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.
Definition and basic properties of relay fusion frames and their operators
Definition 6
Let \(\{K_{i}\}_{i\in \mathbb{I}}\) be a sequence of separable Hilbert spaces and \(\{W_{i}\}_{i\in \mathbb{I}}\) be a family of closed subspaces in H and let \(\{V_{ij}\}_{j\in \mathbb{J}_{i}}\) be a family of closed subspaces in \(K_{i}\) for each \(i\in \mathbb{I}\). Let \(\{v_{ij}\}_{i \in \mathbb{I}, j\in \mathbb{J}_{i}}\) be a family of weights, i.e. \(v_{ij}>0\) for each \(i\in \mathbb{I}, j\in \mathbb{J}_{i}\), and let \(\varLambda _{i}\in \mathcal{B}(H, K_{i})\) for each \(i\in \mathbb{I}\). Then \(\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij})\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}\) is said to be a relay fusion frame, or simply r-fusion frame, if there exist constants \(0< \alpha \leqslant \beta < \infty \) such that
$$ \alpha \Vert f \Vert ^{2} \leqslant \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2}\leqslant \beta \Vert f \Vert ^{2},\quad \forall f \in H. $$
(2)
We call α and β the r-fusion frame bounds.
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].
Lemma 3.1
Let
\(\mathcal{R}\)
be a Bessel r-fusion sequence in
H
with Bessel bound
β. Then, for each sequence
\(\{f_{ij}\}_{i\in \mathbb{I}, j \in \mathbb{J}_{i}}\)
with
\(\{f_{ij}\}\in V_{ij}\)
for all
\({i\in \mathbb{I}, j\in \mathbb{J}_{i}}\), the series
\(\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} \pi _{W_{i}}\varLambda _{i}^{*}f_{ij}\)
converges unconditionally.
Proof
Let \(\mathbb{L}\) and \(\mathbb{M}\) be fixed finite subsets of \(\mathbb{I}\) and \(\mathbb{J}_{i}\), respectively. Let
$$ f=\{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}\in \biggl(\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}\oplus V _{ij} \biggr)_{\ell ^{2}} \quad\text{and}\quad g=\sum _{i\in \mathbb{L}} \sum_{j\in \mathbb{M}}v_{ij} \pi _{W_{i}}\varLambda _{i}^{*}f_{ij}. $$
Then we have
$$\begin{aligned} \Vert g \Vert &= \biggl\Vert \sum _{i\in \mathbb{L}}\sum_{j\in \mathbb{M}}v_{ij} \pi _{W_{i}}\varLambda _{i}^{*}f_{ij} \biggr\Vert \\ &=\sup_{h\in H, \Vert h \Vert =1} \biggl\vert \biggl\langle \sum _{i\in \mathbb{L}}\sum_{j\in \mathbb{M}}v_{ij} \pi _{W _{i}}\varLambda _{i}^{*}f_{ij}, h \biggr\rangle \biggr\vert \\ &=\sup_{h\in H, \Vert h \Vert =1} \biggl\vert \sum _{i\in \mathbb{L}} \sum_{j\in \mathbb{M}} \bigl\langle f_{ij}, v_{ij}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}(h) \bigr\rangle \biggr\vert \\ &\leqslant \sup_{h\in H, \Vert h \Vert =1} \biggl( \sum _{i\in \mathbb{L}}\sum_{j\in \mathbb{M}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(h) \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} \cdot \biggl(\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J} _{i}} \Vert f_{ij} \Vert ^{2} \biggr)^{\frac{1}{2}} \\ &\leqslant \sqrt{\beta } \Vert f \Vert , \end{aligned}$$
and it follows that \(\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}\pi _{W_{i}}\varLambda _{i}^{*}f_{ij}\) converges unconditionally (see [8], page 44). □
Definition 7
Let \(\mathcal{R}\) be an RFF for H. Then the analysis operator for \(\mathcal{R}\) is defined by
$$ T_{\mathcal{R}} : H \mapsto \biggl(\sum_{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}\oplus V_{ij} \biggr)_{\ell ^{2}} \quad\text{with } T_{\mathcal{R}}(f)= \bigl\{ v_{ij}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\} _{i\in \mathbb{I},{j\in \mathbb{J}_{i}}}, \forall f\in H. $$
We call the adjoint \(T_{\mathcal{R}}^{*}\) of the analysis operator the synthesis operator of \(\mathcal{R}\).
Proposition 3.2
Let
\(\mathcal{R}\)
be an RFF for H. Then
$$ T_{\mathcal{R}}^{*}(f)=\sum_{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}v_{ij}\pi _{W_{i}} \varLambda _{i}^{*}f_{ij}, \quad\forall f= \{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}\in \biggl(\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}} \oplus V_{ij} \biggr)_{\ell ^{2}}. $$
Proof
Let \(g\in H\) and \(f=\{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}} \in (\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J} _{i}}\oplus V_{ij} )_{\ell ^{2}}\). Then we compute
$$\begin{aligned} \bigl\langle T_{\mathcal{R}}(g), f \bigr\rangle &= \bigl\langle \bigl\{ v_{ij}\tau _{V _{ij}}\varLambda _{i}\pi _{W_{i}}(g) \bigr\} _{i\in \mathbb{I},{j\in \mathbb{J} _{i}}}, \{f_{ij} \}_{i\in \mathbb{I},{j\in \mathbb{J}_{i}}} \bigr\rangle \\ &= \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}} \bigl\langle g, v_{ij}\pi _{W_{i}}\varLambda _{i}^{*}(f_{ij}) \bigr\rangle = \bigl\langle g, T_{\mathcal{R}}^{*}(f) \bigr\rangle . \end{aligned}$$
□
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.
Theorem 3.3
The following assertions are equivalent:
-
(i)
\(\mathcal{R}\)
is an RFF for H.
-
(ii)
The analysis operator
\(T_{\mathcal{R}}\)
is injective and has closed range.
-
(iii)
The synthesis operator
\(T_{\mathcal{R}}^{*}\)
is bounded, linear and surjective.
Proof
(i) ⇒ (ii) For all \(f\in H\), we have
$$ \bigl\Vert T_{\mathcal{R}}(f) \bigr\Vert ^{2}=\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2}\geqslant \alpha \Vert f \Vert ^{2}, $$
which implies that \(T_{\mathcal{R}}\) is injective and has closed range.
(ii) ⇒ (i) This is obvious.
(ii) ⇔ (iii) This follows immediately from the operator-theoretics results of Hilbert spaces. □
By composing \(T_{\mathcal{R}}\) and \(T_{\mathcal{R}}^{*}\), we obtain the frame operator for \(\mathcal{R}\).
Definition 8
Let \(\mathcal{R}\) be an RFF for H. Then the frame operator \(S_{ \mathcal{R}}\) for \(\mathcal{R}\) is defined by
$$ S_{\mathcal{R}}(f)=T_{\mathcal{R}}^{*}T_{\mathcal{R}}(f)= \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{W_{i}}\varLambda _{i}^{*}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f),\quad \forall f\in H. $$
To prove Proposition 3.5 we need the following theorem that gives the relation between a Bessel r-fusion sequence and the synthesis operator \(T_{\mathcal{R}}^{*}\).
Theorem 3.4
\(\mathcal{R}\)
is a Bessel r-fusion sequence in
H
with bound
β
if and only if the map
$$ \bigl(\{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}} \bigr) \mapsto \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} \pi _{W_{i}}\varLambda _{i}^{*}f_{ij} $$
is a well-defined bounded operator from
\((\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}\oplus V _{ij} )_{\ell ^{2}}\)
to
H
and its norm is less than or equal to
\(\sqrt{\beta }\).
Proof
First assume that \(\mathcal{R}\) is a Bessel r-fusion sequence for H with bound β. By Lemma 3.1, the series \(\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} \pi _{W_{i}}\varLambda _{i}^{*}f_{ij}\) is convergent. Thus \(T_{\mathcal{R}} ^{*} (\{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}} )\) is well defined. A simple calculation as in Lemma 3.1 shows that \(T_{\mathcal{R}}^{*}\) is bounded and that \(\|T_{\mathcal{R}}^{*}\| \leqslant \sqrt{\beta }\).
For the opposite implication, suppose that \(T_{\mathcal{R}}^{*}\) is well defined and that \(\|T_{\mathcal{R}}^{*}\| \leqslant \sqrt{\beta }\). Then
$$\begin{aligned} & \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\quad=\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\langle \pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}} \varLambda _{i} \pi _{W_{i}}(f), f \bigr\rangle \\ &\quad= \bigl\langle T_{\mathcal{R}}^{*} \bigl( \bigl\{ v_{ij}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\} _{i\in \mathbb{I},{j\in \mathbb{J}_{i}}} \bigr), f \bigr\rangle \\ &\quad\leqslant \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} \bigl\Vert T_{\mathcal{R}}^{*} \bigr\Vert \Vert f \Vert . \end{aligned}$$
Now solving for \((\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2}\|\tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f)\|^{2} )^{\frac{1}{2}}\) yields
$$ \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v _{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} \biggr)^{ \frac{1}{2}}\leqslant \bigl\Vert T_{\mathcal{R}}^{*} \bigr\Vert \Vert f \Vert \leqslant \sqrt{ \beta } \Vert f \Vert . $$
□
Given an RFF, Proposition 3.5 states some of the important properties of frame operator \(S_{\mathcal{R}}\).
Proposition 3.5
Let
\(\mathcal{R}\)
be an RFF with frame bounds
α
and
β. Then the frame operator for
\(\mathcal{R}\)
is a bounded, positive, self-adjoint, invertible operator on H with
\(\alpha I_{H} \leqslant S_{\mathcal{R}}\leqslant \beta I_{H}\).
Proof
\(S_{\mathcal{R}}\) is bounded as a composition of two bounded operators. By Theorem 3.4,
$$ \Vert S_{\mathcal{R}} \Vert = \bigl\Vert T_{\mathcal{R}}^{*}T_{\mathcal{R}} \bigr\Vert = \bigl\Vert T_{ \mathcal{R}}^{*} \bigr\Vert ^{2}\leqslant \beta. $$
Since \(S_{\mathcal{R}}^{*}=(T_{\mathcal{R}}^{*}T_{\mathcal{R}})^{*}=T _{\mathcal{R}}^{*}T_{\mathcal{R}}=S_{\mathcal{R}}\), the operator \(S_{\mathcal{R}}\) is self-adjoint. The inequality (2) means that
$$ \alpha \Vert f \Vert ^{2} \leqslant \bigl\langle S_{\mathcal{R}}(f), f \bigr\rangle \leqslant \beta \Vert f \Vert ^{2},\quad \forall f\in H. $$
This shows that \(\alpha I_{H}\leqslant S_{\mathcal{R}}\leqslant \beta I_{H}\) and hence \(S_{\mathcal{R}}\) is a positive, invertible operator on H. □
Proposition 3.6
Let
\(\mathcal{R}\)
be an RFF for H with frame operator
\(S_{\mathcal{R}}\), we have then, for all
\(f\in H\),
$$ f=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\mathcal{R}}^{-1}\pi _{W_{i}}\varLambda _{i}^{*}\tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) =\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2}\pi _{W_{i}} \varLambda _{i}^{*} \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-1}(f). $$
Proof
Since \(S_{\mathcal{R}}\) is invertible, for all \(f\in H\) we have
$$\begin{aligned} f&=S_{\mathcal{R}}^{-1}S_{\mathcal{R}}(f)=\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2}S_{\mathcal{R}} ^{-1}\pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \\ &=S_{\mathcal{R}}S_{\mathcal{R}}^{-1}(f)=\sum _{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-1}(f). \end{aligned}$$
□
The following theorem gives a sufficient condition such that two Bessel r-fusion sequence become RFFs in terms of their analysis operators.
Theorem 3.7
Let
\(\mathcal{R}_{1}=\{(W_{i}, V_{ij}, \varLambda _{i}, v_{ij})\}_{i \in \mathbb{I}, j\in \mathbb{J}_{i}}\)
and
\(\mathcal{R}_{2}=\{(W^{\prime } _{i}, V^{\prime }_{ij}, \varLambda ^{\prime }_{i}, v^{\prime }_{ij})\}_{i\in \mathbb{I}, j \in \mathbb{J}_{i}}\)
be two Bessel r-fusion sequence for H with bounds
\(\beta _{1}\)
and
\(\beta _{2}\), respectively. Let
\(T_{\mathcal{R}_{1}}\)
and
\(T_{\mathcal{R}_{2}}\)
be their analysis operators such that
\(T_{\mathcal{R}_{2}}^{*}T_{\mathcal{R}_{1}}=I_{H}\). Then both
\(\mathcal{R}_{1}\)
and
\(\mathcal{R}_{2}\)
are RFFs.
Proof
For all \(f\in H\), we have
$$\begin{aligned} \Vert f \Vert ^{4}&= \bigl( \bigl\langle T_{\mathcal{R}_{1}}(f), T_{\mathcal{R}_{2}}(f) \bigr\rangle \bigr)^{2} \\ &\leqslant \bigl\Vert T_{\mathcal{R}_{1}}(f) \bigr\Vert ^{2} \bigl\Vert T_{\mathcal{R}_{2}}(f) \bigr\Vert ^{2} \\ &= \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v _{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} \biggr) \biggl( \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{\prime \,2} \bigl\Vert \tau _{V^{\prime }_{ij}}\varLambda ^{\prime }_{i}\pi _{W^{\prime }_{i}}(f) \bigr\Vert ^{2} \biggr) \\ &\leqslant \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2} \biggr)\beta _{2} \Vert f \Vert ^{2}. \end{aligned}$$
This yields
$$ \frac{1}{\beta _{2}} \Vert f \Vert ^{2} \leqslant \sum _{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2}. $$
Similarly we obtain a lower bound for \(\mathcal{R}_{2}\). □
Duality of relay fusion frames
To define the dual frames for RFFs, we need the following technical lemma.
Lemma 3.8
(see [10])
Let
\(A \in \mathcal{B}(H)\)
and
\(V \subseteq H\)
be a closed subspace. Then
$$ \pi _{V}A^{*}=\pi _{V}A^{*}\pi _{\overline{AV}}. $$
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 \(\mathcal{R}\).
Theorem 3.9
Let
\(\mathcal{R}\)
be an RFF for H. Then
\(\underline{\mathcal{R}}\)
is an RFF for
H
and, for all
\(f\in H\),
$$ f=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\underline{\mathcal{R}}}^{-1}\pi _{W_{i}} \underline{\varLambda } _{i}^{*}\underline{\varLambda }_{i}\pi _{W_{i}}(f) =\sum_{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{W_{i}}\underline{\varLambda }_{i}^{*}\underline{ \varLambda }_{i}\pi _{W_{i}}S_{\underline{\mathcal{R}}}^{-1}(f). $$
Proof
We first prove the upper bound. For each \(f\in H\), we have
$$\begin{aligned} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{S_{\mathcal{F}_{\mathcal{K}}}^{-1}V_{ij}}S_{\mathcal{F} _{\mathcal{K}}}^{-1}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} &= \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert S_{\mathcal{F}_{\mathcal{K}}}^{-1}\tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\leqslant \bigl\Vert S_{\mathcal{F}_{\mathcal{K}}}^{-1} \bigr\Vert ^{2}\beta \Vert f \Vert ^{2}. \end{aligned}$$
Now we obtain a lower bound for \(\underline{\mathcal{R}}\). We compute
$$\begin{aligned} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{S_{\mathcal{F}_{\mathcal{K}}}^{-1}V_{ij}}S_{\mathcal{F} _{\mathcal{K}}}^{-1}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} &= \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert S_{\mathcal{F}_{\mathcal{K}}}^{-1}\tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\geqslant \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J} _{i}}v_{ij}^{2}\frac{1}{ \Vert S_{\mathcal{F}_{\mathcal{K}}} \Vert ^{2}} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\geqslant \frac{\alpha }{ \Vert S_{\mathcal{F}_{\mathcal{K}}} \Vert ^{2}} \Vert f \Vert ^{2}. \end{aligned}$$
Further, since \(S_{\widetilde{\mathcal{R}}}\) is invertible, for all \(f\in H\) we have
$$\begin{aligned} f&=S_{\underline{\mathcal{R}}}^{-1}S_{\underline{\mathcal{R}}}(f)=S _{\underline{\mathcal{R}}}S_{\underline{\mathcal{R}}}^{-1}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\underline{\mathcal{R}}}^{-1}\pi _{W_{i}} \underline{\varLambda } _{i}^{*}\tau _{\underline{V}_{ij}} \underline{\varLambda }_{i}\pi _{W_{i}}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\underline{\mathcal{R}}}^{-1}\pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}S_{i}^{-1} \tau _{S_{i}^{-1}V_{ij}}S_{i}^{-1}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\underline{\mathcal{R}}}^{-1}\pi _{W_{i}} \underline{\varLambda } _{i}^{*}\underline{\varLambda }_{i}\pi _{W_{i}}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}\pi _{W_{i}}\underline{\varLambda }_{i}^{*} \underline{\varLambda }_{i} \pi _{W_{i}}S_{\underline{\mathcal{R}}}^{-1}(f). \end{aligned}$$
□
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 \(\mathcal{R}\).
Theorem 3.10
Let
\(\mathcal{R}\)
be an RFF for H. Then
\(\widetilde{\mathcal{R}}\)
is an RFF for
H
and, for all
\(f\in H\),
$$ f=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\widetilde{\mathcal{R}}}^{-1}\pi _{W_{i}} \widetilde{\varLambda } _{i}^{*}\widetilde{\varLambda }_{i}\pi _{W_{i}}(f) =\sum_{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{W_{i}}\widetilde{\varLambda }_{i}^{*}\widetilde{ \varLambda }_{i}\pi _{W_{i}}S_{\widetilde{\mathcal{R}}}^{-1}(f). $$
Proof
It is easy to show that, for all \(f\in H\),
$$\begin{aligned} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{S_{i}^{-1}V_{ij}}S_{i}^{-1}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} &= \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert S_{i}^{-1}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\leqslant \max_{i\in \mathbb{I}} \bigl\{ \bigl\Vert S_{i}^{-1} \bigr\Vert ^{2} \beta \bigr\} \Vert f \Vert ^{2}. \end{aligned}$$
Now we obtain a lower bound for \(\widetilde{\mathcal{R}}\). We compute
$$\begin{aligned} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{S_{i}^{-1}V_{ij}}S_{i}^{-1}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} &= \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert S_{i}^{-1}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\geqslant \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J} _{i}}v_{ij}^{2}\frac{1}{ \Vert S_{i} \Vert ^{2}} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}(f) \bigr\Vert ^{2} \\ &\geqslant \min_{i\in \mathbb{I}} \biggl\{ \frac{\alpha }{ \Vert S_{i} \Vert ^{2}} \biggr\} \Vert f \Vert ^{2}. \end{aligned}$$
Further, since \(S_{\widetilde{\mathcal{R}}}\) is invertible, for all \(f\in H\) we have
$$\begin{aligned} f&=S_{\widetilde{\mathcal{R}}}^{-1}S_{\widetilde{\mathcal{R}}}(f)=S _{\widetilde{\mathcal{R}}}S_{\widetilde{\mathcal{R}}}^{-1}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\widetilde{\mathcal{R}}}^{-1}\pi _{W_{i}} \widetilde{\varLambda } _{i}^{*}\tau _{\widetilde{V}_{ij}} \widetilde{\varLambda }_{i}\pi _{W_{i}}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\widetilde{\mathcal{R}}}^{-1}\pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}S_{i}^{-1} \tau _{S_{i}^{-1}V_{ij}}S_{i}^{-1}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\widetilde{\mathcal{R}}}^{-1}\pi _{W_{i}} \widetilde{\varLambda } _{i}^{*}\widetilde{\varLambda }_{i}\pi _{W_{i}}(f) \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}\pi _{W_{i}}\widetilde{\varLambda }_{i}^{*} \widetilde{\varLambda }_{i} \pi _{W_{i}}S_{\widetilde{\mathcal{R}}}^{-1}(f). \end{aligned}$$
□
Remark 3.11
Recall that there are always many different frames for global relay space \(\mathcal{K}\) and local relay space \(K_{i}\), respectively. For this reason, the global relay dual RFF \(\underline{\mathcal{R}}\) and local relay dual RFF \(\widetilde{\mathcal{R}}\) are not unique.
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.
Theorem 3.12
Let
\(\mathcal{R}\)
be an RFF for H. Then
\(\widehat{\mathcal{R}}\)
is an RFF for H.
Proof
For all \(f\in H\), we have
$$\begin{aligned} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}} S_{\mathcal{R}}^{-1} \pi _{S_{\mathcal{R}}^{-1}W_{i}}(f) \bigr\Vert ^{2} &= \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}} S_{\mathcal{R}}^{-1}(f) \bigr\Vert ^{2} \\ &\leqslant \bigl\Vert S_{\mathcal{R}}^{-1} \bigr\Vert ^{2}\beta \Vert f \Vert ^{2}. \end{aligned}$$
Now we obtain a lower bound for \(\widehat{\mathcal{R}}\). We compute
$$\begin{aligned} \Vert f \Vert ^{4}&= \biggl\vert \biggl\langle \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-1}(f), f \biggr\rangle \biggr\vert ^{2} \\ &= \biggl\vert \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v _{ij}^{2} \bigl\langle \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{ \mathcal{R}}^{-1}(f), \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}f \bigr\rangle \biggr\vert ^{2} \\ &\leqslant \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}} S_{\mathcal{R}}^{-1}\pi _{S_{\mathcal{R}}^{-1}W_{i}}(f) \bigr\Vert ^{2} \biggr) \biggl( \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}} (f) \bigr\Vert ^{2} \biggr) \\ &\leqslant \biggl(\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}}S_{\mathcal{R}}^{-1} \pi _{S_{\mathcal{R}}^{-1}W_{i}}(f) \bigr\Vert ^{2} \biggr)\beta \Vert f \Vert ^{2}, \end{aligned}$$
which implies that
$$ \frac{1}{\beta } \Vert f \Vert ^{2} \leqslant \sum _{i\in \mathbb{I}} \sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i} \pi _{W_{i}} S_{\mathcal{R}}^{-1}\pi _{S_{\mathcal{R}}^{-1}W_{i}}(f) \bigr\Vert ^{2}. $$
□
Theorem 3.13
Let
\(\mathcal{R}\)
be an RFF for H with frame operator
\(S_{\mathcal{R}}\)
and let
\(\widehat{\mathcal{R}}\)
be the canonical dual RFF of
\(\mathcal{R}\)
with frame operator
\(S_{\widehat{\mathcal{R}}}\). Then
\(S_{\mathcal{R}}S_{\widehat{\mathcal{R}}}=I_{H}\)
and
\(T_{\mathcal{R}} ^{*}T_{\widehat{\mathcal{R}}}=I_{H}\)
and, for all
\(f\in H\),
$$ f=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}\pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}\widehat{\varLambda }_{i} \pi _{\widehat{W}_{i}}(f) =\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{\widehat{W}_{i}} \widehat{\varLambda }_{i}^{*}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f). $$
Proof
For all \(f\in H\), we obtain
$$\begin{aligned} S_{\mathcal{R}}S_{\widehat{\mathcal{R}}}(f) &=S_{\mathcal{R}} \sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{S_{\mathcal{R}}^{-1}W_{i}}S_{\mathcal{R}}^{-1}\pi _{W_{i}} \varLambda _{i}^{*}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-1} \pi _{S_{\mathcal{R}}^{-1}W_{i}}(f) \\ &=S_{\mathcal{R}} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij}^{2}S_{\mathcal{R}}^{-1} \pi _{W_{i}} \varLambda _{i}^{*}\tau _{V_{ij}} \varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}} ^{-1}(f) \\ &= \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v _{ij}^{2}\pi _{W_{i}}\varLambda _{i}^{*}\tau _{V_{ij}} \varLambda _{i} \pi _{W _{i}}S_{\mathcal{R}}^{-1}(f) \\ &=S_{\mathcal{R}}S_{\mathcal{R}}^{-1}(f) \\ &=f \end{aligned}$$
and
$$\begin{aligned} T_{\mathcal{R}}^{*}T_{\widehat{\mathcal{R}}}(f) &=T_{\mathcal{R}}^{*} \bigl( \bigl\{ v_{ij}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-1} \pi _{S_{\mathcal{R}}^{-1}W_{i}}(f) \bigr\} _{i\in \mathbb{I},{j\in \mathbb{J}_{i}}} \bigr) \\ &=T_{\mathcal{R}}^{*} \bigl( \bigl\{ v_{ij}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S _{\mathcal{R}}^{-1}(f) \bigr\} _{i\in \mathbb{I},{j\in \mathbb{J}_{i}}} \bigr) \\ &=T_{\mathcal{R}}^{*}T_{\mathcal{R}}S_{\mathcal{R}}^{-1}(f) \\ &=f. \end{aligned}$$
The last assertion of the theorem follows from the previous steps of the proof. □
Moreover, the canonical dual RFFs give rise to expansion coefficients with the minimal norm.
Theorem 3.14
Let
\(\mathcal{R}\)
be an RFF with canonical dual RFF
\(\widehat{\mathcal{R}}\). Then, for any
\(g_{ij}\in V_{ij}\)
satisfying
\(f=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v _{ij}^{2}\pi _{W_{i}}\varLambda _{i}^{*}g_{ij}\), we have
$$ \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}} \Vert g_{ij} \Vert ^{2} =\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v _{ij}^{2} \bigl\Vert \tau _{V_{ij}}\widehat{ \varLambda }_{i}\pi _{\widehat{W}_{i}}(f) \bigr\Vert ^{2} + \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J} _{i}} \bigl\Vert g_{ij}-v_{ij}^{2}\tau _{V_{ij}}\widehat{\varLambda }_{i} \pi _{\widehat{W_{i}}}(f) \bigr\Vert ^{2}. $$
Proof
We compute
$$\begin{aligned} \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{V_{ij}}\widehat{\varLambda }_{i}\pi _{\widehat{W_{i}}}(f) \bigr\Vert ^{2} &= \bigl\langle S_{\widehat{\mathcal{R}}}(f), f \bigr\rangle \\ &= \bigl\langle f, S_{\mathcal{R}}^{-1}(f) \bigr\rangle \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\langle \pi _{W_{i}}\varLambda _{i}^{*}g_{ij}, S_{\mathcal{R}}^{-1}(f) \bigr\rangle \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\langle g_{ij}, \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}} ^{-1}(f) \bigr\rangle \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\langle g_{ij}, \tau _{V_{ij}}\widehat{ \varLambda }_{i} \pi _{\widehat{W}_{i}}(f) \bigr\rangle \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\langle \tau _{V_{ij}}\widehat{\varLambda }_{i}\pi _{\widehat{W}_{i}}(f), g_{ij} \bigr\rangle , \end{aligned}$$
which finishes the proof. □
Example 3.15
In classical frame theory we can always construction a Parseval frame by applying \(S_{\mathcal{F}}^{-\frac{1}{2}}\), where \(S_{\mathcal{F}}\) denote the frame operator of frame \(\mathcal{F}\). For the situation of RFFs is similarly. In fact, for all \(f\in H\),
$$\begin{aligned} \Vert f \Vert ^{2}= \bigl\langle S_{\mathcal{R}}^{-\frac{1}{2}}S_{\mathcal{R}}S _{\mathcal{R}}^{-\frac{1}{2}}(f), f \bigr\rangle &= \biggl\langle \sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v_{ij} ^{2}S_{\mathcal{R}}^{-\frac{1}{2}} \pi _{W_{i}}\varLambda _{i}^{*}\tau _{V _{ij}} \varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{-\frac{1}{2}}(f), f \biggr\rangle \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}S_{\mathcal{R}}^{- \frac{1}{2}}(f) \bigr\Vert ^{2} \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2} \bigl\Vert \tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}} S_{\mathcal{R}}^{- \frac{1}{2}}\pi _{S_{\mathcal{R}}^{-\frac{1}{2}}W_{i}}(f) \bigr\Vert ^{2}; \end{aligned}$$
therefore, \(\{(S_{\mathcal{R}}^{-\frac{1}{2}}W_{i}, V_{ij}, \varLambda _{i}\pi _{W_{i}} S_{\mathcal{R}}^{-\frac{1}{2}}, v_{ij})\}_{i\in \mathbb{I},{j\in \mathbb{J}_{i}}}\) is a Parseval RFF for H.
Example 3.16
We introduce atomic resolutions of an operator R on H. Let \(\mathcal{R}\) be an RFF for H. Suppose that \(\widehat{\mathcal{R}}\) is the canonical dual RFF of \(\mathcal{R}\). Then from Theorem 3.13, we have for all \(f\in H\)
$$ f=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}\pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}\widehat{\varLambda }_{i} \pi _{\widehat{W}_{i}}(f) =\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{\widehat{W}_{i}} \widehat{\varLambda }_{i}^{*}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}(f). $$
This implies that
$$ I_{H}=\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v _{ij}^{2}\pi _{W_{i}}\varLambda _{i}^{*}\tau _{V_{ij}} \widehat{\varLambda } _{i}\pi _{\widehat{W}_{i}} =\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{\widehat{W}_{i}} \widehat{\varLambda }_{i}^{*}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}, $$
and the series are convergent in the weak* sense. Let \(R\in \mathcal{B}(H)\). As can be seen from the discussion above
$$\begin{aligned} R&=\sum_{i\in \mathbb{I}}\sum _{j\in \mathbb{J}_{i}}v _{ij}^{2}\pi _{W_{i}}\varLambda _{i}^{*}\tau _{V_{ij}} \widehat{\varLambda } _{i}\pi _{\widehat{W}_{i}}R =\sum _{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2} \pi _{\widehat{W}_{i}} \widehat{\varLambda }_{i}^{*}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}R \\ &=\sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij} ^{2}R\pi _{W_{i}}\varLambda _{i}^{*} \tau _{V_{ij}}\widehat{\varLambda }_{i} \pi _{\widehat{W}_{i}} = \sum_{i\in \mathbb{I}}\sum_{j\in \mathbb{J}_{i}}v_{ij}^{2}R \pi _{\widehat{W}_{i}} \widehat{\varLambda }_{i}^{*}\tau _{V_{ij}}\varLambda _{i}\pi _{W_{i}}. \end{aligned}$$
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.
Definition 9
Let \(\mathcal{R}_{V}\) and \(\mathcal{R}_{U}\) be two RFFs for H. If there exists \(Q\in \mathcal{B}(\mathcal{K}_{V}, \mathcal{K}_{U})\) such that
$$ T_{\mathcal{R}_{U}}^{*}QT_{\mathcal{R}_{V}}=I_{H}, $$
then \(\mathcal{R}_{U}\) is said to be a Q-dual RFF of \(\mathcal{R}_{V}\).
Definition 10
Let \(P_{(m,n)} : \mathcal{K}_{V} \mapsto \mathcal{K}_{U}\), \(P_{(m,n)} \{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}}=\{ \delta _{\{(m,n),(i,j)\}}f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J} _{i}}\), where \(\delta _{\{(m,n),(i,j)\}}\) is the Kronecker delta. If Q in Definition 9 satisfies
$$ QP_{(m,n)}\mathcal{K}_{V}=P_{(m,n)} \mathcal{K}_{U}, $$
we say that Q is component preserving and \(\mathcal{R}_{U}\) is a component preserving dual RFF of \(\mathcal{R}_{V}\).
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.
Lemma 3.17
Let
\(\mathcal{R}_{V}\)
be an RFF for H. If
\(\mathcal{R}_{U}\)
is a component preserving dual RFF of
\(\mathcal{R}_{V}\), then
\(U_{(m,n)}=AP _{(m,n)}\mathcal{K}_{V}\), for each
\(i\in \mathbb{I}, {j\in \mathbb{J} _{i}}\), where
\(A\in \mathcal{L}_{T_{\mathcal{R}_{V}}}\).
Theorem 3.18
Let
\(\mathcal{R}_{V}\)
be an RFF for H, \(A\in \mathcal{L}_{T_{ \mathcal{R}_{V}}}\)
and
\(U_{(m,n)}=AP_{(m,n)}\mathcal{K}_{V}\), for each
\(i\in \mathbb{I}, {j\in \mathbb{J}_{i}}\). If
\(\mathcal{R}_{U}\)
is a Bessel r-fusion sequence and
$$ Q_{A} : \mathcal{K}_{V} \mapsto \mathcal{K}_{U}, Q_{A}\{f_{ij}\}_{i \in \mathbb{I}, j\in \mathbb{J}_{i}}= \biggl\{ \frac{1}{u_{ij}}AP_{(m,n)} \{f_{ij}\}_{i\in \mathbb{I}, j\in \mathbb{J}_{i}} \biggr\} _{m\in \mathbb{I}, n\in \mathbb{J}_{i}} $$
is a well-defined bounded operator, then
\(\mathcal{R}_{U}\)
is a
\(Q_{A}\)-component preserving the dual RFF of
\(\mathcal{R}_{V}\).
Remark 3.19
As discussed in [14], Remark 3.6, 3.10, we can always find the conditions for \(\mathcal{R}_{U}\) being a Bessel r-fusion sequence and for \(Q_{A}\) being a well defined bounded operator. In particular, the hypotheses \(\mathcal{R}_{U}\) to be a Bessel r-fusion sequence and \(Q_{A}\) to be a well-defined bounded operator cannot be avoided.