2.1. Discovery of Fusion Frames on Hilbert Spaces and Generalization of the Index Set
Theorem 3.  Let  be a-tight frame for H with the frame transform θ, and let  be an orthogonal projection. Then  and , where  is the orthonormal basis for H.
 Proof.  Since 
 is 
a-tight frame for 
H, 
 and 
 on 
, i.e., 
. Then for any 
,
          therefore, 
 or 
.
Since 
, we obtain
By the arbitrariness of f, it follows that .
In addition, from 
 and 
, we have
Its self-adjointness is obvious. Thus, , which fully verifies that P acts as an orthogonal projection in the relevant context.    □
 Let 
 be 
-tight frame sequence for 
H with the frame transforms 
 (where 
i is the number of frames and 
j is the dimension of the frames), and let 
 be the orthogonal projections. Then 
, and
Therefore, 
 is a fusion frame if and only if there exist constants 
 so that
Obviously, since , when I is a finite set,  must be a Bessel fusion sequence.
Our questions are, do such traditional frames and fusion frames exist? And, can I here be extended to an infinite set, i.e.,  Our answers are affirmative and we now give example below.
Example 1.  Let  with  and , Then  is a s-tight fusion frame for H, where  is the standard orthonormal basis for H.
 Proof.  For any 
, we have
          so 
 are 
-tight frames for 
H. Since
          i.e., 
, 
. Then
          which means that 
 satisfying 
 and
Therefore,  is a s-tight fusion frame.
However,  cannot be 1-uniform fusion frame or w-uniform fusion frame, because the constant series  diverges forever unless I is a finite set.    □
 We present it in the form of a certain theorem.
Theorem 4.  Let H be a Hilbert space. Then there exists a sequence of -tight frames  for H such that  is a tight fusion frame for H, where  are the frame transforms of ,  are the orthogonal projections.
 Remark 1.  (1) Example 1 makes use of the convergence of positive-term series. For instance, if we take , then , and  forms a Parseval fusion frame, and in this case, .
In fact, for the positive term-series considered here, it is sufficient to establish convergence, there is no need to compute the exact sum. Moreover, many series can only be analyzed in terms of convergence, without an explicit evaluation of their sums. Even in such cases,  still constitutes a (tight) fusion frame.
(2) The conclusion only holds for traditional tight frames, since the relationship between the frame transform and orthogonal projection exists exclusively in the tight-frame setting.
(3) Example 1 and Theorem 4 establish a close connection between traditional frames and fusion frames, and further realize the closed subspaces  of H by taking . Even more significantly, the index set I is generalized to infinite sets.
   2.2. Orthogonal Projections in the Fusion Frames
The orthogonal projector P is a self-adjoint operator satisfying , i.e., . It projects element of H onto a closed subspace, and the projection is orthogonal to the complement of that subspace, that is, the inner product of the residual with the subspace is zero. Below, we present some simple examples in  and .
Example 2.  In , let . Then for any  and for any ,  Example 3.  In , let . Thenfor any  where the coefficients are the Fourier coefficients.  Example 4.  In , let  be a fixed interval, and let  A fusion frame is a subspace frame. Its central role lies in providing stable signal representation and reconstruction through subspace decomposition and weight balancing, while preserving the characteristics of each data source. Fusion frames thus serve as an important theoretical tool in multimodal and multi-sensor data processing. The following examples illustrate the role of orthogonal projection in the fusion frames and in the projection process as a whole.
Example 5.  Let , ,  (i.e.,  is the x-axis and  is the y-axis),  Then for any , , , , . Thus, the weighted sum is Therefore,  is a Parseval fusion frame.
If  fails, and only  and  are retained, then the weighted sum is , which cannot control . However, if a third subspace  is added (i.e.,  is the line  ()). Let , then  and the weighted sum is Therefore,  is a fusion frame, and the lower and upper fusion frame bounds are , , respectively. In other words, when  fails, f can still be reconstructed through  and , which demonstrates the robustness of the fusion frame.
In addition, we can also add two subspaces ,  (i.e.,  is the line ,  is the line ),  Then,and the weighted sum is Therefore,  is a fusion frame. In other words, f can still be reconstructed through , , and .
 Orthogonal projection is also a fundamental tool for constructing tight frames. For instance, redundant frames can be transformed into Parseval frames by means of orthogonal projection. This property ensures lossless reconstruction in signal processing.
Example 6.  Let , , , , . Then for any , the weighted sum is Therefore,  is a fusion frame, but not a Parseval fusion frame.
Since  can be linearly represented by  and , it is therefore a redundant frame. Next, we transform it into a Parseval frame. we derive the analysis operator T of f. Since Thus, T is a symmetric matrix, and Projecting  onto the original subspaces yields After adjusting the weights , the weighted sum is obtained as Thus, by projecting the subspaces of the redundant frame via  and adjusting the weights, the redundant frame is converted into a Parseval fusion frame.
   2.3. Research on Dual Fusion Frames on Hilbert Space
In this section, we begin with an example to explore how traditional dual frames in Hilbert spaces recover original signals in cases of data loss demonstrating the importance and research significance of dual frames.
Example 7.  Let , , . Then so  is a tight frame for  with frame bound , where  is the frame transform of X and S is its frame operator. Let the original signal vector be . Then the frame coefficients are , , . Suppose that during transmission, the original signal loses , how can we recover the original signal?
We try to recover it using the alternate dual frame  of X.
Since  is lost, let us set , , . Then . Since Y is the alternate dual frame of X, they are mutually alternate dual frames and  or . Then we have It is solved that , , , . So , , , andi.e., the alternate dual frame Y of X can recover the original signal. The canonical dual frame  of X can also recover the original signal.
Since , . Then we obtain This means that the canonical dual frame Z of X can also recover the original signal.
Meanwhile, let  (). Thenand We obtain , i.e., , and of course , where .
It follows that the frames X and U are disjointed, meaning that any alternate dual frame of a frame can be expressed as the sum of its canonical dual frame and a frame can disjoint from itself, and the canonical dual frame is the “minimal” dual frame.
 Next, we study dual fusion frames on Hilbert spaces.
Definition 5  (see [
19]). 
Let  be a fusion frame for H, the analysis operator is defined bywhere . It can easily be shown that the synthesis operator , which is defined to be the adjoint operator, is given byThe fusion frame operator S for  is defined by If  be a fusion frame for H with fusion frame bound A and B, then the associated fusion frame operator S is a self-adjoint, positive, and invertible operator on H, and .
 It follows from Definition 5 that  is a B-tight fusion frame if and only if , and is a Parseval fusion frame if and only if .
Obviously, if 
 is a fusion frame for 
H with fusion frame operator 
S, then 
 is a Parseval fusion frame for 
H. In fact,
Theorem 5  (see [
1]). 
Let T be a bounded linear operator on H, and let V be a closed subspace of H. Then . Moreover if T is a unitary operator, then , where P is the orthogonal projection from H to V. It follows from Theorem 5 that 
, i.e., 
. So, if 
 is the fusion frame with fusion frame operator 
S, then
Therefore, for any 
, 
, and
 is called the canonical dual fusion frame of .
Corollary 1.  Let  be the fusion frame for H with fusion frame operator S, and let  be its the canonical dual fusion frame. Then .
 Proof.  Therefore, the frame operator of the canonical dual fusion frame is the inverse of its own frame operator. It is obvious that if  is a parseval fusion frame, i.e., , then its canonical dual fusion frame is itself.    □
 Definition 6  (see [
16]). 
Let  and  be fusion frames for H. If for any ,then  is called an alternate dual fusion frame of , where S is the fusion frame operator of . Remark 2.  (1) The canonical dual fusion frame of  must be its alternate dual fusion frame. In particular, when  and , these two dual fusion frames are the same fusion frame.
(2) Traditional alternate dual frames are mutual alternate duals, while alternate dual fusion frames are not dual to each other.
In fact, if  is the alternate dual fusion frame of , then  for any , where  is the fusion frame operator of . If  is an alternate dual fusion frame of , then  for any , where  is the fusion frame operator of .
In particular, if , then the alternate dual fusion frames are mutual alternate duals.
 The following are the necessary and sufficient conditions under which they are mutual alternate duals and their stability.
Theorem 6.  Let  and  be fusion frames for H, let  and  be their analysis operators respectively, let  and  be their synthesis operators, respectively, and let  and  be their frame operators respectively. Then
(1)  and  are mutual alternate duals if and only if .
(2) If  and  is alternate dual fusion frames of , then  is also a fusion frame for H.
 Proof.  (1) For any 
, we have
Therefore, 
 and 
 are mutual alternate duals if and only if
          if and only if
          by the arbitrariness of 
f, we obtain 
.
(2) Since 
, 
 and 
 are mutual alternate duals. Combining with (1) of Theorem 6, for any 
, we have
Therefore,
          that is to say,
Moreover, since 
 and 
 are fusion frames for 
H, there exist constants 
 such that 
. Therefore,
          and
Similarly, since
          we have
          so 
 is also a fusion frame with frame operator 
 and 
.    □
 Theorem 7.  Let  be a fusion frame for H with fusion frame operator , let  and  be its canonical dual fusion frame and alternate dual fusion frame respectively. Then , where  is the fusion frame operator of .
 Proof.  Since for any 
,
          and
          we have
Replacing 
f with 
,
Therefore,
          and
          that is to say,
Moreover, since
          we obtain
Since  is the frame operator of the canonical dual fusion frame of , Theorem 7 reveals the relationship between the canonical dual fusion frame and the alternate dual fusion frame, especially the relationship between their respective fusion frame operators.    □