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C-Fusion Frame



M.H. Faroughi and R. Ahmadi
 
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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.

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  How to cite this article:

M.H. Faroughi and R. Ahmadi, 2008. C-Fusion Frame. Journal of Applied Sciences, 8: 2881-2887.

DOI: 10.3923/jas.2008.2881.2887

URL: https://scialert.net/abstract/?doi=jas.2008.2881.2887
 

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 {fi}iεI be sequence of members of H. We say that {fi}iεI is a frame for H if there exist 0 < A ≤ B < ∞ such that for all h ε H,

Image for - C-Fusion Frame

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 {fi}iεI a Bessel sequence. If {fi}iεI is a Bessel sequence then the following operators are bounded,

Image for - C-Fusion Frame

Definition 2: For a countable index set I, let {Wi}iεI be a family of closed subspace in H and let {vi}iεI be a family of weights, i.e., vi>0 for all i ε I. Then {(Wi,vi)}iεI is a fusion frame for H if there exist 0 < C ≤ D < ∞ such that for all h ε H:

Image for - C-Fusion Frame

where, Image for - C-Fusion Frame is the orthogonal projection onto the subspace Wi. We call C and D the fusion frame bounds.

The family {(Wi,vi)}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 = ⊕iεIWi. If {(Wi,vi)}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,

Image for - C-Fusion Frame

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.
Ru is closed, if and only if, Ru is closed.
u is subjective, if and only if, there exists c > 0 such that for each h ε H

c||h||≤||u*(h)||

Lemma 2: Let u be a self-adjoint bounded operator on H. Let

Image for - C-Fusion Frame

and

Image for - C-Fusion Frame

Then, mu, Mu ε σ(u).

Theorem 1: Let u: K→H be a bounded operator with closed range Ru. Then there exists a bounded operator ux: H→K for which uuxf = f, F ε Ru.

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

The operator ux 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 = uxy.

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 πF(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,

Image for - C-Fusion Frame

(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 Image for - C-Fusion Frame

It can be verified that L2(X, H, F) is a Hilbert space with inner product defined by:

Image for - C-Fusion Frame

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:

Image for - C-Fusion Frame

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 TF: L2(X, F) by

Image for - C-Fusion Frame

By the remark (5),

TF: L2(X, F)→H

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

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

Image for - C-Fusion Frame

Where:

Image for - C-Fusion Frame

For each h ε H and f ε L2(X, F) we have:

Image for - C-Fusion Frame

Hence for each h ε H, TF*(h) = vπF(h).
So TF* = vπF.
Therefore, SF: H → H is also a vector-valued integral which for each h ε H, we have

Image for - C-Fusion Frame

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 TF* = TG*, which is equivalent with

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

πF (h) = πG (h), a.e.

for all h ε H.

Remark 3: Let TF = 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 TO = 0. Let h ε H. Since, vπF(h)εL2(X, F), so

Image for - C-Fusion Frame

Thus,

πF(x) (h) = 0, a.e.

Therefore,

πF (h) = πO (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

Image for - C-Fusion Frame

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

<TFTF*(h), h > = ||vπF(h)||2,

AF,y and BF,y are optimal scalars which satisfy

AF,y≤TFTF*(h)≤ BF,y

So (F, v) is a c-fusion frame for H if and only if AF,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 TF is surjective.

Proof: Let AF,y>0 Since, for each h ε H

Image for - C-Fusion Frame

Therefore,

TF: L2(X, F)→H

is surjective.
Now let TF be surjective. Let

TFx: H → L2(X, F)

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

Image for - C-Fusion Frame

so

Image for - C-Fusion Frame

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:

(i) Image for - C-Fusion Frame
(ii) Image for - C-Fusion Frame
(iii) F is a c-fusion frame for H if and only if (u oF, v) is a c-fusion 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.

Image for - C-Fusion Frame

Hence

Image for - C-Fusion Frame

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

Proof: Let SF = TFTF* be invertible. We have

Image for - C-Fusion Frame

so, AF,y>0. Now let AF,y>0. So, by the Lemma 3, TF is surjective. Then there exist A>0 such that

Image for - C-Fusion Frame

Hence

Image for - C-Fusion Frame

Theorem 4: Let {Hi}iεI be a collection of Hilbert space and H = ⊕iHi. 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)⊆Hi. Let each finite subset of X be measurable. Then, for each h ε H

Image for - C-Fusion Frame

Proof: Let

Image for - C-Fusion Frame

Let {fn} be a sequence of members of K which tends to f ε H. Given ε>0, we can find N>0 such that ||fN—f||<ε There exists a finite Z⊆X such that for each finite Z⊆Y⊆Y,

Image for - C-Fusion Frame

We have

Image for - C-Fusion Frame

But

Image for - C-Fusion Frame

So, K is a closed subspace of H. Now, let h ε K Since, for each t ε X

Image for - C-Fusion Frame

Image for - C-Fusion Frame Since
and AF,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

Image for - C-Fusion Frame

and let

Image for - C-Fusion Frame

Then, (F, v) is a c-fusion frame for H if and only if

Image for - C-Fusion Frame

is a continuous frame for H.

Proof: For each h ε H we have

Image for - C-Fusion Frame

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

Image for - C-Fusion Frame

measurable and Image for - C-Fusion Frame Choosing such mapping is always possible, because let Image for - C-Fusion Frame be an orthonormal basis for F(x). We can suppose that Image for - C-Fusion Frameis pairwise disjoint (we can consider {x}xI). Let Image for - C-Fusion Frame and λ be the counting measure on Y. Then we can define f: XxY → H by

Image for - C-Fusion Frame

and

f(x, i) = 0 otherwise
Then, for each x ε X
A(x) = B(x) = 1
By the Theorem 3

Image for - C-Fusion Frame

Then, and for each x ε X,

Image for - C-Fusion Frame

Since, u is surjective, there is C > 0 such that

Image for - C-Fusion Frame

So,

Image for - C-Fusion Frame

Similarly, we have

Image for - C-Fusion Frame

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 = TFTF*. Then:

(i)  We have the following retrieval formulas

Image for - C-Fusion Frame

and

Image for - C-Fusion Frame

(ii) In the retrieval formula

Image for - C-Fusion Frame

Image for - C-Fusion Frame has least norm among all of the retrieval formulas.

(iii) For each h ε H,

Image for - C-Fusion Frame

Proof:

(i) Since (F, v) is a c-fusion frame, SF is an invertible operator. By the Theorem 4, we have

Image for - C-Fusion Frame

Also, we have

Image for - C-Fusion Frame

(ii) Let f ε L2(X, F) and

h = TF (f)

Thus, for each k ε H we have

Image for - C-Fusion Frame

Therefore

Image for - C-Fusion Frame

So,

Image for - C-Fusion Frame

Hence

Image for - C-Fusion Frame

Since, F is a c-fusion frame,

Image for - C-Fusion Frame

But,

Image for - C-Fusion Frame

So,

Image for - C-Fusion Frame

and (ii) is proved.

(iii) Let f ε L2(X, F). Since, TFx is the unique solution of minimal norm of TF(f) = h so by

(ii),

Image for - C-Fusion Frame

Therefore,

Image for - C-Fusion Frame

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,

Image for - C-Fusion Frame

(ii) For each h ε H,

Image for - C-Fusion Frame

(iii) For each h , k ε H,

Image for - C-Fusion Frame

(iv) For each h ε H,

Image for - C-Fusion Frame

(v) For each orthonormal bases

{ei}iεI and {λj}jεJ

for H we have

Image for - C-Fusion Frame

(iv) For each orthonormal bases {ei}iεI for H and i ε I,

Image for - C-Fusion Frame

Proof: (i) → (ii) Let h, k ε H. We have

Image for - C-Fusion Frame

Hence, Image for - C-Fusion Frame

(ii) → (iii) It is evident by the proof of (i) → (ii).
(iii) → (i) For each h, k ε H, we have

Image for - C-Fusion Frame

Thus Image for - C-Fusion Frame
(iv) → (i) Let L: H → H be defined by

Image for - C-Fusion Frame

It clear that L is linear. Since

Image for - C-Fusion Frame

that, L ε B(H). For each h ε H, we have

Image for - C-Fusion Frame

Hence, for each h ε H,

Image for - C-Fusion Frame

(iii) → (iv) is evident.
(v) → (iii) We have

Image for - C-Fusion Frame

(vi) → (v) it is similar with the proof of (v) → (iii).

REFERENCES
1:  Benedetto, J.J., A.M. Powell and O. Yilmaz, 2004. Sigma-Delta quantization and finite frames. Acoustics Speech Signal Process., 3: 937-940.

2:  Bolcskel, H., F. Hlawatsch and H.G. Feichyinger, 1998. Frame-Theoretic analysis of oversampled filter bank. IEEE Trans. Signal Process, 46: 3256-3268.
CrossRef  |  Direct Link  |  

3:  Candes, E.J. and D.L. Donoho, 2004. New tight frames of curvelets and optimal representation of objects with piecwise C2 singularities. Comm. Pure Applied Math., 57: 219-266.
CrossRef  |  Direct Link  |  

4:  Casazza, P.G. and J. Kovacevic, 2003. Equal-norm tight frames with erasures. Adv. Comput. Math., 18: 387-430.
CrossRef  |  Direct Link  |  

5:  Casazza, P.G. and G. Kutyniok, 2004. Frame of subspaces. Contemporery Math., 345: 87-114.
Direct Link  |  

6:  Christensen, O., 2002. An introduction to frames and Riesz bases. Birkhauser, Boston.

7:  Daubechies, I. and A. Grossmann and Y. Meyer, 1986. Painless nonorthogonal expansions. J. Math. Phys., 27: 1271-1283.
CrossRef  |  Direct Link  |  

8:  Duffin, R.J. and A.C. Schaeffer, 1952. A class of nonharmonik fourier series. Trans. Am. Math. Soc., 72: 341-366.
Direct Link  |  

9:  Hassibi, B., B. Hochwald, A. Shokrollahi and W. Sweldens, 2001. Representation theory for high-rate multiple-antenna code design. IEEE Trans. Inform. Theory., 47: 2335-2367.
CrossRef  |  Direct Link  |  

10:  Pedersen, G.K., 1989. Analysis Now. 1st Edn. Springer Verlag, New York.

11:  Rudin, W., 1973. Functional Analysis. 1st Edn. Tata McGraw Hill Editions, New York.

12:  Rudin, W., 1986. Real and Complex Analysis. 1st Edn. McGraw Hill International Editions, New York.

13:  Sakai, S., 1998. C*-Algebras and W*-Algebras. 1st Edn. Springer Verlag, New York.

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