Relay fusion frames in Banach spaces

: The relay fusion frames have been recently introduced in Hilbert spaces to model sensor relay networks and distributed sensor relay systems, which are deeply connected with compressed sensing. In this article, we introduce the notions of relay fusion frames and Banach relay fusion frames in Banach spaces and study certain attractive properties of relay fusion frames in this more general setting. In a particular sense, Schauder frames can be shown to be a special case of relay fusion frames. Moreover, the stability issue of relay fusion frames and Banach relay fusion frames will be addressed


Introduction
The concept of frames in a separable Hilbert space was originally introduced by Duffin and Schaeffer in the context of the non-harmonic Fourier series [1]. They satisfy the well-known property of perfect reconstruction, in that any vector of the Hilbert space can be synthesized back from its inner products with the frame vectors [2]. Fusion frames were launched by Casazza and Kutyniok in [3] and further developed in their joint article [4] with Li et al. Fusion frames are an emerging topic of frame theory with applications to communications and distributed processing. Relay fusion frames, or simply, R-fusion frames, are recent developments of fusion frames and g-frames that provide a natural mathematical framework for three-stage (or, more generally, hierarchical) data processing. The notion of an R-fusion frame was introduced in [5] with the main idea already presented in [6] and has become a tool in the implementation of distributed relay systems and distributed sparse recovery [7]. The main purpose of this article is to introduce some of the natural generalizations of R-fusion frame theory to Banach spaces and study some important properties and stability of R-fusion frames in this more general setting.
As well known, the extension of frame theory to Banach Spaces is an important issue, not only from a theoretical perspective but also from a number of applications involving signals in Banach Spaces [8][9][10]. Mathematically, the extension of frame theory to Banach spaces is highly nontrivial and is deeply related to some well-known problems in Banach space theory. The classical frames have been extended to Banach spaces in a variety of ways. One way is to extend the frames of Banach spaces to the compression of unconditional bases of larger Banach spaces [11,12]. This definition captures the unconditional convergence property of frame operators, which seems to be the core of Hilbert space frame theory. Another way is to generalize the frame inequality, such as [13,14]. The third method is to generalize the dual frame pairs in Hilbert spaces and the Schauder bases in Banach spaces [15,16]. Various frames for Banach spaces have played a key role in the development of wavelet theory and Gabor or Fourier analysis. In addition to the theoretical appeal, the frames in the L p spaces and other Banach function spaces are effective tools for modeling various natural signals and images [17,18]. In general, unlike Hilbert Spaces, Banach spaces do not necessarily have inner product and orthogonality, and most Banach Spaces do not have reflexivity, so it is relatively difficult to study frames in Banach Spaces. One way to overcome these difficulties is to use operator theory in Banach Spaces [19].
R-fusion frame is a new signal representation method that uses collections of relay subspaces instead of vectors to represent signals. Such a representation provides significant flexibility compared to classical frame representations. In this article we define the concept of R-fusion frames in Banach spaces and make a detailed study of R-fusion frames in this more general setting. The rest of the article is organized as follows: In Section 2, we give the basic definitions and certain preliminaries to be used in the article. R-fusion frames for Banach spaces will be introduced in Section 3, and some basic properties about R-fusion frames in this context will be developed in this section. The investigation of stability of R-fusion frames in Banach spaces is in Section 4. Example 1. Let c 0 be the space of sequences converging to zero and c be the space of convergent sequences. Then c 0 and c are solid BK-spaces. Moreover, we note that every We will need to work with X d -frames.
‖are equivalent, i.e., there exist constants α β 0 < ≤ < ∞ such that In this case, we say that g i { } is an α β , ( )-X d -frame for X.
If X is a Hilbert space and X d 2 ( ) = ℓ , (2.1) means that g i { } is a frame, and in this case, it is well known that there exists a sequence f i { } in X such that Similar reconstruction formulas are not always available in the Banach space setting. This is the reason behind the following definition [14].
Definition 3. Let X be a Banach space and X d be a sequence space. Given a bounded linear operator S X X : d → and an X d -frame g i { }, we say that g S , In the following example, we construct a new Banach frame for p ℓ with respect to ∈ is an p ℓ -frame for p ℓ with bounds 1 and δ. For any x X d ∈ and i ∈ , define reconstruction operator S X X : is a Banach frame for p ℓ with respect to p ℓ .
In 1989, Gröchenig extended the concept of classical frames to Banach spaces and called it atomic decomposition. We recall the original definition of atomic decomposition that appears in [14].
Definition 4. Let X be a Banach space and let X d be an associated Banach space of scalar-valued sequences.
In 1999, Casazza et al. introduced the concept of the Schauder frames to the Banach spaces in [16]. The definition of Schauder frames is a natural extension of the dual frame pairs in Hilbert spaces and Schauder bases in Banach spaces.
The definition of atomic decomposition expresses the desire to obtain a series expansion of x X ∈ like the frames in Hilbert spaces. On the other hand, the definition of Banach frames opens the way for the possibility of reconstruct formulas. Equation (2.2) can be regarded as a "generalized reconstruction formula," because it tells us how to restore the original element x X ∈ based on the coefficients g x i { ( )}. It is well known that there are separable Banach spaces without bases. Casazza et al. proved in [16] that every separable Banach space has a Banach frame. From this point of view, it can be said that the concept of Banach frames is very satisfactory because they always exist.

Relay fusion frames in Banach spaces
In this section, we introduce the relay fusion frames in Banach spaces, and we discuss some of their useful properties in this setting.
. Suppose that q i is a continuous linear projection on Y i and p i { } is a sequence of continuous linear projections on X and let ) }is called a relay fusion frame for X, or simply R-fusion frame for X, if there exists a solid BK-space X d such that In this case, we say that v Ω , Δ , Λ , )-R-fusion frame for X. The numbers α β , are called R-fusion frame bounds or simply "bounds." In the following remark, we point out the relationship among R-fusion frame and fusion frame and g-frame.
then Definition 6 covers the g-frames in Banach spaces. The special case, where Y i = , i ∈ , gives rise to the X d -frames.
For the existence of R-fusion frames in Banach spaces, we have the following example.
is an R-fusion frame for Y . This scenario shows that R-fusion frame model can be carried out in both directions. Now consider the analysis space of the R-fusion frames in Banach spaces. Define The following theorem will be essential for the research.
Theorem 1. The analysis space Σ with the norm It is easy to observe that for every λ ∈ , λz λ z Σ Σ | | ‖ ‖ = ‖ ‖ . Therefore, Σ is a normed space. Now we show that Σ is complete. Consider the mapping η X η y y y : Σ , , Σ.
For every i ∈ , define mapping Obviously, P i is a projection. By the continuity of coordinate functionals π X : converges to λ. Thanks to the continuity of coordinate functionals π X : It can be seen from the aforementioned proof that for every n ∈ , w w Σ We state some of the important properties of analysis operator Θ: ) }be an R-fusion frame for X. Then the analysis operator Θ is a bounded, injective, linear operator with closed range and X X Θ : which implies that Θ is a bounded, injective, linear operator with closed range and X X Θ : The generalized reconstruction formula (2.2) motivates the following definition. According to this definition, we can easily obtain the following result. The Schauder frames can be displayed as a special case of the r-fusion frames in a specific sense. We will elaborate on this in the following proposition. We first recall the definition of a biorthogonal system.
)-X d -frame for X and g x , are isometry operators and g i { } and x i { } are biorthogonal. Then is a continuous linear projection on X. Further, )-R-fusion frame for X. Hence, the conclusion follows. □ For further research, we need the following result. For convenience, Zi d , ‖⋅‖ is abbreviated as i ‖⋅‖ .
∈ are all BK-spaces, then We recall the formal definition of g-frames in Banach spaces.
Definition 10. We call a sequence X Y i Λ , : ∈ an α β , ( )-g-frame for X with respect to Y i if there exist a solid BK-space X d and positive constants α β 0 < ≤ < ∞ such that for all x X ∈ , In this case we say that Y W j Λ , : , and suppose that they are λ μ , ( )-bounded. Then the following conditions are equivalent.
∈ is a g-frame for X with respect to Y i : is a g-frame for X with respect to W i j : , Proof. Note that for every x X ∈ and each i ∈ , we have R TS x Y Since X d is a solid space, by (3.4), which proves the claim. □ We can obtain two equivalent descriptions of the R-fusion frames in Banach spaces as follows by using Theorem 4.
. Suppose that q i is a continuous projection on Y i and p i { } is a family of continuous projection on X and let v i { } be a family of weights, i.e., v 0 If Y W j Λ , : ∈ is a g-frame for X with respect to i Δ : is a g-frame for X with respect to W i j : , . Suppose that q i is a continuous projection on Y i and p i { } is a family of continuous projection on X and let v i { } be a family of weights, i.e., v 0 )-fusion frame for each Y i , i ∈ , and they are λ μ , ( )-bounded, then the following statements are equivalent.
∈ is a g-frame for X with respect to i Δ : : , is a g-frame for X with respect to W i j : ,

Perturbation
The problem studied in this section is the stability of the R-fusion frames in Banach spaces, that is, whether a set of elements close to the R-fusion frame is itself an R-fusion frame. We define λ μ c , , ( )-perturbation by employing the canonical Paley-Wiener type definition.
. Suppose that τ i is a continuous projection on Y i and π i { } is a family of continuous projection on X and let u i { } be a family of weights, i.e., u 0 . If for all x X ∈ , vq p x u τ π x λ vq p x μ u τ π x c x Λ Γ Λ Γ ,