Journal of Applied Sciences

Year: 2008 | Volume: 8 | Issue: 16 | Page No.: 2881-2887
DOI: 10.3923/jas.2008.2881.2887

C-Fusion Frame

M.H. Faroughi and R. Ahmadi

Abstract: In this study, we shall generalized the concept of fusion frame, namely, c - fusion frames, which is continuous version of the fusion frames. We give characterization of c - fusion frames and show that many basic properties can be derived within this general context.

Keywords: bessel, Hilbert space, Operator, fusion frame and frame

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 H₁.

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 high-rate constellation with full diversity in multiple-antenna 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 = ⊕_iεIW_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:

c||h||≤||u*(h)||

Lemma 2: Let u be a self-adjoint 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^xf = f, F ε R_u.

Also, u*: H → K has closed range and (u*)^† = (u^†)*.

The operator u^x is called the pseudo-inverse 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^xy.

Now we introduce the concept of c-fusion 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 c-fusion frame for H if there exist 0 < A ≤ B < ∞ such that for all h ε H,

(F, v) is called a tight c-fusion 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 c-fusion mapping for H.

Definition 3: Let F: X→Ĥ . Let L²(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 L²(X, H, F) is a Hilbert space with inner product defined by:

Remark 1: For brevity, we shall denote L²(X, H, F) by L²(X, F). Let (F, v) be a Bessel c-fusion mapping, f ε L²(X, F) and h ε H. Then:

So we may define:

Definition 5: Let (F, v) be a Bessel c-fusion mapping for H. We define the c-fusion pre-frame operator T_F: L²(X, F) by

By the remark (5),

T_F: L²(X, F)→H

is a bounded linear mapping. Its adjoint T_F*: H→L²(X, F) will be called c-fusion analysis operator and S_F = T_F oT_F* will be called c-fusion frame operator.

Remark 2: Let (F, v) be a Bessel c-fusion mapping for H. Then T_F: L²(X, F)→H is indeed a vector-valued integral, which we shall denote by:

Where:

For each h ε H and f ε L²(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 vector-valued integral which for each h ε H, we have

Definition 6: Let (F, v) and (G, v) are Bessel c-fusion 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 c-fusion mapping and T_O = 0. Let h ε H. Since, vπ_F(h)εL²(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 c-fusion mapping F for H, we shall denote

Remark 4: Let F is a Bessel c-fusion mapping for H. Since, for each h ε H.

<T_FT_F*(h), h > = ||vπ_F(h)||²,

A_F,y and B_F,y are optimal scalars which satisfy

A_F,y≤T_FT_F*(h)≤ B_F,y

So (F, v) is a c-fusion frame for H if and only if A_F,y>0.

Lemma 3: Let (F, v) is a Bessel c-fusion mapping for H. Then F is c-fusion 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: L²(X, F)→H

is surjective.
Now let T_F be surjective. Let

T_F^x: H → L²(X, F)

be its pseudo-inverse. Since, for each h ε H

so

Theorem 3: Let (F, v) be a Bessel c-fusion 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 c-fusion mapping for K. Then:

Proof:

Hence

Lemma 4: Let (F, v) be a Bessel c-fusion mapping for H. Then the frame operator S_F = T_FT_F* is invertible if and only if F is a c-fusion frame for H.

Proof: Let S_F = T_FT_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 = ⊕_iH_i. Let (F, v) be a Bessel c-fusion 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 c-fusion 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 c-fusion frame for H if and only if (u oF, v) is a c-fusion frame for K.

Proof: Let F be a c-fusion frame for H. Let (Y, λ) be a σ-finite 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 c-fusion frame for (u oF, v). The proof of the converse is similar.

Theorem 7: Let (F, v) be a c-fusion frame for H. Let h ε H and SF = T_FT_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 c-fusion frame, S_F is an invertible operator. By the Theorem 4, we have

Also, we have

(ii) Let f ε L²(X, F) and

h = T_F (f)

Thus, for each k ε H we have

Therefore

So,

Hence

Since, F is a c-fusion frame,

But,

So,

and (ii) is proved.

(iii) Let f ε L²(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 c-fusion 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).

REFERENCES