$p$-frames

This paper will generalize $b$-frames, a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product, we will use a new product called the $b$-dual product, which is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames and $K$-$b$-frames and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will express the reconstruction formula of the elements of the Hilbert space. We will also study the stability and preservation of both $b$-frames and $K$-$b$-frames and to do so, we will give the equivalent of the adjoint operator according to the $b$-dual product.

Phase retrieval is a fundamental problem in signal processing and imaging, where the goal is to recover a signal from its magnitude measurements.This overview paper provides a brief survey of the history, theory and algorithms of phase retrieval topic. We discuss the importance of phase retrieval frames in various fields and highlight key milestones in its development.

The work is devoted to the characteristic properties analysis of the sea surface image on video stream frames, which makes it possible to make a decision about the detection of object images on them in the absence of a priori information about objects, which is important in the development of navigation systems, security and surveillance systems. In this paper, it is proposes to analyze the values of the normalized cross-correlation coefficients of a given image fragment with higher dimension fragments on sequential video stream frames to identify the distinctive properties of the agitated sea surface and floating objects images. The computational experiments were carried out and have shown that the analysis of the results of calculating the frame fragments cross-correlations allows us to estimate the amount of displacement and distortion of a given image fragment. The results of computational experiments demonstrate the presence of differences in the values of the corresponding cross-correlation coefficients for the sea surface images with different agitation degrees, containing and not containing an image of the object. Based on the analysis of the proposed correlation properties of the sea surface images, a decisive rule for selecting fragments of frames containing an image of an object is formulated, the use of which, in many cases, allows to detection of fragments of object images correctly.

In this work‎, ‎we generalize the equalities from ‎ g‎-frame and g‎-Bessel sequences to continuous g‎-frame and continuous g‎-Bessel sequences in Hilbert spaces‎. ‎This generalization enables us to obtain the relationship between continuous g-frames and their alternate dual on a measure space‎.

In this paper, the concept of K-g-frames in Hilbert pro-C∗-modules is introduced, and their basic properties are examined. The conditions under which family of operators can form a K-g-frame on Hilbert pro-C∗-modules are derived. Moreover g-atomic systems in Hilbert proC∗-modules are introduced and their properties are examined.

The purpose of the paper is to analyze frames $\{f_k\}_{k \in \mathbb{Z}}$ having the form $\{T^{k}f_{0}\}_{k \in \mathbb{Z}}$; so called iteration operator frames for some bounded linear operator $T$ and a fixed vector $f_0$. We state the duality of such frames with respect to their generators. Moreover, we characterize all duals of iteration operator frames with the same structure. We also show that the duals of two iteration operator frames are close to each other provided that the original frames are sufficiently close to each other and vise versa.

Some generalizations of Besselian, Hilbertian systems and frames in nonseparable Banach spaces with respect to some nonseparable Banach space $K$ of systems of scalars are considered in this work. The concepts of  uncountable $K$-Bessel, $K$-Hilbert systems, $K$-frames and  $K^{*} $-Riesz bases in nonseparable Banach spaces are introduced. Criteria of uncountable $K$-Besselianness, $K$-Hilbertianness for systems, $K$-frames and unconditional $K^{*} $-Riesz basicity are found, and the relationship between them is studied. Unlike before, these new facts about Besselian and Hilbertian systems in Hilbert and Banach spaces are proved without using a conjugate system and, in some cases, a completeness of a system. Examples of $K$-Besselian systems which are not minimal are given. It is proved that every $K$-Hilbertian systems is minimal. The case where $K$ is an space of systems of coefficients of uncountable unconditional basis of some space is also considered.

In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from a Hilbert $C^{ast}$-module $mathcal{H}$ to  itself denoted by $End_{mathcal{A}}^{ast}(mathcal{H}) $.  We give some properties relating to some constructions of integral $K$-operator frames and to operators preserving  integral $K$-operator frame and we establish some new results.

In recently years, frames in Krein spaces had been considered. The paper presents a family of generators for a Krein space by their frames. These generators are dual frames and operator dual frames corresponding to a given frame in a Krein space. We characterize all generalized dual frames of a primary frame. Also, approximately dual frames in a Krein space are introduced and, we study the relation between approximately dual frames and operator duals in a Krein space. Finally, perturbation of frames in this space is considered.

In this paper, we intend to prove the applied properties of woven p-frames from a functional analysis point of view. To this end, first, we prove the stability of the Banach woven p-frames under small perturbations. The results of the classical perturbation for bases led us to examine the perturbation of Banach woven p-frames and provide the new and weaker conditions that guarantee the desired stability. In the following, we give some ways to identify and build woven p-frames. To this end, we give the conditions under which a p-frame and a nite family of bounded invertible operators form a woven p-frame.

 Let U be a Hilbert A-module and L(U) the set of all adjointable A-linear maps on U. Let K = {Λx ∈ L(U, Vx) : x ∈ X } and L = {Γx ∈ L(U, Vx) : x ∈ X } be two continuous g-frames for U, K is said to be similar with L if there exists an invertible operator J ∈ L(U) such that Γx = ΛxJ, for all x ∈ X . In this paper, we define the concepts of closeness and nearness between two continuous g-frames. In particular, we show that K and L are near, if and only if they are similar.

In this paper, we introduce $(p,q)g-$Bessel multipliers in Banach spaces and we show that under some conditions a $(p,q)g-$Bessel multiplier is invertible. Also, we show the continuous dependency of $(p,q)g-$Bessel multipliers on their parameters.

In this paper, we reintroduce the concept of controlled K-frames and then, we show that this definition is equivalent with the concept that has been recently introduced in cite{Nouri}. Meanwhile, we correct one of the results which were obtained in the mentioned paper. In the sequel, we construct some new controlled K-frames by some operator theory tools. Finally, we provide some conditions under which the sum of two controlled K-frames remains a controlled K-frame.

In this note, the notion of generalized continuous K- frame in a Hilbert space is defined. Examples have been given to exhibit the existence of generalized continuous $K$-frames. A necessary and sufficient condition for the existence of a generalized continuous $K$-frame in terms of its frame operator is obtained and a characterization of a generalized continuous $K$-frame for $ mathcal{H} $ with respect to $ mu $ is given. Also, a sufficient condition for a generalized continuous $K$-frame is given.  Further, among other results, we prove that generalized continuous $K$-frames are invariant under a linear  homeomorphism. Finally, keeping in mind the importance of perturbation theory in various branches of applied mathematics, we study perturbation of $K$-frames and obtain conditions for the stability of generalized continuous $K$-frames.

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert-Schmidtframes. Also show that the canonical dual of a perturbed Hilbert-Schmidt frame is a perturbationof the canonical dual (alternative dual respectively) of the original Hilbert-Schmidt frame and dis-cuss best approximation in the set of all dual Hilbert-Schmidt frames. Next, we apply the wovenprinciple to Hilbert-Schmidt frames and study the stability of weaving Hilbert-Schmidt frames underperturbations. Finally, we present sufficient conditions under which perturbations preserve weavingHilbert-Schmidt frames and weaving dual Hilbert-Schmidt frames.

This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued functions, and for the related functors ${mathfrak R}^*$ and ${mathfrak Z}^*$ corresponding to the bounded functions.  Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.

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.

In this paper we introduce continuous g-Bessel multipliers in Hilbert spaces and investigate some of their properties. We provide some conditions under which a continuous g-Bessel multiplier is a compact operator. Also, we show the continuous dependency of continuous g -Bessel multipliers on their parameters.