# Subfusion Frames

Authors: Z. Amiri; M. A. Dehghan; E. Rahimi
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

Fusion frames are generalizations of frames in Hilbert spaces which were introduced by Casazza et al. (2008). In the present paper, we study the relations between fusion frames and subfusion frame operators. Specially, we introduce new construction of subfusion frames and derive new results.


---

## Body

## 1. Introduction

Frames were first introduced in 1924 by Duffin and Schaeffer [1]. Daubechies et al. in [2] found a fundamental new application. Nice properties of frames make them very useful in filter banks, sigma-delta quantization, signal and image processing. The theory of frames has been generalized rapidly, and various generalizations of frames have been proposed recently. Later general frame theory of subspaces was introduced by Casazza and Kutyniok [3] and Fornasier [4] as a natural generalization of the frame theory in Hilbert spaces. Since frames, particular frames of subspaces, are applied to signal processing, image processing, data compression, and sampling theory, we consider frames of subspaces on Hilbert spaces and extend some of the known results about bases and frames to frames of subspaces. Recently, the frames of subspaces have been renamed as fusion frames. This notion has been intensely studied earlier and several new applications have been discovered. The reader is referred to the works by Casazza and Kutyniok [3] and Găvruţa [5]. There exists a variety of applications which cannot be modeled naturally by one ordinary frame, for example, wireless sensor network [6], sensor geophones in geophysics measurement and studies [7], and the physiological structure of visual and hearing system [8].Letν={(Vi,αi)}i∈I be a fusion frame for H, Wi be a closed subspace of Vi and βi≤αi for all i∈I. If ω={(Wi,βi)}i∈I is a fusion frame for H, then ω is called a subfusion frame of ν. If ν and ω are Bessel fusion sequences for H, then ω is called a Bessel subfusion sequence of ν.In [9], the authors introduced Bessel subfusion sequences and subfusion frames and they investigated the relationship between their operation. Also, the definition of the orthogonal complement of subfusion frames and the definition of the completion of Bessel fusion sequences were provided, and several results related with these notions were shown.A notion related to subfusion frames has been brought in [10], which is called frame of subspaces refinement (shortly: FSR). A subfusion frame ω={(Wi,βi)}i∈I of ν={(Vi,αi)}i∈I is a FSR if αi=βi for all i∈I. Therefore, an FSR is a special subfusion frame and the authors have studied the excess of FSR in [10].In the present paper, we study the relations between fusion frames and subfusion frame operators. We also obtain some results about subfusion frame operators that these results are not true for fusion frame operators.This paper is organized as follows. Section2 briefly review the concept of frames, subfusion frames, and their properties. Section 3 includes some results of operator obtained of Bessel subfusion sequences. In [11], the authors tried to show that the frame operator for a pair of fusion frames is bounded below and invertible, but we show this is not true. We further prove that the frame operator for a pair of subfusion frames is bounded below and invertible. Also, we will study operators for a pair of Bessel subfusion sequences. In Section 4, we study some constructions of subfusion frames. Finally, Section 5 contains a discussion on dual subfusion frames. In [5], it has been shown that the dual fusion frame is a fusion frame. In this section, through an example, we show if ω={(Wi,βi)}i∈I be a subfusion frame of ν={(Vi,αi)}i∈I, then {(Sω-1Wi,βi)}i∈I is not necessarily a subfusion frame of {(Sν-1Vi,αi)}i∈I, and we show that under some conditions, {(Sω-1Wi,βi)}i∈I is a subfusion frame of {(Sν-1Vi,αi)}i∈I.Through this paper,H is a separable Hilbert space, I is a countable index set, and {Vi}i∈I is a sequence of closed subspaces of H.


## 2. Review of Frames, Fusion Frames, and Subfusion Frames

In this section, we recall some definitions and basic properties of frames, fusion frames and subfusion frames. For more information, we refer the reader to [3, 9, 12, 13].Definition 2.1.
A sequence{fi}i∈I of elements in H is a frame for H if there exist positive constants A and B (lower and upper frame bounds, resp.) such that
(2.1)A‖f‖2≤∑i∈I‖〈f,fi〉‖2≤B‖f‖2,∀f∈H.Definition 2.2.
Let{Vi}i∈I be a family of closed subspaces of a Hilbert space H and let {αi}i∈I be a family of weights, that is, αi>0 for all i∈I. Then ν={(Vi,αi)}i∈I is a fusion frame, if there exist positive constants C and D (lower and upper fusion frame bounds, resp.) such that
(2.2)C‖f‖2≤∑‍i∈Iαi2‖πVi(f)‖2≤D‖f‖2,∀f∈H,
where πVi is the orthogonal projection onto the subspace Vi. A fusion frame ν is called λ-tight fusion frame if C=D=λ, Parseval fusion frame if C=D=1, and α-uniform fusion frame if α=αi for all i∈I. If the second part of the above inequality is satisfied, then ν is called a Bessel fusion sequence for H with bound D.
Similar to ordinary frames, the fusion frame operatorSν is defined by
(2.3)Sν(f)=∑‍i∈Iαi2πVif,∀f∈H.Sν is a linear, positive, self-adjoint, and invertible operator and we have
(2.4)CI≤Sν≤DI.
A family of bounded operators{Ti}i∈I on H is called a resolution of identity on H if f=∑i∈ITif, for all f∈H.Proposition 2.3.
Let{(Vi,αi)}i∈I be a fusion frame for H, Wi be a closed subspace of Vi and βi≤αi for all i∈I. Then {(Wi,βi)}i∈I is a Bessel fusion sequence for H.Proof.
SinceWi is a closed subspace of Vi, πWiπVif=πViπWif=πWif and ∥πWif∥2≤∥πVif∥2 for all f∈H and for all i∈I. Hence,
(2.5)∑‍i∈Iβi2‖πWif‖2≤∑‍i∈Iαi2‖πVif‖2
implies that {(Wi,βi)}i∈I is a Bessel fusion sequence.There are examples such that{(Vi,αi)}i∈I is a fusion frame, Wi is a closed subspace of Vi and βi≤αi for all i∈I, while {(Wi,αi)}i∈I is not a fusion frame.Example 2.4.
Let{(Vi,αi)}i∈ℕ be a fusion frame for H and V1≠H. Define
(2.6)Wi={V1ifi=1,0otherwise.
Since span¯i∈ℕ{Wi}=V1≠H, {(Wi,αi)}i∈I is not a fusion frame for H [3, Lemma 3.4].Definition 2.5.
Letν={(Vi,αi)}i∈I be a fusion frame for H, Wi be a closed subspace of Vi and βi≤αi for all i∈I. If ω={(Wi,βi)}i∈I is a fusion frame for H, then ω is called a subfusion frame of ν. If ν and ω are Bessel fusion sequences for H, then ω is called a Bessel subfusion sequence of ν [9].


## 3. Operators between a Pair of Bessel Subfusion Sequences

In this section, we will study operators for a pair of Bessel subfusion sequences. Alternate dual frames and Bessel fusion sequences are important in the literature of frame theory because of their important role in applications. The notions of operators for a pair of Bessel fusion sequences and alternative dual of a fusion frame inH are defined by Găvruţa in [5].Letν={(Vi,αi)}i∈I and ω={(Wi,βi)}i∈I be two Bessel fusion sequences for H. Then the frame operator for them is defined by
(3.1)Sωνf=∑‍i∈IαiβiπWiπVif,∀f∈H.
Moreover, if ν={(Vi,αi)}i∈I is a fusion frame for H, with fusion frame operatore Sν, then ω is called an alternate dual of ν, if we have
(3.2)f=∑‍i∈IαiβiπWiSν-1πVif,∀f∈H.Sων is bounded and Sων=Sνω*. Recently, Khosravi and Musazadeh in [11, Proposition  2.9] tried to show that Sων is bounded below and invertible. “Let ν={(Vi,αi)}i∈I be a fusion frame with fusion frame bounds C and D and fusion frame operator Sν for H. Let ω={(Wi,βi)}i∈I be an alternate dual fusion frame for ν with required positivity. Then we have
(3.3)CI≤Sων≤DI,
and also Sων is invertible.”In the following example, we show thatSων is neither invertible nor bounded below.Example 3.1.
Set(3.4)V1=〈(1,0,0)〉,V2=〈(1,1,0)〉,V3=〈(0,1,0)〉,V4=〈(0,0,1)〉,
and α1=α3=α4=1,α2=2. Define πVi:ℝ3→Vi by
(3.5)πV1(a,b,c)=(a,0,0),πV2(a,b,c)=(a+b2,a+b2,0),πV3(a,b,c)=(0,b,0),πV4(a,b,c)=(0,0,c).
For any f=(a,b,c) in ℝ3.Computation show thatπVi, i={1,2,3,4} are projection (πVi is projection if πVi2=πVi [14]). We have
(3.6)Sν(f)=(a,0,0)+(a+b,a+b,0)+(0,b,0)+(0,0,c)=(2a+b,2b+a,c).
Then
(3.7)Sν=[210120001],Sν-1=[23-130-13230001].
Now, let
(3.8)W1=〈(0,1,0)〉,W2=R3,W3=〈(1,0,0)〉,W4=〈(0,0,1)〉,
and β1=β3=3,β2=32,β4=1. Define πWi:ℝ3→Wi by
(3.9)πW1(a,b,c)=(0,b,0),πW2(a,b,c)=(a,b,c),πW3(a,b,c)=(a,0,0),πW4(a,b,c)=(0,0,c).
Obviously, ω={(Wi,βi)}i∈I is an alternate dual of ν={(Vi,αi)}i∈I, where I={1,2,3,4}. Then, we have
(3.10)∑‍i=14αiβiπWiSν-1πVi(f)=3πW1[2a3-a30]+6πW2[a+b6a+b60]+3πW3[-b32b30]+πW4[00c]=[0-a0]+[a+ba+b0]+[-b00]+[00c]=(a,b,c)=f.
Also
(3.11)Sων=∑‍i=14αiβiπWiπVi(f)=α2β2πW2πV2(f)+α4β4πW4πV4(f)=6πW2[a+b2a+b20]+πW4[00c]=3[a+ba+b0]+[00c]=[3(a+b)3(a+b)c].
Then Sων=[330330001]. Therefore, Sων is not invertible. If f=(1,-1,0), then there is not a positive number C such that
(3.12)C〈f,f〉≤〈Sωνf,f〉.Next, we show that under some conditions,Sων is invertible and bounded below.Proposition 3.2.
Letν={(Vi,αi)}i∈I be a fusion frame with fusion frame bounds C and D and fusion frame operator Sν for H. Let ω={(Vi,βi)}i∈I be an alternate dual fusion frame for ν with required positivity. Then we have
(3.13)CI≤Sων≤DI,
and Sων is invertible.Proof.
Letf be an arbitrary element of H. Then we have
(3.14)‖f‖2=〈f,f〉=〈∑‍i∈IαiβiπViSν-1πVi(f),f〉=∑‍i∈Iαiβi〈Sν-1πVif,πVif〉≤1C∑‍i∈Iαiβi〈πVif,πVif〉=1C〈∑‍i∈IαiβiπVif,f〉=1C〈Sωνf,f〉.
Similarly, we have 〈Sωνg,f〉≤D∥f∥2. Hence, Sων is injective and SωνH is closed in H(3.15)Range(Sων)=Range(Sων)¯=(N(Sων*))⊥=(N(Sων))⊥=H.
Then Sων is invertible.Now, we are ready to describe the operator for a pair of subfusion frames. In this case, positivity, invertibility, and boundedness properties of these operators have been checked.Proposition 3.3.
Letω={(Wi,βi)}i∈I be a Bessel subfusion frame of ν={(Vi,αi)}i∈I. Then Sων is self-adjoint and positive.Proof.
For anyf∈H(3.16)Sνωf=∑‍i∈IαiβiπViπWif=∑‍i∈IαiβiπWif=∑‍i∈IαiβiπWiπVif=Sωνf,
then Sνω=Sων. Also,
(3.17)〈Sωνf,f〉=∑‍i∈Iαiβi〈πWif,f〉=∑‍i∈Iαiβi‖πWif‖2.
Then Sων is positive.Lemma 3.4.
Letω={(Wi,βi)}i∈I be Bessel subfusion sequence of ν={(Vi,αi)}i∈I. If Sων is bounded below, then ω is a subfusion frame of ν.Proof.
Suppose that there exist a numberλ>0 such that for all f∈H(3.18)λ‖f‖≤‖Sωνf‖,
Then we have
(3.19)λ‖f‖≤‖Sωνf‖=sup⁡g∈H,∥g∥=1‖〈∑‍i∈IαiβiπWiπVif,g〉‖≤sup⁡g∈H,‖g‖=1(∑‍i∈Iβi2‖πWif‖2)1/2(∑‍i∈Iαi2‖πVig‖2)1/2≤D(∑‍i∈Iβi2‖πWif‖2)1/2,
where D is upper bound of ν. Hence,
(3.20)λ2D‖f‖2≤∑‍i∈Iβi2‖πWif‖2≤∑‍i∈Iαi2‖πVif‖2.Corollary 3.5.
Letω={(Wi,βi)}i∈I be a Bessel subfusion sequence of ν={(Vi,αi)}i∈I and Sων be invertible. Then ω is a subfusion frame of ν.Proof.
SinceSων is invertible, Sων is a below bounded. Then ω is a subfusion frame of ν.Lemma 3.6.
Letω={(Wi,βi)}i∈I be Bessel subfusion frame of ν={(Vi,αi)}i∈I. Then
(3.21)Sω≤Sων≤Sν.Proof.
(3.22)
〈
(
S
ω
ν
-
S
ω
f
)
,
f
〉
=
〈
∑
‍
i
∈
I
α
i
β
i
π
W
i
f
-
∑
‍
i
∈
I
β
i
2
π
W
i
f
,
f
〉
=
∑
‍
i
∈
I
β
i
(
α
i
-
β
i
)
〈
π
W
i
f
,
f
〉
=
∑
‍
i
∈
I
β
i
(
α
i
-
β
i
)
∥
π
W
i
f
∥
2
,
then Sω≤Sων. We have
(3.23)〈(Sν-Sωνf),f〉=〈∑‍i∈Iαi2πVif-∑‍i∈IαiβiπWif,f〉=∑‍i∈Iαi(〈αiπVif,f〉-〈βiπWif,f〉)=∑‍i∈Iαi(αi‖πVif‖2-βi‖πVif‖2),
since βi∥πVif∥2≤αi∥πVif∥2,Sων≤Sν. Hence, Sω≤Sων≤Sν.Now we show that ifω be a subfusion frame of ν, then Sων is invertible.Corollary 3.7.
Letω={(Wi,αi)}i∈I be a subfusion frame of ν={(Vi,αi)}i∈I. Then Sων is invertible.Proof.
For allf∈H(3.24)Sωνf=∑‍i∈Iαi2πWiπVif=∑‍i∈Iαi2πWif=Sωf,
then Sων=Sω. Since Sω is invertible, hence, Sων is invertible.Lemma 3.8.
Ifω={(Wi,βi)}i∈I be a subfusion frame of ν={(Vi,αi)} such that C is lower bound of ω and D is upper bound of ν, then
(3.25)CI≤Sων≤DI,
and Sων is invertible.Proof.
By Lemma3.6, we have
(3.26)CI≤Sων≤DI.
Hence, Sων is injective, SωνH is closed in H. We have
(3.27)Range(Sων)=Range(Sων)¯=(N(Sων*))⊥=(N(Sων))⊥=H.
Hence, Sων is onto and therefor, invertible on H. On the other hand, since CI≤Sων≤DI, we have 1/D≤∥Sων-1∥≤1/C.Remark 3.9.
By this lemma we have the following reconstruction formulas:(3.28)f=∑‍i∈IαiβiπWiSων-1f=∑‍i∈IαiβiSων-1πWif,
for all f∈H. Therefore, two families of bounded operators {αiβiπWiSων-1}i∈I and {αiβiSων-1πWi}i∈I are resolutions of the identity.Theorem 3.10.
Letν={(Vi,αi)}i∈I be a fusion frame with upper bound D and ω={(Wi,βi)}i∈I be a Bessel subfusion sequence of ν. Suppose that there exists a number λ>D such that λ∥f∥≤∥(Sν-Sων)f∥, for all f∈H. Then Sων is invertible and also ω is a subfusion frame of ν.Proof.
Letf be an arbitrary element of H. Then we have
(3.29)‖Sωνf‖=‖(Sων-Sν+Sν)f‖≥‖(Sων-Sν)f‖-‖Sνf‖≥(λ-D)‖f‖.
So Sων is bounded below, hence Sων is invertible and there exists a number B>0 such that B∥f∥≤∥Sωνf∥, for all f∈H. Then
(3.30)‖Sωνf‖≤D(∑‍i∈Iβi2‖πWif‖2)1/2.
Hence,
(3.31)B2D‖f‖2≤∑‍i∈Iβi2‖πWif‖2≤∑‍i∈Iαi2‖πVif‖2≤D‖f‖2.


## 4. Construction of Subfusion Frames

In this section, we study some new constructions of subfusion frames. Dealing with Bessel subfusion frames is important, since there are easy ways to turn such a family into  subfusion frames. One way is to just add the subspaceW0=V0=H to the families. We follow some ways that obtain new subfusion frames.Lemma 4.1.
Let{(Wi,αi)}i∈I be a fusion frame for H with bound C and D, and Let {Vi}i∈I be a family of closed subspace H such that Wi⊂Vi for all i∈I. If there exists λ>0 such that
(4.1)∑‍i∈Iαi2‖πVif-πWif‖2≤λ‖f‖2,
for all f∈H, then {(Wi,αi)}i∈I is a subfusion frame of {(Vi,αi)}i∈I.Proof.
By using the triangle inequality, for allf∈H, we have
(4.2)‖{αiπVif}i∈I‖l2≤‖{αi(πVif-πWif)}i∈I‖l2+‖{αiπWif}i∈I‖l2≤λ‖f‖+D‖f‖=D(1+λD)‖f‖;
therefore,
(4.3)C‖f‖2≤∑‍i∈Iαi2‖πWif‖2≤∑‍i∈Iαi2‖πVif‖2≤D(1+λD)2‖f‖2.Theorem 4.2.
Letω={(Wi,αi)}i∈I be a fusion frame for H, and Let {Vi}i∈I be a family of closed subspace in H such that Wi⊂Vi for i∈I. If U:H→H defined by
(4.4)U(f)=∑‍i∈Iαi2(πVif-πWif)∀f∈H,
is a bounded operator, then ω is a subfusion frame of {(Vi,αi)}i∈I.Proof.
LetC and D be the frame bounds for ω. Then ∥Sω∥≤D. A simple computation show that U is self-adjoint. So if T:H→H is defined by T=Sω+U, then T is a bounded, linear and self-adjoint operator. Therefore,
(4.5)‖T‖=sup⁡‖f‖≤1‖〈Tf,f〉‖=sup⁡‖f‖≤1∑‍αi2‖πVif‖2,∑i∈Iαi2‖πVif‖2≤‖T‖‖f‖2≤(‖Sω‖+‖U‖)‖f‖2≤(D+‖U‖)‖f‖2,
for all f∈H. Hence,
(4.6)C‖f‖2≤∑‍i∈Iαi2‖πWif‖2≤∑‍i∈Iαi2‖πVif‖2≤(D+‖U‖)‖f‖2.Proposition 4.3.
Letω={(Wi,βi)}i∈I be a Bessel subfusion sequence of ν={(Vi,αi)}i∈I and Sων be below bounded. Then(i){(SωνWi,βi)}i∈I is a subfusion frame of {(SωνVi,αi)}i∈I.(ii){(Sων-1Wi,βi)}i∈I is a subfusion frame of {(Sων-1Vi,αi)}i∈I.Proof.
The proof is straight forward.Lemma 4.4.
Let{Wi}i∈I and {Vi}i∈I be a closed subspaces of H and J⊂I. Suppose that
(4.7)Wi={Vii∈J,〈0〉i∈I-J.
If ω={(Wi,αi)}i∈I is a fusion frame for H with bound C,D and c=∑i∈I-Jαi2, then ω is a subfusion frame of ν={(Vi,αi)}i∈I with bound C,D+c.Proof.
For allf∈H, we have
(4.8)∑i∈Iαi2‖πVif‖2=∑‍i∈I-Jαi2‖πVif‖2+∑‍i∈Jαi2‖πVif‖2=∑‍i∈I-Jαi2‖πVif‖2+∑‍i∈Jαi2‖πWif‖2≤D‖f‖2+(∑‍i∈I-Jαi2)‖f‖2≤(D+c)‖f‖2.
then ω is a subfusion frame of ν.


## 5. Dual Subfusion Frames

Letν={(Vi,αi)}i∈I be a fusion frame, then the family {(Sν-1Vi,αi)}i∈I is called the dual fusion frame. In [5], it has been shown that the dual fusion frame is a fusion frame. In this section, through an example, we show that if ω={(Wi,βi)}i∈I be a subfusion frame of ν={(Vi,αi)}i∈I, then {(Sω-1Wi,βi)}i∈I is not necessarily a subfusion frame of {(Sν-1Vi,αi)}i∈I.Example 5.1.
SupposeV1=〈e1〉,V2=V3=R3. We define πVi:R3→Vi by πV1(a,b,c) = (a,0,0),πV2 = πV3(a,b,c) = (a,b,c).{(V1,α1),(V2,α2),(V3,α3)} is a fusion frame. We have
(5.1)Sν=[α12+α22+α32000α22+α32000α22+α32].
Also we assume that W1=〈e1〉,W2=〈(1,1,0)〉,W3=R3.We defineπWi:R3→Wi by πW1(a,b,c) = (a,0,0),πW2(a,b,c) = ((a+b)/2,(a+b)/2,0),πW3(a,b,c) = (a,b,c). The set {(W1,β1),(W2,β2),(W3,β3)} is a fusion frame and
(5.2)Sω=[β12+β222+β32β2220β222β222+β32000β32].
Set α12=α32=1,α22=3, β12=β32=1 and β22=2. It is clear that {(Sω-1Wi,βi)}i∈I is not a subfusion frame of {(Sν-1Vi,αi)}i∈I where I={1,2,3}.Now by another example we show that{(Sν-1Vi,αi)}i∈I is not subfusion frame of {(Sω-1Wi,βi)}i∈I.Example 5.2.
SupposeV1=〈e1〉,V2=V3=R3. We define πVi:R3→Vi by πV1(a,b,c) = (a,0,0),πV2(a,b,c) = πV3(a,b,c) = (a,b,c).{(Vi,αi)}i∈I where I={1,2,3} is a fusion frame, We assume that W1=〈e1〉,W2=〈e2〉,W3=R3. We define πWi:R3→Wi by
(5.3)πW1(a,b,c)=(a,0,0),πW2(a,b,c)=(0,b,0),πW3(a,b,c)=(a,b,c).
The set{(W1,β1),(W2,β2),(W3,β3)}is a fusion frame and
(5.4)Sω=[β12+β32000β22+β32000β32].
Set α12=α32=1,α22=3, β12=β32=1 and β22=3. A simple computation shows that {(Sν-1Vi,αi)}i∈I is not a subfusion frame of {(Sω-1Wi,βi)}i∈I.Proposition 5.3.
Letω={(Wi,βi)}i be a subfusion frame of ν={(Vi,αi)}i∈I. If Vi⊥Vj and Wi⊥Wj for all i≠j, then {(Sω-1Wi,βi)}i∈I is a subfusion frame of {(Sν-1Vi,αi)}i∈I.Proof.
Setf∈Vj and f=αj(g)g∈Vj,f=αj(g)=∑i∈Iαi2πVi(g)∈SνVj then Vj⊂SνVj for all j∈I. If f∈SνVj,f=∑i∈αi2πVi(g)=αj(g)g∈Vj, then SνVj⊂Vj. Hence, Sν-1Vj=Vj.



---

*Source: 603580-2012-06-26.xml*