In this section, we first introduce the definition of relay fusion frames and then we will show that it also provide an associated analysis and synthesis operator, a frame operator and a dual object.
3.1 Definition and basic properties of relay fusion frames and their operators
The family
\(\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij})\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}\) is called an
α-
tight r-fusion frame, if the constants
α and
β can be chosen so that
\(\alpha =\beta \), a
Parseval r-fusion frame provided that
\(\alpha =\beta =1\). If
\(\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij})\}_{i \in \mathbb{I}, j\in \mathbb{J}_{i}}\) satisfies the second inequality in Eq. (
2), then it is said to be a
Bessel
r-
fusion sequence in
H with Bessel r-fusion bound
β.
If we take
\(K_{i}=H, V_{ij}=W_{i}, \varLambda _{i}=I_{H}\) and
\(v_{ij}=w_{i}\) for all
\(i\in \mathbb{I}, j\in \mathbb{J}_{i}\), then we get from Definition
6 the fusion frame
\(\{(W_{i}, w_{i})\} _{i\in \mathbb{I}}\) for
H and thus r-fusion frame can be viewed as a generalization of fusion frame.
Similarly, let
\(W_{i}=H, V_{ij}=K_{i}\) and
\(v_{ij}=1\) for all
\(i\in \mathbb{I}, j\in \mathbb{J}_{i}\), then inequality (
2) can be restated as the following form which is, as defined in [
21], the g-frames:
$$ \alpha \Vert f \Vert ^{2} \leqslant \sum _{i\in \mathbb{I}} \bigl\Vert \varLambda _{i}(f) \bigr\Vert ^{2}\leqslant \beta \Vert f \Vert ^{2}, \quad \forall f \in H. $$
Consequently, g-frames can be thought of as a special class of r-fusion frames. The special case, where
\(K_{i}=\mathbb{C}\),
\(i\in \mathbb{I}\), gives rise to the classical frames.
The representation space employed in classical frame theory and fusion frame theory equal
\(\ell ^{2}(\mathbb{I})\) and
\((\sum_{i\in \mathbb{I}}\oplus W_{i} )_{\ell ^{2}}\), respectively. However, in r-fusion frame theory an input signal
\(f \in H\) is represented by the collection of vector coefficients that can be thought of as to represent the projection onto each subspace of
\(\mathit{local}\ \mathit{relay}\ \mathit{spaces}\)
\(K_{i}\),
\(i\in \mathbb{I}\). Hence, the representation space employed in this framework defined by
$$ \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}} \oplus V_{ij} \biggr)_{\ell ^{2}} = \biggl\{ \{f_{ij}\}_{i\in \mathbb{I}, j \in \mathbb{J}_{i}} | f_{ij}\in V_{ij} \text{ and } \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}} \Vert f_{ij} \Vert ^{2}< \infty \biggr\} , $$
with inner product given by
$$ \bigl\langle \{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}, \{g _{ij} \}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}} \bigr\rangle =\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}\langle f _{ij}, g_{ij}\rangle , $$
with respect to the pointwise operations is a Hilbert space.
We can give an intuitive explanation about r-fusion frames. Let us assume that we want to transmit the wireless signal f belonging to a vector space W from a transmitter \(\mathcal{X}\) to a receiver \(\mathcal{Y}\). If the distance between \(\mathcal{X}\) and \(\mathcal{Y}\) is too far, the wireless signal f will come to nothing before reaching the receiver \(\mathcal{Y}\). However, in the case we set up a relay station \(\mathcal{Z}\) between \(\mathcal{X}\) and \(\mathcal{Y}\), this situation will clear away. By transmitting from the relay stations, the restriction that the ordinary receiver and the transmitter cannot be connected due to the distance can be solved.
In the sequel, we will denote \(\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij}) \}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}\) by \(\mathcal{R}\), simply. We abbreviate r-fusion frame to RFF.
Before define the analysis operator for an RFF, we state the following lemma, which is analogous to Lemma 3.9 in [
4].
We call the adjoint \(T_{\mathcal{R}}^{*}\) of the analysis operator the synthesis operator of \(\mathcal{R}\).
In an analogous way as in frame and fusion frame theory we can give the following well-known relations between an RFF and the associated analysis and synthesis operator.
By composing \(T_{\mathcal{R}}\) and \(T_{\mathcal{R}}^{*}\), we obtain the frame operator for \(\mathcal{R}\).
To prove Proposition
3.5 we need the following theorem that gives the relation between a Bessel r-fusion sequence and the synthesis operator
\(T_{\mathcal{R}}^{*}\).
Given an RFF, Proposition
3.5 states some of the important properties of frame operator
\(S_{\mathcal{R}}\).
The following theorem gives a sufficient condition such that two Bessel r-fusion sequence become RFFs in terms of their analysis operators.
3.2 Duality of relay fusion frames
To define the dual frames for RFFs, we need the following technical lemma.
3.2.1 Global relay dual of relay fusion frames
Let \(\mathcal{R}=\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij})\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}\) be an RFF for H. We consider \(\mathit{global}\ \mathit{relay}\ \mathit{space}\)
\(\mathcal{K}= (\sum_{i\in \mathbb{I}}\oplus K_{i} )_{\ell ^{2}}\) and let \(\mathcal{F}_{\mathcal{K}}\) be a frame for \(\mathcal{K}\), where every \(K_{i}\) is local relay space. We use \(S_{\mathcal{F}_{ \mathcal{K}}}\) to denote the frame operator for \(\mathcal{K}\). Let \(\underline{V}_{ij}=S_{\mathcal{F}_{\mathcal{K}}}^{-1}V_{ij}\) and \(\underline{\varLambda}_{i}=S_{\mathcal{F}_{\mathcal{K}}}^{-1}\tau _{V _{ij}}\varLambda _{i}\). We now prove that \(\underline{\mathcal{R}}=\{(W _{i}, \underline{V}_{ij}, \underline{\varLambda}_{i}, v_{ij})\}_{i \in \mathbb{I},{j\in \mathbb{J}_{i}}}\) is an RFF for H and we call \(\underline{\mathcal{R}}\) the \(\mathit{global}\ \mathit{relay}\ \mathit{dual}\) RFF of \(\mathcal{R}\).
3.2.2 Local relay dual of relay fusion frames
Let \(\widetilde{V}_{ij}=S_{i}^{-1}V_{ij}\) and \(\widetilde{\varLambda } _{i}=S_{i}^{-1}\tau _{V_{ij}}\varLambda _{i}\), where \(S_{i}\) denote the frame operators with respect to \(K_{i}\) for each \(i\in \mathbb{I}\) and we call every \(S_{i}\)
\(\mathit{local}\ \mathit{relay}\ \mathit{frame}\ \mathit{operator}\). We now prove that \(\widetilde{\mathcal{R}}=\{(W _{i}, \widetilde{V}_{ij}, \widetilde{\varLambda }_{i}, v_{ij})\}_{i \in \mathbb{I},{j\in \mathbb{J}_{i}}}\) is also an RFF for H and we call \(\widetilde{\mathcal{R}}\) the \(\mathit{local}\ \mathit{relay}\ \mathit{dual}\) RFF of \(\mathcal{R}\).
3.2.3 Canonical dual of relay fusion frames
Now let \(\widehat{W}_{i}=S_{\mathcal{R}}^{-1}W_{i}\) and \(\widehat{\varLambda }_{i}=\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-1}\), where \(S_{\mathcal{R}}\) is the frame operator for \(\mathcal{R}\). We prove that \(\widehat{\mathcal{R}}=\{(\widehat{W}_{i}, V_{ij}, \widehat{\varLambda }_{i}, v_{ij})\}_{i\in \mathbb{I},{j\in \mathbb{J} _{i}}}\) is also an RFF for H and we call \(\widehat{\mathcal{R}}\) the \(\mathit{canonical}\ \mathit{dual}\) RFF of \(\mathcal{R}\) for H.
Moreover, the canonical dual RFFs give rise to expansion coefficients with the minimal norm.
3.3 Q-dual relay fusion frames
The concept of Q-dual fusion frames for finite-dimensional Hilbert spaces and any separable Hilbert spaces were introduced in [
13,
14], respectively. In this subsection we transfer some definitions and results of Q-dual fusion frames to the situation of RFFs. For more information about Q-dual fusion frames, we refer to [
13,
14].
Throughout this subsection, the symbols \(\mathcal{K}_{V}, \mathcal{K}_{U}, \mathcal{R}_{V}, \mathcal{R}_{U}\), and \(\mathcal{L}_{T_{\mathcal{R}_{V}}}\) refer, respectively, to the spaces \((\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}} \oplus V_{ij} )_{\ell ^{2}}\), \((\sum_{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}\oplus U_{ij} )_{\ell ^{2}}\), the families \(\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij})\}_{i\in \mathbb{I}, j \in \mathbb{J}_{i}}\), \(\{(M_{i}, U_{ij}, \varGamma _{i}, u_{ij})\}_{i \in \mathbb{I}, j\in \mathbb{J}_{i}}\) and the collection of bounded left inverses of \(T_{\mathcal{R}_{V}}\).
In analogy with the fusion frame case (see [
14], Definition 3.1, 3.3), we introduce the following terminology.
To simplify the exposition, we just formulate the following results which are analogous to Lemma 3.4, 3.5 of [
14] with the proofs carrying over with small changes, so we omit them.