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. □