The most important purpose of this article is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by matrix methods. To this end, first we obtain more structures of the canonical K-dual. % and survey optimal K-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the K-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. Finally, we exhibit several examples that illustrate our results and the advantage of using the new matrix with respect to previous approaches in existence and construction.
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.
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.
K-frame theory was recently introduced to reconstruct elements from the range of a bounded linear operator K in a separable Hilbert space. This significant property is worthwhile especially in some problems arising in sampling theory. Some equalities and inequalities have been established for ordinary frames and their duals. In this paper, we continue and extend these results to obtain several important equalities and inequalities for K-frames. Moreover, by applying Jensen's operator inequality we obtain some new inequalities for $K$-frames.
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.
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.