Continuous Frames in Hilbert Space

Generalized frames (in short, $g$-frames) are a natural generalization of standard frames in separable Hilbert spaces. Motivated by the concept of weaving frames in separable Hilbert spaces by Bemrose, Casazza, Grochenig, Lammers and Lynch in the context of distributed signal processing, we study weaving properties of $g$-frames. Firstly, we present necessary and sufficient con\-ditions for weaving $g$-frames in Hilbert spaces. We extend some results of \cite Bemrose, Casazza, Grochenig, Lammers and Lynch, and Casazza and Lynch regarding conversion of standard weaving frames to $g$-weaving frames. Some Paley-Wiener type perturbation results for weaving $g$-frames are obtained. Finally, we give necessary and sufficient conditions for weaving $g$-Riesz bases.

In this paper, we first introduce the notion of controlled weaving [Formula: see text]-[Formula: see text]-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving [Formula: see text]-[Formula: see text]-frames in separable Hilbert spaces. Also, a characterization of controlled weaving [Formula: see text]-[Formula: see text]-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled [Formula: see text]-[Formula: see text]-woven frames give ordinary weaving [Formula: see text]-frames and vice-versa.

In this paper, we first introduce the notation of weaving continuous fusion frames in separable Hilbert spaces. After reviewing the conditions for maintaining the weaving [Formula: see text]-fusion frames under the bounded linear operator and also, removing vectors from these frames, we will present a necessarily and sufficient condition about [Formula: see text]-woven and [Formula: see text]-fusion woven. Finally, perturbation of these frames will be introduced.

In this paper, we study continuous frames in Hilbert spaces using a family of linearly independent vectors called coherent state (CS) and applying it in any physical space. To accomplish this goal, the standard theory of frames in Hilbert spaces, using discrete bases, is generalized to one where the basis vectors may be labeled using discrete, continuous or a mixture of the two types of indices.  A comprehensive analysis of such frames is presented and illustrated by the examples drawn from a toy example Sea Star and the affine group. Key words: Frame, continuous frame, unitary representation, coherent state (CS), sea star, affine group. &nbsp

Some equalities and inequalities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>g</mml:mi></mml:math>-Bessel sequences in Hilbert spaces

Disjointness of frames in Hilbert spaces is closely related with superframes in Hilbert spaces and it also plays an important role in construction of superframes and frames, which were introduced and studied by Han and Larson. \(G\)-frame is a generalization of frame in Hilbert spaces, which covers many recent generalizations of frame in Hilbert spaces. In this paper, we study the \(g\)-frames in Hilbert spaces. We focus on the characterizations of disjointness of \(g\)-frames and constructions of \(g\)-frames. All types of disjointness are firstly characterized in terms of disjointness of frames induced by \(g\)-frames, then are characterized in terms of certain orthogonal projections. Finally we use disjoint \(g\)-frames to construct \(g\)-frames.

The material presented in this book naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. For the second part the most general results were presented in Chapter 21, in the setting of generalized shift-invariant systems on an LCA group.The current chapter is in a certain sense a natural continuation of both tracks. We consider connections between frame theory and abstract harmonic analysis and show how we can construct frames in Hilbert spaces via the theory for group representations. In special cases the general approach will bring us back to the Gabor systems and wavelet systems. The abstract framework adds another new aspect to the theory: we will not only obtain expansions in Hilbert spaces but also in a class of Banach spaces.

In this paper, we introduce and characterize controlled dual frames in Hilbert spaces. We also investigate the relation between bounds of controlled frames and their related frames. Then, we define the concept of approximate duality for controlled frames in Hilbert spaces. Next, we introduce multiplier operators of controlled frames in Hilbert spaces and investigate some of their properties. Finally, we show that the inverse of a controlled multiplier operator is also a controlled multiplier operator under some mild conditions.

Abstract. Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames [40] and the unitary extension principle for wavelet frames [38]. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in [38], e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.

Continuous frames and fusion frames were considered recently as generalizations of frames in Hilbert spaces. In this paper, for any continuous fusion frame, we obtain a new family of inequalities which are parametrized by a parameter $$\lambda \in \mathbb {R}$$ . By suitable choices of $$\lambda $$ , one obtains the previous results as special cases. Moreover, these new inequalities involve the expressions $$\langle S_Yh,h \rangle $$ , $$\Vert S_Yh\Vert $$ , etc., where $$S_Y$$ is a “truncated form” of the continuous fusion frame operator.

G-frames, which were proposed recently as generalized frames in Hilbert spaces, share many similar properties with frames, but not all the properties of them are similar. Christensen presented that every Riesz frame contains a Riesz basis. In this paper, the authors showed that not all g-Riesz frames contain a g-Riesz basis, but they obtained that every g-Riesz frame contains an exact g-frame. They also gave a necessary and su±cient condition for a g-Riesz frame in a Hilbert space. From this, they might get the characterization of Riesz frames. Lastly the authors considered the stability of a g-Riesz frame for a Hilbert space under perturbations. These properties of g-Riesz frames in Hilbert spaces are not similar to those of Riesz frames.

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

G-frames, which include many generalizations of frames such as frames of subspaces or fusion frames, oblique frames, and pseudo-frames, are natural generalizations of frames in Hilbert spaces. They have some properties similar to those of frames in Hilbert spaces, but not all of their properties are similar. For example, exact frames are equivalent to Riesz bases, but exact g-frames are not equivalent to g-Riesz bases. Some authors have extended the equalities and inequalities for frames and dual frames to g-frames and dual g-frames in Hilbert spaces. In this paper, we establish some new equalities and inequalities for g-Bessel sequences or g-frames in Hilbert spaces. We also give a necessary and sufficient condition that the equality occurs in one of these inequalities. Our results generalize and improve the remarkable results which had been obtained by Balan, Casazza and Gavruta.

Frame theory is recently an active research area in mathematics, computer science and engineering with many exciting applications in a variety of different fields. This theory has been generalized rapidly and various generalizations of frames in Hilbert spaces. In this papers we study the notion of dual continuous $K$-frames in Hilbert spaces. Also we etablish some new properties.

The theory of continuous frames in Hilbert spaces is extended, by using the concepts of measure spaces, in order to get the results of a new application of operator theory. The $K$-frames were introduced by G$breve{mbox{a}}$vruta (2012) for Hilbert spaces to study atomic systems with respect to a bounded linear operator. Due to the structure of $K$-frames, there are many differences between $K$-frames and standard frames. $K$-frames, which are a generalization of frames, allow us in a stable way, to reconstruct elements from the range of a bounded linear operator in a Hilbert space. In this paper, we get some new results on the continuous $K$-frames or briefly c$K$-frames, namely some operators preserving and some identities for c$K$-frames. Also, the stability of these frames are discussed.