Some Properties of Relay Fusion Frames in Finite Dimensions

Abstract

In this paper, we analyze the relationship between relay fusion frames and standard fusion frames in finite-dimensional real Hilbert spaces. We propose an optimal design method for tight relay fusion frames in the setting of orthogonal subspaces. Additionally, we prove the existence of non-trivial relay operators and establish stability results for both subspaces and relay operators, showing that small perturbations preserve the relay fusion frame property with frame bounds converging to the original ones. We also present a sufficient condition for constructing relay fusion frames from scaled operators of existing fusion frames and show that invertible relay operators induce fusion frames.

1. Introduction

In the field of signal processing and harmonic analysis, the rapid development of distributed systems has raised higher demands for efficient signal perception and robust recovery. Traditional single sensor systems often encounter bottlenecks such as hardware costs, sampling rates, energy consumption, or physical limitations in practical applications. A single high quality sensor may be expensive, inaccessible, or unable to handle high complexity signals [1]. To address this challenge, Casazza and Kutyniok introduced the concept of a fusion frame in the early 21st century, providing a powerful mathematical tool for distributed signal processing [2].

1.1. Existing Work and Research Gap

A fusion frame decomposes a Hilbert space into a family of closed subspaces, where each subspace corresponds to a local sensing node. By orthogonal projections onto these subspaces, the original signal is split into multiple local components, which are then measured by low quality sensors and fused via a frame operator to reconstruct the global signal [3]. The core advantage of the fusion frames lies in its ability to utilize redundant information between subspaces, ensuring stable, robust, and accurate signal recovery, thereby overcoming the limitations of single sensor systems [4]. This framework has been successfully applied in diverse fields, including synthetic aperture radar imaging, magnetic resonance imaging reconstruction, doppler signal denoising, and wireless sensor networks [5].

However, practical distributed systems often involve more complex constraints that traditional fusion frames cannot fully address [6]. Different local sensors may have varying reliability, sensitivity, or importance, and local measurements may not be directly comparable due to differences in hardware, data format, or sampling dimensions, often requiring linear transformation to a unified space before fusion. In scenarios like multichannel imaging or remote sensing, local signals may need to be relayed through intermediate processing to match the fusion frame. Moreover, many real-world applications, including sensor networks, distributed computing, and multi-channel imaging, operate in finite dimensions, making explicit constructions and bounds crucial for implementation.

To address these gaps, the relay fusion frame is proposed as a natural and practical extension of the classical fusion frame [6]. By introducing a key component, the relay operator, it enhances the flexibility and adaptability of distributed signal processing. This extension retains the core advantages of the fusion framework and simultaneously meets the practical requirements of heterogeneous distributed systems [7]. To date, most research on relay fusion frames has concentrated on infinite-dimensional spaces, while finite-dimensional cases have received relatively little attention. Therefore, this paper analyzes and studies relay fusion frames in finite-dimensional spaces. We establish the basic properties of such frames, including analysis operators and synthesis operators, and derive reconstruction formulas. We further analyze the relationship between relay fusion frames and standard fusion frames, with a focus on discussing the optimal fusion frame bounds. Finally, we present stability results regarding perturbations of subspaces and relay operators.

1.2. Our Contribution

Our contribution to the theory of relay fusion frames is four-fold. First, the optimal design of tight relay fusion frames for orthogonal subspaces (Theorem 1) that maximizes the frame bound ratio under a trace constraint is a problem not studied in prior relay fusion frame literature. Second, we establish an existence theorem for non-trivial relay operators (Theorem 4), which constructively shows that non-scalar, non-projection relay operators always exist for any spanning set of subspaces. Third, we provide stability theorems (Theorems 3 and 6) that specifically address coupled perturbations of both subspaces and relay operators, a scenario not covered by classical fusion frame perturbation results. And fourth, we give a sufficient condition for constructing relay fusion frames via scaled operators (Theorem 2) and the induced fusion frame result for invertible relay operators (Theorem 5), which bridge relay fusion frames with standard fusion frames in a novel way.

2. Preliminaries

Let ${\mathbb{R}}^n$ be an n-dimensional real Hilbert space equipped with the standard Euclidean inner product $⟨·,·⟩$ and the induced norm $\parallel · \parallel$. For a linear operator $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$, its adjoint is denoted by $T^{*}$. The operator norm of T is $\parallel T\parallel ={sup}_{\parallel x\parallel =1}\parallel Tx\parallel$. For a subspace $X\subseteq {\mathbb{R}}^n$, the orthogonal projection onto X is denoted by $P_X$. The identity operator on ${\mathbb{R}}^n$ is denoted by $I_n$ or simply I.

A central concept in signal processing is that of a frame [8], which provides a redundant but stable representation of signals. Traditional frame theory revolutionized harmonic analysis by allowing for non-unique representations of vectors in Hilbert spaces [9]. This redundancy proved invaluable for noise reduction and error correction [10].

Definition 1.

A family of vectors ${\left\{f_j\right\}}_{j=1}^J\subset {\mathbb{R}}^n$ is a frame for ${\mathbb{R}}^n$ if there exist constants $0<A\le B<\infty$ such that

$${A\parallel g\parallel }^2\le \sum _{j=1}^J|⟨g,f_j⟩{|}^2\le B{\parallel g\parallel }^2,\forall g\in {\mathbb{R}}^n.$$

The constants A and B are called the frame bounds.

Fusion frames extend this idea from vectors to subspaces, modeling a system of distributed sensors, each capturing a component of the signal in a specific subspace. In addition to the practical application requirements, the fusion frames also make significant contributions in the theoretical aspect. For instance, Balazs et al. investigate the matrix representation of operators on Hilbert spaces using fusion frames and present applications to solving operator equations, overlapped convolution, and non-standard wavelet representation of operators [11]. Ellouz et al. establish the existence of non-orthogonal fusion frames for a family of analytic perturbed operators with eigenvalues of finite multiplicity [12]. For more information about the fusion frames, refer to references [13,14,15].

Definition 2.

Let ${\left\{X_i\right\}}_{i=1}^m$ be a family of subspaces of ${\mathbb{R}}^n$, and let ${\left\{x_i\right\}}_{i=1}^m\subset \mathbb{R}$ be a family of positive weights. The family ${\left\{(X_i,x_i)\right\}}_{i=1}^m$ is called a fusion frame for ${\mathbb{R}}^n$ if there exist constants $0<A\le B<\infty$ such that

$${A\parallel g\parallel }^2\le \sum _{i=1}^m{x}_i^2\parallel P_{X_i}{\left(g\right)\parallel }^2\le B{\parallel g\parallel }^2,\forall g\in {\mathbb{R}}^n.$$

(1)

The constants A and B are called the fusion frame bounds. The fusion frame is called tight if $A=B$, and Parseval if $A=B=1$.

To model more complex distributed processing where local measurements undergo linear transformations before fusion, the concept of a fusion frame is generalized [6].

Definition 3.

Let ${\left\{X_i\right\}}_{i=1}^m$ be a family of subspaces of ${\mathbb{R}}^n$. Let ${\left\{x_i\right\}}_{i=1}^m\subset \mathbb{R}$ be positive weights, and let ${\mathbf{T}}_i\in {\mathbb{R}}^{d_i\times n}$ be an operator for each $i\in \{1,\dots ,m\}$. The family ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ is called a relay fusion frame for ${\mathbb{R}}^n$ if there exist constants $0<C\le D<\infty$ such that

$${C\parallel g\parallel }^2\le \sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2\le D{\parallel g\parallel }^2,\forall g\in {\mathbb{R}}^n.$$

(2)

The operators ${\left\{{\mathbf{T}}_i\right\}}_{i=1}^m$ are called the relay operators. The constants C and D are the relay fusion frame bounds.

Remark 1.

If ${\mathbf{T}}_i=I_n$ for all i, a relay fusion frame reduces to a standard fusion frame. In addition, if each subspace $X_i$ is one dimensional, it reduces to a standard vector frame.

The analysis of relay fusion frames is facilitated by associated linear operators. For this, we consider the Hilbert space of sequences

$$\mathbb{F}:=\underset{i=1}{\overset{m}{⨁}}{\mathbb{R}}^{d_i}=\left\{\mathbf{y}={\left(y_i\right)}_{i=1}^m\mid y_i\in {\mathbb{R}}^{d_i}\right\},$$

equipped with the inner product ${⟨\mathbf{y},\mathbf{z}⟩}_{\mathbb{F}}={\sum }_{i=1}^m{⟨y_i,z_i⟩}_{{\mathbb{R}}^{d_i}}$ and norm ${\parallel \mathbf{y}\parallel }_{\mathbb{F}}^2={\sum }_{i=1}^m{\parallel y_i\parallel }^2$.

Definition 4.

Given a relay fusion frame ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ for ${\mathbb{R}}^n$, its analysis operator is defined by

$$\mathcal{F}:{\mathbb{R}}^n\to \mathbb{F},\mathcal{F}\left(g\right)=\left(x_1{\mathbf{T}}_1{P}_{X_1}\left(g\right),x_2{\mathbf{T}}_2{P}_{X_2}\left(g\right),\dots ,x_m{\mathbf{T}}_m{P}_{X_m}\left(g\right)\right),\forall g\in {\mathbb{R}}^n.$$

The relay fusion frame condition (2) is equivalent to

$${C\parallel g\parallel }^2\le {\parallel \mathcal{F}\left(g\right)\parallel }_{\mathbb{F}}^2\le D{\parallel g\parallel }^2,\forall g\in {\mathbb{R}}^n,$$

which implies $\mathcal{F}$ is a bounded, injective linear operator. The Hilbert space $\mathbb{F}$ is called the analysis space of relay fusion frame ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$.

Definition 5.

The synthesis operator ${\mathcal{F}}^{*}:\mathbb{F}\to {\mathbb{R}}^n$ is the adjoint of $\mathcal{F}$. For any $\mathbf{y}=(y_1,\dots ,y_m)\in \mathbb{F}$, it acts as

$${\mathcal{F}}^{*}\left(\mathbf{y}\right)=\sum _{i=1}^m{x}_i{P}_{X_i}{\mathbf{T}}_i^{*}{y}_i.$$

The composition of these operators yields the central object for reconstruction.

Definition 6.

The frame operator $S:{\mathbb{R}}^n\to {\mathbb{R}}^n$ associated with the relay fusion frame is defined by

$$S={\mathcal{F}}^{*}\mathcal{F}.$$

Equivalently, for any $g\in {\mathbb{R}}^n$,

$$S\left(g\right)=\sum _{i=1}^m{x}_i^2{P}_{X_i}{\mathbf{T}}_i^{*}{\mathbf{T}}_i{P}_{X_i}\left(g\right).$$

A direct calculation shows that $⟨S\left(g\right),g⟩={\parallel \mathcal{F}\left(g\right)\parallel }_{\mathbb{F}}^2$. Therefore, the relay fusion frame condition (2) is equivalent to

$$CI\le S\le DI,$$

where the inequality is in the sense of self-adjoint operators. This implies that S is a positive, invertible operator on ${\mathbb{R}}^n$. Its inverse $S^{-1}$ satisfies $D^{-1}I\le S^{-1}\le C^{-1}I$. The invertibility of the frame operator provides a standard reconstruction formula, which is the foundation for recovering a signal from its relay fused measurements.

Proposition 1.

Let ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ be a relay fusion frame for ${\mathbb{R}}^n$ with frame operator S. Then, for any $g\in {\mathbb{R}}^n$,

$$g=\sum _{i=1}^m{x}_i^2{S}^{-1}{P}_{X_i}{\mathbf{T}}_i^{*}{\mathbf{T}}_i{P}_{X_i}\left(g\right)=\sum _{i=1}^m{x}_i^2{P}_{X_i}{\mathbf{T}}_i^{*}{\mathbf{T}}_i{P}_{X_i}{S}^{-1}\left(g\right).$$

(3)

Furthermore, the family ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ is a Parseval relay fusion frame if and only if $S=I$.

Remark 2.

The reconstruction Formula (3) demonstrates how the global signal g can be stably recovered from the local, relayed measurements ${\left\{x_i{\mathbf{T}}_i{P}_{X_i}\left(g\right)\right\}}_{i=1}^m$ via the inverse frame operator $S^{-1}$. This mirrors the classical frame reconstruction but incorporates the subspace projections and relay transformations.

3. Main Results

In this section, we focus on the core properties of relay fusion frames, including their construction conditions, optimal frame bounds, perturbation stability, the existence of non-trivial relay operators, and the connection with traditional fusion frames. We begin with a setting where the subspaces are mutually orthogonal, which leads to explicit constructions and optimal tight relay fusion frame designs.

Theorem 1.

Let $F={\left\{(X_i,x_i)\right\}}_{i=1}^m$ be a fusion frame for ${\mathbb{R}}^n$ with optimal frame bounds $0<a_F\le b_F<\infty$. Assume that ${\left\{X_i\right\}}_{i=1}^m$ are mutually orthogonal and $n_i=dim\left(X_i\right)$. Suppose that ${\lambda }_i>0$ and ${\mathbf{T}}_i\in {\mathbb{R}}^{d_i\times n}$ satisfy ${\mathbf{T}}_i^{*}{\mathbf{T}}_i={\lambda }_i{P}_{X_i}$. Then the following statements hold:

1. 

$R={\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ is a relay fusion frame with frame operator $S_R={\sum }_{i=1}^m{x}_i^2{\lambda }_i{P}_{X_i}$.

2. 

Let $a_R,b_R$ be the optimal bounds of R, ${\gamma }_R=a_R/b_R$, and $tr\left(S_R\right)=t>0$. The unique choice maximizing ${\gamma }_R$ is ${\lambda }_i^{*}={\displaystyle \frac{t}{x_i^2{\sum }_{j=1}^m{n}_j}}$, making $R^{*}$ tight with ${\alpha }_{R^{*}}={\beta }_{R^{*}}={\displaystyle \frac{t}{{\sum }_j{n}_j}}$. Moreover, for $t=a_F+b_F$, ${\gamma }_{R^{*}}=1$.

3. 

${\gamma }_{R^{*}}=1\ge {\gamma }_F={\displaystyle \frac{a_F}{b_F}}$, with equality if and only if F is tight.

Proof. 

1. For any $g\in {\mathbb{R}}^n$, write $g={\sum }_{i=1}^m{g}_i$ with $g_i=P_{X_i}\left(g\right)$. Then we have ${\parallel g\parallel }^2={\sum }_{i=1}^m{\parallel g_i\parallel }^2$ and $P_{X_i}\left(g\right)=g_i$. Since ${\mathbf{T}}_i^{*}{\mathbf{T}}_i={\lambda }_i{P}_{X_i}$,

$$\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2=⟨{\mathbf{T}}_i{g}_i,{\mathbf{T}}_i{g}_i⟩={\lambda }_i{\parallel g_i\parallel }^2.$$

Therefore,

$$\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2=\sum _{i=1}^m{x}_i^2{\lambda }_i{\parallel g_i\parallel }^2.$$

Since ${\lambda }_i>0$ and the numbers ${\left\{x_i^2{\lambda }_i\right\}}_{i=1}^m$ are finite and positive, set

$$a_R=\underset{i}{min}{x}_i^2{\lambda }_i,b_R=\underset{i}{max}{x}_i^2{\lambda }_i.$$

Then for any $g\in {\mathbb{R}}^n$,

$$a_R{\parallel g\parallel }^2\le \sum _{i=1}^m{x}_i^2{\lambda }_i\parallel g_i{\parallel }^2\le b_R{\parallel g\parallel }^2.$$

Thus $R={\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ is a relay fusion frame with bounds $a_R,b_R$. The frame operator $S_R$ satisfies

$$⟨S_{R}g,g⟩=\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right){\parallel }^2=\sum _{i=1}^m{x}_i^2{\lambda }_i{\parallel P_{X_i}\left(g\right)\parallel }^2=⟨\left(\sum _{i=1}^m{x}_i^2{\lambda }_i{P}_{X_i}\right)g,g⟩,$$

so $S_R={\sum }_{i=1}^m{x}_i^2{\lambda }_i{P}_{X_i}$.

2. Let $u_i=x_i^2{\lambda }_i>0$. Then the optimal bounds are $a_R={min}_i{u}_i$ and $b_R={max}_i{u}_i$, and ${\gamma }_R=a_R/b_R$. The trace constraint is

$$\mathrm{tr}(S_R)=\sum _{i=1}^m{n}_i{u}_i=t>0.$$

We maximize ${\gamma }_R$ subject to this constraint. The frame bound ratio ${\gamma }_R\le 1$ for all $\left\{u_i\right\}$, with equality if and only if all $u_i=c$, where c is a constant. This is a necessary condition for the maximum. If $u_i\ne u_j$ for some $i,j$, then ${min}_i{u}_i/{max}_i{u}_i<1$. For equality, set $u_i=c$ for all i. The trace constraint gives

$$c\sum _{i=1}^m{n}_i=t\Rightarrow c={\displaystyle \frac{t}{{\sum }_{j=1}^m{n}_j}}.$$

Since $u_i=x_i^2{\lambda }_i$, we obtain ${\lambda }_i^{*}=c/x_i^2={\displaystyle \frac{t}{x_i^2{\sum }_{j=1}^m{n}_j}}$, which is the unique optimal choice, and equality is sufficient for achieving the maximum. the corresponding relay fusion frame $R^{*}$ has frame operator

$$S_{R^{*}}=\sum _{i=1}^m{x}_i^2{\lambda }_i^{*}{P}_{X_i}=\sum _{i=1}^m{\displaystyle \frac{t}{{\sum }_{j=1}^m{n}_j}}{P}_{X_i}={\displaystyle \frac{t}{{\sum }_{j=1}^m{n}_j}}I,$$

so $R^{*}$ is tight with ${\alpha }_{R^{*}}={\beta }_{R^{*}}={\displaystyle \frac{t}{{\sum }_{j=1}^m{n}_j}}$ and ${\gamma }_{R^{*}}=1$. The choice ${\lambda }_i^{*}$ is unique because equality in ${\gamma }_R=1$ forces all $u_i$ equal, and the constraint then determines their common value. The optimal ${\lambda }_i^{*}$ is unique because the only way to achieve ${\gamma }_R=1$ (the global maximum) is to set all $u_i=c$. Any other choice of $\left\{{\lambda }_i\right\}$ results in $u_i\ne u_j$ for some $i,j$, so ${\gamma }_R<1$. The trace constraint uniquely determines c, hence uniquely determines ${\lambda }_i^{*}$. For $t=a_F+b_F$, we trivially have ${\gamma }_{R^{*}}=1$.

3. The original fusion frame has ${\gamma }_F=a_F/b_F={min}_i{x}_i^2/{max}_i{x}_i^2\le 1$. The optimal relay fusion frame has ${\gamma }_{R^{*}}=1$, so ${\gamma }_{R^{*}}\ge {\gamma }_F$. Equality holds if and only if ${min}_i{x}_i^2/{max}_i{x}_i^2=1$, i.e., all $x_i^2$ equal, meaning F is tight. □

Remark 3.

The mathematical mechanism of the frame bound ratio improvement is imbalance compensation. The original fusion frame has an unbalanced ratio ${\gamma }_F<1$ due to heterogeneous weights $x_i^2$ and subspace dimensions $n_i$. The optimal ${\lambda }_i^{*}$ compensates for this imbalance by scaling the relay operator, resulting in a tight frame operator and the maximum possible ratio ${\gamma }_R=1$.

Remark 4.

The general form of ${\mathbf{T}}_i$ satisfying ${\mathbf{T}}_i^{*}{\mathbf{T}}_i={\lambda }_i{P}_{X_i}$ is

$${\mathbf{T}}_i=B_i{P}_{X_i},$$

where $B_i:X_i\to {\mathbb{R}}^{d_i}$ is any operator such that $B_i^{*}{B}_i={\lambda }_i{I}_{X_i}$. Here, $I_{X_i}$ denotes the identity operator on $X_i$. Equivalently, ${\displaystyle \frac{1}{\sqrt{{\lambda }_i}}}{B}_i$ is an isometric embedding of $X_i$ into ${\mathbb{R}}^{d_i}$.

Example 1.

Let $B_i=\sqrt{{\lambda }_i}i{d}_{X_i}$, where $i{d}_{X_i}:X_i\to {\mathbb{R}}^n$ is the inclusion map. Then ${\mathbf{T}}_i=\sqrt{{\lambda }_i}{P}_{X_i}$.

Example 2.

Let $U_i:X_i\to {\mathbb{R}}^{d_i}$ be any isometric embedding. Then $B_i=\sqrt{{\lambda }_i}{U}_i$ works, giving ${\mathbf{T}}_i=\sqrt{{\lambda }_i}{U}_i{P}_{X_i}$.

Example 3.

Choose an orthonormal basis $\{e_{i,1},\dots ,e_{i,n_i}\}$ for $X_i$. For ${\mathbb{R}}^{d_i}$ containing an orthogonal set $\{g_{i,1},\dots ,g_{i,n_i}\}$ with $\parallel g_{i,k}\parallel =\sqrt{{\lambda }_i}$, define $B_i\left(e_{i,k}\right)=g_{i,k}$. Then $B_i^{*}{B}_i={\lambda }_i{I}_{X_i}$.

The following theorem establishes a connection between standard fusion frames and relay fusion frames in finite dimensions, providing a sufficient condition for constructing relay fusion frames from scaled operators of existing fusion frames.

Theorem 2.

Let $F={\left\{(X_i,x_i)\right\}}_{i=1}^m$ be a fusion frame for ${\mathbb{R}}^n$ with optimal frame bounds $0<a_F\le b_F<\infty$. Suppose ${\mathbf{T}}_i$ are bounded linear operators with lower and upper bounds $m_i,M_i$ on $X_i$. Define the scaled operators ${\mathbf{B}}_i=c_i{\mathbf{T}}_i$ with $c_i>0$. If there exist constants $c,C>0$ satisfying $c\le c_i{m}_i$ and $c_i{M}_i\le C$ for all i, then the system ${\left\{(X_i,{\mathbf{B}}_i,x_i)\right\}}_{i=1}^m$ is a relay fusion frame for ${\mathbb{R}}^n$ with frame bounds $c^2{\alpha }_F$ and $C^2{\beta }_F$.

Proof. 

For any $g\in {\mathbb{R}}^n$, let $g_i=P_{X_i}\left(g\right)\in X_i$. Then

$$\parallel {\mathbf{B}}_i{g}_i\parallel =c_i\parallel {\mathbf{T}}_i{g}_i\parallel \mathrm{and}{m}_i\parallel g_i\parallel \le \parallel {\mathbf{T}}_i{g}_i\parallel \le M_i\parallel g_i\parallel .$$

Multiplying by $c_i$,

$$c_i{m}_i\parallel g_i\parallel \le \parallel {\mathbf{B}}_i{g}_i\parallel \le c_i{M}_i\parallel g_i\parallel .$$

Using $c\le c_i{m}_i$ and $c_i{M}_i\le C$,

$$c\parallel g_i\parallel \le \parallel {\mathbf{B}}_i{g}_i\parallel \le C\parallel g_i\parallel .$$

Squaring and multiplying by $x_i^2$,

$$c^2{x}_i^2\parallel g_i{\parallel }^2\le x_i^2\parallel {\mathbf{B}}_i{g}_i{\parallel }^2\le C^2{x}_i^2{\parallel g_i\parallel }^2.$$

Summing over i,

$$c^2\sum _{i=1}^m{x}_i^2\parallel g_i{\parallel }^2\le \sum _{i=1}^m{x}_i^2\parallel {\mathbf{B}}_i{P}_{X_i}{\left(g\right)\parallel }^2\le C^2\sum _{i=1}^m{x}_i^2{\parallel g_i\parallel }^2.$$

Now use the fusion frame bounds for F,

$$a_F{\parallel g\parallel }^2\le \sum _{i=1}^m{x}_i^2\parallel g_i{\parallel }^2\le b_F{\parallel g\parallel }^2.$$

Therefore

$$c^2{a}_F{\parallel g\parallel }^2\le \sum _{i=1}^m{x}_i^2\parallel {\mathbf{B}}_i{P}_{X_i}{\left(g\right)\parallel }^2\le C^2{b}_F{\parallel g\parallel }^2.$$

□

Now, we establish a perturbation result for relay fusion frames, which are mathematical models for sensor networks with data relaying capabilities.

Theorem 3.

Let ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ be a relay fusion frame for ${\mathbb{R}}^n$ with frame bounds $a_R,b_R>0$. For each i, let $X_i^{\epsilon }$ be a subspace such that $\parallel P_{X_i}-P_{X_i^{\epsilon }}\parallel \le \epsilon$, and define

$${\mathbf{T}}_i^{\epsilon }={\mathbf{T}}_i+{\mathbf{T}}_i(P_{X_i}-P_{X_i^{\epsilon }}).$$

Let $M={\sum }_{i=1}^m{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\parallel }^2$. If $\epsilon >0$ is sufficiently small such that $\sqrt{a_R}-\epsilon \sqrt{M}>0$, then ${\left\{(X_i^{\epsilon },{\mathbf{T}}_i^{\epsilon },x_i)\right\}}_{i=1}^m$ is also a relay fusion frame, and there exists a constant $C>0$ such that

$$|a_{R^{\epsilon }}-a_R|\le C\epsilon ,|b_{R^{\epsilon }}-b_R|\le C\epsilon ,$$

where $a_{R^{\epsilon }},b_{R^{\epsilon }}$ are frame bounds for the perturbed system.

Proof. 

Let ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ be a relay fusion frame with analysis operator $\mathcal{F}$ and frame bounds $a_R,b_R>0$. For each i, define $D_i=P_{X_i}-P_{X_i^{\epsilon }}$ with $\parallel D_i\parallel \le \epsilon$, and let ${\mathbf{T}}_i^{\epsilon }={\mathbf{T}}_i+{\mathbf{T}}_i{D}_i$. Then the perturbed analysis operator satisfies

$${\mathcal{F}}_{\epsilon }\left(g\right)={\left(x_i{\mathbf{T}}_i^{\epsilon }{P}_{X_i^{\epsilon }}\left(g\right)\right)}_{i=1}^m={\left(x_i{\mathbf{T}}_i{P}_{X_i}{P}_{X_i^{\epsilon }}\left(g\right)\right)}_{i=1}^m=\mathcal{F}\left(g\right)-{\left(x_i{\mathbf{T}}_i{P}_{X_i}{D}_i\left(g\right)\right)}_{i=1}^m,$$

so ${\mathcal{F}}_{\epsilon }=\mathcal{F}-\Delta \mathcal{F}$ with $\parallel \Delta \mathcal{F}\parallel \le \epsilon \sqrt{M}$, where $M={\sum }_{i=1}^m{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\parallel }^2<\infty$. For sufficiently small $\epsilon$, we have for all $g\in {\mathbb{R}}^n$,

$$\parallel {\mathcal{F}}_{\epsilon }\left(g\right)\parallel \ge \parallel \mathcal{F}\left(g\right)\parallel -\parallel \Delta \mathcal{F}\left(g\right)\parallel \ge (\sqrt{a_R}-\epsilon \sqrt{M})\parallel g\parallel ,$$

$$\parallel {\mathcal{F}}_{\epsilon }\left(g\right)\parallel \le \parallel \mathcal{F}\left(g\right)\parallel +\parallel \Delta \mathcal{F}\left(g\right)\parallel \le (\sqrt{b_R}+\epsilon \sqrt{M})\parallel g\parallel .$$

Squaring these inequalities gives frame bounds $a_{R^{\epsilon }}={(\sqrt{a_R}-\epsilon \sqrt{M})}^2$ and $b_{R^{\epsilon }}={(\sqrt{b_R}+\epsilon \sqrt{M})}^2$ for the perturbed system. Consequently,

$$|a_{R^{\epsilon }}-a_R|\le 2\sqrt{a_R}\sqrt{M}\epsilon +M{\epsilon }^2,|b_{R^{\epsilon }}-b_R|\le 2\sqrt{b_R}\sqrt{M}\epsilon +M{\epsilon }^2.$$

For sufficiently small $\epsilon$, we can choose $C>2max(\sqrt{a_R},\sqrt{b_R})\sqrt{M}$ such that

$$|a_{R^{\epsilon }}-a_R|\le C\epsilon ,|b_{R^{\epsilon }}-b_R|\le C\epsilon .$$

□

If the relay operator ${\mathbf{T}}_i$ is a scalar or a multiple of the projection onto the subspace $X_i$, then it is said to be trivial. A central question is whether meaningful, non-trivial designs exist for given subspaces and weights. The following theorem provides a affirmative answer.

Theorem 4.

Let ${\left\{X_i\right\}}_{i=1}^m$ be a family of subspaces of ${\mathbb{R}}^n$ with $span{\left\{X_i\right\}}_{i=1}^m={\mathbb{R}}^n$, and let ${\left\{x_i\right\}}_{i=1}^m\subset \mathbb{R}$ be a family of positive weights. Then there exist non-trivial relay operators ${\left\{{\mathbf{T}}_i\right\}}_{i=1}^m$ for some $d_i\in \mathbb{N}$ such that ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ is a relay fusion frame for ${\mathbb{R}}^n$.

Proof. 

For each $i\in \{1,\dots ,m\}$, let $k_i=dim{X}_i\ge 1$ and $d_i=k_i$. Fix an orthonormal basis ${\mathcal{B}}_i=\{v_{i1},v_{i2},\dots ,v_{i{k}_i}\}$ for $X_i$. Extend ${\mathcal{B}}_i$ to an orthonormal basis $\mathcal{B}=\{v_{i1},\dots ,v_{i{k}_i},v_{i(k_i+1)},\dots ,v_{in}\}$ of ${\mathbb{R}}^n$. Define ${\mathbf{T}}_i\in {\mathbb{R}}^{d_i\times n}$ by its action on the basis $\mathcal{B}$,

$${\mathbf{T}}_i{v}_{ij}=\left\{\begin{array}{cc}{e}_j\in {\mathbb{R}}^{d_i}\hfill & \mathrm{if}1\le j\le k_i,\hfill \\ {\displaystyle \frac{1}{\sqrt{n}}}{e}_1\in {\mathbb{R}}^{d_i}\hfill & \mathrm{if}j=k_i+1,\hfill \\ 0\in {\mathbb{R}}^{d_i}\hfill & \mathrm{if}j>k_i+1,\hfill \end{array}\right.$$

where $e_j$ denotes the j-th standard basis vector of ${\mathbb{R}}^{d_i}$. Clearly, ${\mathbf{T}}_i$ is not a scalar and not a scalar multiple of $P_{X_i}$. Thus, ${\left\{{\mathbf{T}}_i\right\}}_{i=1}^m$ are non-trivial.

For any $g\in {\mathbb{R}}^n$, expand g in the orthonormal basis $\mathcal{B}$ of ${\mathbb{R}}^n$,

$$g=\sum _{j=1}^n⟨g,v_{ij}⟩v_{ij}\Rightarrow P_{X_i}\left(g\right)=\sum _{j=1}^{k_i}⟨g,v_{ij}⟩v_{ij}.$$

Applying ${\mathbf{T}}_i$,

$${\mathbf{T}}_i{P}_{X_i}\left(g\right)=\sum _{j=1}^{k_i}⟨g,v_{ij}⟩{\mathbf{T}}_i{v}_{ij}=\sum _{j=1}^{k_i}⟨g,v_{ij}⟩e_j.$$

Therefore,

$$\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2=\sum _{j=1}^{k_i}{⟨g,v_{ij}⟩}^2.$$

By Bessel inequality, ${\sum }_{j=1}^{k_i}{⟨g,v_{ij}⟩}^2\le {\parallel g\parallel }^2$. Summing over all i and multiplying by $x_i^2$, we have

$$\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2\le \sum _{i=1}^m{x}_i^2{\parallel g\parallel }^2=\left(\sum _{i=1}^m{x}_i^2\right) {\parallel g\parallel }^2$$

Let $b={\sum }_{i=1}^m{x}_i^2$. Since $m<\infty$ and $x_i>0$, $b<\infty$. Thus, the upper bound holds.

Define the quadratic form $Q:{\mathbb{R}}^n\to \mathbb{R}$ by

$$Q\left(g\right)=\sum _{i=1}^m{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel }^2.$$

We need to show $Q\left(g\right)>0$ for all $g\ne 0$. Suppose for contradiction that there exists $g_0\in {\mathbb{R}}^n$, $g_0\ne 0$, such that $Q\left(g_0\right)=0$. Then

$$\sum _{i=1}^m{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\left(g_0\right)\parallel }^2=0.$$

Since $x_i^2>0$ for all i, this implies $\parallel {\mathbf{T}}_i{P}_{X_i}\left(g_0\right)\parallel =0$ for all i. By construction of ${\mathbf{T}}_i$, ${\mathbf{T}}_i{P}_{X_i}\left(g_0\right)=0$ if and only if $P_{X_i}\left(g_0\right)=0$. Thus, $P_{X_i}\left(g_0\right)=0$ for all i, meaning $g_0\perp X_i$ for all i. But $\mathrm{span}\left\{X_i\right\}={\mathbb{R}}^n$, so the only vector orthogonal to all $X_i$ is $g_0=0$, a contradiction. Therefore, $Q\left(g\right)>0$ for all $g\ne 0$. Since ${\mathbb{R}}^n$ is finite-dimensional, the unit sphere $S^{n-1}=\{g\in {\mathbb{R}}^n: \parallel g\parallel =1\}$ is compact. The continuous function $Q\left(g\right)$ attains its minimum $a={min}_{g\in S^{n-1}}Q\left(g\right)>0$ on $S^{n-1}$. For any $g\ne 0$, let $h=g/\parallel g\parallel \in S^{n-1}$, hence

$$Q\left(g\right)=Q(\parallel g\parallel h)=\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{(\parallel g\parallel h)\parallel }^2={\parallel g\parallel }^2\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(h\right)\parallel }^2\ge a{\parallel g\parallel }^2.$$

Thus, the lower bound holds. □

Example 4.

Let $n=3$, $m=2$. Define $X_1=span\{e_1,e_2\}=\{(x,y,0):x,y\in \mathbb{R}\}$, $X_2=span\{e_2,e_3\}=\{(0,y,z):y,z\in \mathbb{R}\}$, and weights $x_1=1$, $x_2=1$. Clearly, $span\{X_1,X_2\}={\mathbb{R}}^3$. Let

$${\mathbf{T}}_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\end{array}\right],{\mathbf{T}}_2=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 2\end{array}\right].$$

For any $g=(g_1,g_2,g_3)\in {\mathbb{R}}^3$, we have ${\mathbf{T}}_1{P}_{X_1}\left(g\right)=(g_1,2g_2)$ and ${\mathbf{T}}_2{P}_{X_2}\left(g\right)=(g_2,2g_3)$. Then the frame operator gives

$$S\left(g\right)=\sum _{i=1}^2{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel }^2=(g_1^2+4g_2^2)+(g_2^2+4g_3^2)=g_1^2+5g_2^2+4g_3^2.$$

Thus for all $g\ne 0$, we have

$${\parallel g\parallel }^2\le S\left(g\right)\le 5{\parallel g\parallel }^2.$$

Hence ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^2$ is a relay fusion frame with bounds $a=1$, $b=5$.

The following result shows that under the condition of invertible operators, a relay fusion frame induces a fusion frame by a natural scaling of its local bounds.

Theorem 5.

If ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ is a relay fusion frame for ${\mathbb{R}}^n$, and ${\mathbf{T}}_i$ is invertible, then $\left\{\right(X_i,x_i\parallel {\mathbf{T}}_i{\parallel \left)\right\}}_{i=1}^m$ is a fusion frame for ${\mathbb{R}}^n$.

Proof. 

Since $\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel \le \parallel {\mathbf{T}}_i\parallel \parallel P_{X_i}\left(g\right)\parallel$, we have

$$\sum _{i=1}^m(x_i\parallel {\mathbf{T}}_i{\parallel )}^2\parallel P_{X_i}{\left(g\right)\parallel }^2\ge \sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2\ge a{\parallel g\parallel }^2.$$

Using the invertibility of ${\mathbf{T}}_i$, we have $\parallel P_{X_i}\left(g\right)\parallel \le \parallel {\mathbf{T}}_i^{-1}\parallel \parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel$. Then

$$\sum _{i=1}^m(x_i\parallel {\mathbf{T}}_i{\parallel )}^2\parallel P_{X_i}{\left(g\right)\parallel }^2\le \sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{\parallel }^2\parallel {\mathbf{T}}_i^{-1}{\parallel }^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2=\sum _{i=1}^m{x}_i^2{\kappa }_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel }^2.$$

Since there are finitely many i, let ${\kappa }_{max}={max}_i\parallel {\mathbf{T}}_i\parallel \parallel {\mathbf{T}}_i^{-1}\parallel$. Then

$$\sum _{i=1}^m{x}_i^2{\kappa }_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2\le {\kappa }_{max}^2\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2\le {\kappa }_{max}^{2}b{\parallel g\parallel }^2.$$

Therefore, $\left\{\right(X_i,x_i\parallel {\mathbf{T}}_i{\parallel \left)\right\}}_{i=1}^m$ is a fusion frame with bounds a and $b{\kappa }_{max}^2$. □

In practical applications, the subspaces and relay transformations may only be known approximately or are subject to measurement errors. The following result guarantees that the relay fusion frame structure is robust, ensuring that a sufficiently small perturbation of each subspace and relay operator yields a new relay fusion frame whose frame bounds are close to the original ones. For more research on perturbation theory, please refer to references [16,17,18].

Theorem 6.

Let ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ be a relay fusion frame for ${\mathbb{R}}^n$ with frame bounds $0<a\le b<\infty$. Suppose each subspace $X_i$ is perturbed to a subspace $X_i^{\prime }$ of ${\mathbb{R}}^n$, and each relay operator ${\mathbf{T}}_i\in {\mathbb{R}}^{d_i\times n}$ is perturbed to an operator ${\mathbf{T}}_i^{\prime }\in {\mathbb{R}}^{d_i\times n}$. Assume there exists a constant $\delta >0$ such that

$$∥P_{X_i}-P_{X_i^{\prime }}∥\le \delta and∥{\mathbf{T}}_i-{\mathbf{T}}_i^{\prime }∥\le \delta \forall i\in \{1,2,\dots ,m\}.$$

Let $K={\sum }_{i=1}^m{x}_i^2(1 + \parallel {\mathbf{T}}_i\parallel \left)\right(2\parallel {\mathbf{T}}_i\parallel + 1).$ If δ is sufficiently small such that that $a-K\delta >0$, then ${\left\{(X_i^{\prime },{\mathbf{T}}_i^{\prime },x_i)\right\}}_{i=1}^m$ is also a relay fusion frame for ${\mathbb{R}}^n$. Moreover, its frame bounds $a^{\prime }\le b^{\prime }$ satisfy $a^{\prime }\to a$ and $b^{\prime }\to b$ as $\delta \to 0$.

Proof. 

Let ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^m$ be a relay fusion frame for ${\mathbb{R}}^n$ with bounds $0<a\le b<\infty$. For any $g\in {\mathbb{R}}^n$, define

$$S\left(g\right)=\sum _{i=1}^m{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(g\right)\parallel }^2,S^{\prime }\left(g\right)=\sum _{i=1}^m{x}_i^2{\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}\left(g\right)\parallel }^2.$$

By assumption, there exists $\delta >0$ such that

$$\parallel P_{X_i}-P_{X_i^{\prime }}\parallel \le \delta ,\parallel {\mathbf{T}}_i-{\mathbf{T}}_i^{\prime }\parallel \le \delta \forall i.$$

Using the inequality ${|\parallel A\parallel }^2-{\parallel B\parallel }^2|\le \parallel A-B\parallel (\parallel A\parallel +\parallel B\parallel )$, we have

$$\left|\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}{\left(g\right)\parallel }^2-{\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel }^2\right|\le \parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}\left(g\right)-{\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel \left(\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}\left(g\right)\parallel +\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel \right).$$

Now,

$$\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}\left(g\right)-{\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel \le \delta \parallel g\parallel +\parallel {\mathbf{T}}_i\parallel \delta \parallel g\parallel =\delta (1+\parallel {\mathbf{T}}_i\parallel )\parallel g\parallel .$$

Also,

$$\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}\left(g\right)\parallel \le \parallel {\mathbf{T}}_i^{\prime }\parallel \parallel g\parallel \le (\parallel {\mathbf{T}}_i\parallel +\delta )\parallel g\parallel ,\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel \le \parallel {\mathbf{T}}_i\parallel \parallel g\parallel .$$

Thus,

$$\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}\left(g\right)\parallel +\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel \le (2\parallel {\mathbf{T}}_i\parallel +\delta )\parallel g\parallel .$$

Moreover,

$$\left|\parallel {\mathbf{T}}_i^{\prime }{P}_{X_i^{\prime }}{\left(g\right)\parallel }^2-{\parallel {\mathbf{T}}_i{P}_{X_i}\left(g\right)\parallel }^2\right|\le \delta (1+\parallel {\mathbf{T}}_i\parallel \left)\right(2\parallel {\mathbf{T}}_i{\parallel +\delta )\parallel g\parallel }^2.$$

Multiplying by $x_i^2$ and summing over i, we obtain

$$|S^{\prime }\left(g\right)-S\left(g\right)|\le \delta \sum _{i=1}^m{x}_i^2(1+\parallel {\mathbf{T}}_i\parallel \left)\right(2\parallel {\mathbf{T}}_i{\parallel +\delta )\parallel g\parallel }^2.$$

(4)

For $\delta \le 1$, let

$$K=\sum _{i=1}^m{x}_i^2(1+\parallel {\mathbf{T}}_i\parallel \left)\right(2\parallel {\mathbf{T}}_i\parallel +1).$$

Then $|S^{\prime }{\left(g\right)-S\left(g\right)|\le K\delta \parallel g\parallel }^2$ for all $g\in {\mathbb{R}}^n$.

From the frame condition,

$${a\parallel g\parallel }^2\le S\left(g\right)\le b{\parallel g\parallel }^2.$$

Combining with the bound from (4), we obtain

$$S\left(g\right)-{K\delta \parallel g\parallel }^2\le S^{\prime }\left(g\right)\le S\left(g\right)+K\delta {\parallel g\parallel }^2.$$

Therefore,

$${(a-K\delta )\parallel g\parallel }^2\le S^{\prime }\left(g\right)\le (b+K\delta ){\parallel g\parallel }^2.$$

Choose $\delta$ small enough so that $a-K\delta >0$. Then ${\left\{(X_i^{\prime },{\mathbf{T}}_i^{\prime },x_i)\right\}}_{i=1}^m$ is a relay fusion frame with bounds $a^{\prime }$ and $b^{\prime }$ satisfying

$$a^{\prime }\ge a-K\delta ,b^{\prime }\le b+K\delta .$$

Let $a^{\prime }$ and $b^{\prime }$ be the optimal frame bounds for the perturbed system, i.e.,

$$a^{\prime }=\underset{\parallel g\parallel =1}{inf}{S}^{\prime }\left(g\right),b^{\prime }=\underset{\parallel g\parallel =1}{sup}{S}^{\prime }\left(g\right).$$

From the inequality $S^{\prime }\left(g\right)\le S\left(g\right)+K\delta {\parallel g\parallel }^2$, we have for any unit vector g,

$$S^{\prime }\left(g\right)\le S\left(g\right)+K\delta .$$

Taking the infimum over $\parallel g\parallel =1$ on the right-hand side gives

$$a^{\prime }\le \underset{\parallel g\parallel =1}{inf}(S\left(g\right)+K\delta )=a+K\delta .$$

Similarly, from $S^{\prime }\left(g\right)\ge S\left(g\right)-K\delta {\parallel g\parallel }^2$,

$$S^{\prime }\left(g\right)\ge S\left(g\right)-K\delta ,$$

and taking the supremum yields

$$b^{\prime }\ge \underset{\parallel g\parallel =1}{sup}(S\left(g\right)-K\delta )=b-K\delta .$$

Further, we obtain

$$a-K\delta \le a^{\prime }\le a+K\delta ,b-K\delta \le b^{\prime }\le b+K\delta .$$

Thus, $|a^{\prime }-a|\le K\delta$ and $|b^{\prime }-b|\le K\delta$, so $a^{\prime }\to a$ and $b^{\prime }\to b$ as $\delta \to 0$. □

Remark 5.

This result considers simultaneous perturbations of subspaces and relay operators as a practical scenario for real-world systems. The number K is derived from measurable system parameters such as $x_i$ and the norms of $\parallel {\mathbf{T}}_i\parallel$, and it remains computable for finite m. The frame bound convergence rate $K\delta$ is uniform over all g in the finite-dimensional space ${\mathbb{R}}^n$, an exclusive property of finite dimensions. In contrast, infinite-dimensional perturbation results only address separate perturbations and offer no explicit error rates [19].

4. Examples

In this section, we provide two explicitly examples to illustrate the practical application and utility of relay operators within fusion frames. We first construct an example where the relay fusion frame bounds improve both the lower and upper bounds of the original fusion frame.

Example 5.

Let $\mathcal{H}={\mathbb{R}}^2$, $\mathbb{I}=\{1,2,3\}$. Define subspaces and weights $X_1=span\left\{(1,0)\right\}$, $X_2=span\left\{(0,1)\right\}$, $X_3=span\{(1,1)/\sqrt{2}\}$, $x_i=1$ for all $i\in \mathbb{I}$. For $f=(x,y)\in \mathcal{H}$, we have

$$\sum _{i\in \mathbb{I}}{x}_i^2{\parallel P_{X_i}\left(f\right)\parallel }^2=x^2+y^2+{\displaystyle \frac{{(x+y)}^2}{2}}.$$

Expressing $f=(cos\theta ,sin\theta )$, we get

$$\sum _{i\in \mathbb{I}}{x}_i^2{\parallel P_{X_i}\left(f\right)\parallel }^2={\displaystyle \frac{3}{2}}+{\displaystyle \frac{1}{2}}sin2\theta ,$$

so the optimal bounds are ${\alpha }_F=1$ and ${\beta }_F=2$. For each i, define $J_i:X_i\to \mathbb{R}$: $J_1\left((x,0)\right)=x$, $J_2\left((0,y)\right)=y$, $J_3(z(1,1)/\sqrt{2})=z$. Choose self-adjoint positive definite operators ${\Sigma }_1={\displaystyle \frac{5}{6}},{\Sigma }_2={\displaystyle \frac{5}{6}},{\Sigma }_3=2.$ Then ${\lambda }_{min}\left({\Sigma }_i\right)={\Sigma }_i$, ${\lambda }_{max}\left({\Sigma }_i\right)={\Sigma }_i$, and

$${\lambda }_{min}=min\left\{{\displaystyle \frac{5}{6}},{\displaystyle \frac{5}{6}},2\right\}={\displaystyle \frac{5}{6}},{\lambda }_{max}=max\left\{{\displaystyle \frac{5}{6}},{\displaystyle \frac{5}{6}},2\right\}=2.$$

Define relay operators

$${\mathbf{T}}_1(x,y)=\sqrt{6/5}x,{\mathbf{T}}_2(x,y)=\sqrt{6/5}y,{\mathbf{T}}_3(x,y)={\displaystyle \frac{x+y}{2}}.$$

For $f=(cos\theta ,sin\theta )$, we obtain

$$\sum _{i\in \mathbb{I}}{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\left(f\right)\parallel }^2={\displaystyle \frac{6}{5}}+{\displaystyle \frac{1}{4}}sin2\theta .$$

The optimal bounds are ${\alpha }_R=1.2$ and ${\beta }_R=1.7$. Thus, ${\alpha }_R=1.2>{\alpha }_F=1$ and ${\beta }_R=1.7<{\beta }_F=2$.

Example 6.

Let $\mathcal{H}={\mathbb{R}}^3$ with the standard inner product. Define the following subspaces and weights as

$$X_1=span\left\{e_1\right\},v_1=1,X_2=span\left\{e_2\right\},v_2=1,X_3=span\{e_1,e_3\},v_3=1,$$

where $\{e_1,e_2,e_3\}$ is the standard orthonormal basis. For any $f=(x,y,z)\in \mathcal{H}$, we have

$$\sum _{i=1}^3{x}_i^2{\parallel P_{X_i}\left(f\right)\parallel }^2=2x^2+y^2+z^2.$$

Thus ${\left\{(X_i,x_i)\right\}}_{i=1}^3$ is a fusion frame with bounds $\alpha =1$ and $\beta =2$.

Suppose there exist positive scalars $c_1,c_2,c_3>0$ such that $\left\{(X_i,c_i{x}_i)\right\}$ is a Parseval fusion frame, i.e.,

$$\sum _{i=1}^3{\left(c_i{x}_i\right)}^2{P}_{X_i}=I.$$

In matrix form,

$$c_1^2\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)+c_2^2\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)+c_3^2\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right).$$

This yields the equations

$$c_1^2+c_3^2=1,c_2^2=1,c_3^2=1.$$

From $c_3^2=1$ and $c_1^2+1=1$, we get $c_1=0$, contradicting positivity. Hence, no such scaling exists. Equivalently, there are no scalar rescalings that make the fusion frame Parseval.

Now choose ${\mathbf{T}}_1:X_1\to \mathbb{R}$ and ${\mathbf{T}}_2:X_2\to \mathbb{R}$ by

$${\mathbf{T}}_1\left(e_1\right)=\sqrt{b_1},{\mathbf{T}}_2\left(e_2\right)=\sqrt{b_2},$$

with $b_1,b_2>0$ to be determined. For $X_3$, define ${\mathbf{T}}_3:X_3\to {\mathbb{R}}^2$ by specifying its action on the basis $\{e_1,e_3\}$,

$${\mathbf{T}}_3\left(e_1\right)=(\sqrt{a},0),{\mathbf{T}}_3\left(e_3\right)=(0,\sqrt{d}),$$

where $a,d>0$. Then for $f=(x,y,z)$, we have

$$\begin{array}{cc}\hfill \sum _{i=1}^3{x}_i^2{\parallel {\mathbf{T}}_i{P}_{X_i}\left(f\right)\parallel }^2& =b_1{x}^2+b_2{y}^2+\left(a{x}^2+d{z}^2\right)\hfill \\ & =(b_1+a)x^2+b_2{y}^2+d{z}^2.\hfill \end{array}$$

We require this to equal ${\parallel f\parallel }^2=x^2+y^2+z^2$ for all $x,y,z$. Thus,

$$b_1+a=1,b_2=1,d=1.$$

Choose $a={\displaystyle \frac{1}{2}}$, $b_1={\displaystyle \frac{1}{2}}$, $b_2=1$, $d=1$. Then

$$\sum _{i=1}^3{x}_i^2\parallel {\mathbf{T}}_i{P}_{X_i}{\left(f\right)\parallel }^2=\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\right)x^2+y^2+z^2={\parallel f\parallel }^2.$$

Thus ${\left\{(X_i,{\mathbf{T}}_i,x_i)\right\}}_{i=1}^3$ is a Parseval relay fusion frame.

The fusion frame ${\left\{(X_i,x_i)\right\}}_{i=1}^3$ in ${\mathbb{R}}^3$ defined in Example 6 cannot be made Parseval by any rescaling of weights, but admits a Parseval relay fusion frame with relay operators. This demonstrates relay operators provide additional flexibility beyond scalar adjustments, enabling tightness even when scalability fails.

5. Conclusions

In this paper, we have studied the relay fusion frames in finite dimensions. Our main results indicate that the relay fusion frames possesses rich structural characteristics. For the case of mutually orthogonal subspaces, we presented a clear construction method and proved that an optimal tight relay fusion frame can be designed. Compared with the underlying fusion frame, its frame bound ratio is improved. We have demonstrated a general sufficient condition under which the scaling relay operator can generate a relay fusion frame from the existing fusion frame. Furthermore, we have established a perturbation theorem, which ensures that the properties and bounds of the relay fusion frames are stable under small perturbations in the subspace and the relay operators. We also answered a fundamental question on existence by proving that non-trivial relay operators can always be constructed for any collection of subspaces spanning the space. Finally, we proved that under the condition of invertible relay operators, the relay fusion frame naturally induces a standard fusion frame.