INTRODUCTION
Throughout this study H will be a Hilbert space and Ĥ will be the
collection of all closed subspace of H, respectively. Also, (X, μ)
will be a measure space and v:X→[0, ∞) a measurable mapping
such that v ≠ 0 a.e. We shall denote the unit closed ball of H by H_{1}.
Frames was first introduced at (Duffin and Schaeffer, 1952) in the context
of nonharmonik Fourier series. Outside of signal processing, frames did
not seem to generate much interest until the ground breaking work of Daubechies
et al. (1986). Since then the theory of frames began to be more widely
studied. During the last 20 years the theory of frames has been growing
rapidly, several new applications have been developed. For example, besides
traditional application as signal processing, image processing, data compression
and sampling theory, frames are now used to mitigate the effect of losses
in pocket based communication systems and hence to improve the robustness
of data transmission (Casazza and Kovacevic, 2003) and to design highrate
constellation with full diversity in multipleantenna code design (Hassibi
et al., 2001). In Bolcskel et al. (1998), Benedetto et
al. (2004) and Candes and Donoho (2004) some applications have been
developed.
The fusion frames were considered by Casazza et al. (2000) in
connection with distributed processing and are related to the construction
of global frames. The fusion frame theory is in fact more delicate due
to complicated relations between the structure of the sequence of weighted
subspaces and the local frames in the subspaces and due to the extreme
sensitivity with respect to changes of the weights.
In this study, we shall extend the fusion frames to their continuous
versions in measure spaces.
PRELIMINARIES AND METHODS
This topics can be found by Christensen (2002).
Definition 1: Let {f_{i}}_{iεI} be sequence
of members of H. We say that {f_{i}}_{iεI} is
a frame for H if there exist 0 < A ≤ B < ∞ such that for
all h ε H,
The constants A and B are called frame bounds. If A, B can be chosen
so that A = B, we call this frame an A—tight frame and if A= B =
1 it is called a parseval frame. If we only have the upper bound, we call
{f_{i}}_{iεI} a Bessel sequence. If {f_{i}}_{iεI}
is a Bessel sequence then the following operators are bounded,
Definition 2: For a countable index set I, let {W_{i}}_{iεI}
be a family of closed subspace in H and let {v_{i}}_{iεI}
be a family of weights, i.e., v_{i}>0 for all i ε I.
Then {(W_{i},v_{i})}_{iεI} is a fusion
frame for H if there exist 0 < C ≤ D < ∞ such that for
all h ε H:
where, is the orthogonal projection onto the subspace W_{i}. We call
C and D the fusion frame bounds.
The family {(W_{i},v_{i})}_{iεI} is called
a C—tight fusion frame, if in above inequality the constants C and
D can be chosen so that C = D, a parseval fusion frame provided C = D
= 1 and an orthonormal fusion basis if H = ⊕_{iεI}W_{i}.
If {(W_{i},v_{i})}_{iεI} possesses an
upper fusion frame bound, but not necessarily a lower bound, we call it
is a Bessel fusion sequence with Bessel fusion bound D.
The theory of frames has a continuous version as follows: Let (X, μ)
be a measure space. Let f: X→H be weakly measurable (i.e., for all
h ε H, the mapping x → < f(x), h > is measurable).
Then f is called a continuous frame for H if there exist 0 < A ≤
B < ∞ such that, for all h ε H,
The following lemmas can be found in operator theory text books (Pedersen
and Gert, 1989; Rudin, 1973, 1986; Sakai, 1998) which we shall use then
in the text.
Lemma 1: Let u : H→K be a bounded operator. Then:
• 
u = u* and uu* = u^{2}. 
• 
R_{u} is closed, if and only if, R_{u} is closed.

• 
u is subjective, if and only if, there exists c > 0 such that
for each h ε H 
ch≤u*(h)
Lemma 2: Let u be a selfadjoint bounded operator on H. Let
and
Then, m_{u}, M_{u} ε σ(u).
Theorem 1: Let u: K→H be a bounded operator with closed range
R_{u}. Then there exists a bounded operator u^{x}: H→K
for which uu^{x}f = f, F ε R_{u}.
Also, u*: H → K has closed range and (u*)^{†} = (u^{†})*.
The operator u^{x} is called the pseudoinverse of u.
Theorem 2: Let u: K→H be a bounded surjective operator. Given
y ε H, the equation ux = y has a unique solution of minimal norm,
namely, x = u^{x}y.
Now we introduce the concept of cfusion frame and shall show some its
properties.
Definition 4: Let F: X→Ĥ be such that for each h ε
H, the mapping x a π_{F(X)}(h) is measurable (i.e., F is
weakly measurable). We say that (F, v) is a cfusion frame for H if there
exist 0 < A ≤ B < ∞ such that for all h ε H,
(F, v) is called a tight cfusion frame for H if A, B can be chosen so
that A = B and parseval if A = B = 1. If just the right hand inequality
satisfies then we say that (F, v) is a Bessel cfusion mapping for H.
Definition 3: Let F: X→Ĥ . Let L^{2}(X, H, F)
be the class of all measurable mapping f: X→H. such that for each
x ε X and
f(x) ε F(x) and
It can be verified that L^{2}(X, H, F) is a Hilbert space with
inner product defined by:
Remark 1: For brevity, we shall denote L^{2}(X, H, F)
by L^{2}(X, F). Let (F, v) be a Bessel cfusion mapping, f ε
L^{2}(X, F) and h ε H. Then:
So we may define:
Definition 5: Let (F, v) be a Bessel cfusion mapping for H. We
define the cfusion preframe operator T_{F}: L^{2}(X,
F) by
By the remark (5),
T_{F}: L^{2}(X, F)→H
is a bounded linear mapping. Its adjoint T_{F}*: H→L^{2}(X,
F) will be called cfusion analysis operator and S_{F} = T_{F}
oT_{F}* will be called cfusion frame operator.
Remark 2: Let (F, v) be a Bessel cfusion mapping for H. Then
T_{F}: L^{2}(X, F)→H is indeed a vectorvalued integral,
which we shall denote by:
Where:
For each h ε H and f ε L^{2}(X, F) we have:
Hence for each h ε H, T_{F}*(h) = vπ_{F}(h).
So T_{F}* = vπ_{F}.
Therefore, S_{F}: H → H is also a vectorvalued integral
which for each h ε H, we have
Definition 6: Let (F, v) and (G, v) are Bessel cfusion mapping
for H. We say (F, v) and (G, v) are weakly equal if T_{F}* = TG*,
which is equivalent with
vπ_{F} (h) = vπG (h), a.e.
for all h ε H Since, v≠ 0 a.e,. (F, v) and (G, v) are weakly
equal if
π_{F} (h) = π_{G} (h), a.e.
for all h ε H.
Remark 3: Let T_{F} = 0. Now, Let O: X→Ĥ be
defined by:
O(x) = {0},
for almost all x ε X. Then (O, v) is a Bessel cfusion mapping
and T_{O} = 0. Let h ε H. Since, vπ_{F}(h)εL^{2}(X,
F), so
Thus,
π_{F(x)} (h) = 0, a.e.
Therefore,
π_{F} (h) = π_{O} (h), a.e.
Hence, (F, v) and (G, v) are weakly equal.
RESULTS AND DISCUSSION
Definition 7: For each Bessel cfusion mapping F for H, we shall
denote
Remark 4: Let F is a Bessel cfusion mapping for H. Since, for
each h ε H.
<T_{F}T_{F}*(h), h > = vπ_{F}(h)^{2},
A_{F,y} and B_{F,y} are optimal scalars which satisfy
A_{F,y}≤T_{F}T_{F}*(h)≤
B_{F,y}
So (F, v) is a cfusion frame for H if and only if A_{F,y}>0.
Lemma 3: Let (F, v) is a Bessel cfusion mapping for H. Then F
is cfusion frame for H if and only if T_{F} is surjective.
Proof: Let A_{F,y}>0 Since, for each h ε H
Therefore,
T_{F}: L^{2}(X, F)→H
is surjective.
Now let T_{F} be surjective. Let
T_{F}^{x}: H → L^{2}(X, F)
be its pseudoinverse. Since, for each h ε H
so
Theorem 3: Let (F, v) be a Bessel cfusion mapping for H, and
K be a Hilbert space. Let u: H → K be a bounded bijective operator
and (u oF, v) is a Bessel cfusion mapping for K. Then:
(i) 

(ii) 

(iii) 
F is a cfusion frame for H if and only if (u oF, v) is a cfusion
frame for K. 
Proof:
(i) 
It is straightforward. 
(ii) 
For each k ε K, we have 
(iii) 
It is clear from (ii) and Lemma 3. 
Hence
Lemma 4: Let (F, v) be a Bessel cfusion mapping for H. Then the
frame operator S_{F} = T_{F}T_{F}* is invertible
if and only if F is a cfusion frame for H.
Proof: Let S_{F} = T_{F}T_{F}* be invertible.
We have
so, A_{F,y}>0. Now let A_{F,y}>0. So, by the Lemma
3, T_{F} is surjective. Then there exist A>0 such that
Hence
Theorem 4: Let {H_{i}}_{iεI} be a collection
of Hilbert space and H = ⊕_{i}H_{i}. Let (F, v)
be a Bessel cfusion mapping for H such that for each i ε I there
exist at most one x ε X such that F(x)⊆H_{i}. Let
each finite subset of X be measurable. Then, for each h ε H
Proof: Let
Let {f_{n}} be a sequence of members of K which tends to f ε
H. Given ε>0, we can find N>0 such that f_{N}—f<ε
There exists a finite Z⊆X such that for each finite Z⊆Y⊆Y,
We have
But
So, K is a closed subspace of H. Now, let h ε K^{⊥}
Since, for each t ε X
Since
and A_{F,y}>0. H = 0.
Theorem 5: Let (X, μ) and (Y, λ) be two σ—finite
measure space and let f: XxY→H, F: X→Ĥ be weakly measurable
mappings. Let for each x ε X, f(x,..): Y→F(x) be measurable
and for every x ε F(x), f(x,.) is a continuous frame for H. Let
and let
Then, (F, v) is a cfusion frame for H if and only if
is a continuous frame for H.
Proof: For each h ε H we have
and the theorem is proved.
Theorem 6: Let (X, μ) be a σ—finite measure space and
K be a Hilbert space. Let u: H → K be a bijective linear operator. Let
F: X→Ĥ and be weakly measurable. Then, (F, v) is a cfusion frame
for H if and only if (u oF, v) is a cfusion frame for K.
Proof: Let F be a cfusion frame for H. Let (Y, λ) be a σfinite
measure space and let
f: XxY → H
be such that for each
x ε X, f(x,.):Y → F(x)
with
measurable and Choosing such mapping is always possible, because let be an orthonormal basis for F(x). We can suppose that is
pairwise disjoint (we can consider {x}xI). Let and λ be the counting measure on Y. Then we can define f: XxY →
H by
and
f(x, i) = 0 otherwise
Then, for each x ε X
A(x) = B(x) = 1
By the Theorem 3
Then, and for each x ε X,
Since, u is surjective, there is C > 0 such that
So,
Similarly, we have
Therefore by the Theorem 3 be a cfusion frame for (u oF, v). The proof
of the converse is similar.
Theorem 7: Let (F, v) be a cfusion frame for H. Let h ε
H and SF = T_{F}T_{F}*. Then:
(i) We have the following retrieval formulas
and
(ii) In the retrieval formula
has least norm among all of the retrieval formulas.
(iii) For each h ε H,
Proof:
(i) Since (F, v) is a cfusion frame, S_{F} is an invertible operator.
By the Theorem 4, we have
Also, we have
(ii) Let f ε L^{2}(X, F) and
h = T_{F} (f)
Thus, for each k ε H we have
Therefore
So,
Hence
Since, F is a cfusion frame,
But,
So,
and (ii) is proved.
(iii) Let f ε L^{2}(X, F). Since, T_{F}^{x}
is the unique solution of minimal norm of T_{F}(f) = h so by
(ii),
Therefore,
Theorem 8: Let (F, v) and (G, v) be Bessel cfusion mapping for
H. Then the following assertions are equivalent:
(i) For each h ε H,
(ii) For each h ε H,
(iii) For each h , k ε H,
(iv) For each h ε H,
(v) For each orthonormal bases
{e_{i}}_{iεI} and {λ_{j}}_{jεJ}
for H we have
(iv) For each orthonormal bases {e_{i}}_{iεI}
for H and i ε I,
Proof: (i) → (ii) Let h, k ε H. We have
Hence,
(ii) → (iii) It is evident by the proof of (i) → (ii).
(iii) → (i) For each h, k ε H, we have
Thus
(iv) → (i) Let L: H → H be defined by
It clear that L is linear. Since
that, L ε B(H). For each h ε H, we have
Hence, for each h ε H,
(iii) → (iv) is evident.
(v) → (iii) We have
(vi) → (v) it is similar with the proof of (v) → (iii).