Some properties of dual and approximate dual of fusion frames

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by  Ali Akbar Arefijamaal

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1 Some properties of dual and approximate dual
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of fusion frames

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Ali Akbar Arefijamaal and Fahimeh Arabyani Neyshaburi
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Abstract. In this paper we extend the notion of approximate dual to fusion
[ framesandpresentsomeapproachestoobtaindualandapproximatealternate

dual fusion frames. Also, we study the stability of dual and approximate
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alternate dual fusion frames.
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4 MathematicsSubjectClassification(2010).Primary42C15;Secondary41A58.
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0 Key words: Fusion frames; alternate dual fusion frames; approximate alter-
. nate duals; Riesz fusion bases.
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1. Introduction and preliminaries
:
v
i FusionframetheoryisanaturalgeneralizationofframetheoryinseparableHilbert
X
spaces, which is introduced by Casazza and Kutyniok in [5] and also Fornasier
r
a in [9]. Fusion frames are applied to signal processing, image processing, sampling
theory,filterbanksandavarietyofapplicationsthatcannotbemodeledbydiscrete
frames [13, 16].
Let I be a countable index set, recallthat a sequence f is a frame in a
i i∈I
{ }
separable Hilbert space if there exist constants 0<A B < such that
H ≤ ∞
2 2 2
A f f,f B f , (f ). (1.1)
i
k k ≤ |h i| ≤ k k ∈H
i∈I
X
The constantsAandB arecalledthe lower andupper frame bounds,respectively.
Itissaidthat f isaBesselsequence iftherightinequalityin(1.1)issatisfied.
i i∈I
{ }
Given a frame f , the frame operator is defined by
i i∈I
{ }
Sf = f,f f , (f ).
i i
h i ∈H
i∈I
X
It is a bounded, invertible, and self-adjoint operator [8]. The family S−1f
i i∈I
{ }
is also a frame for , the so called the canonical dual frame. In general, a Bessel
H
2 A. A. Arefijamaal and F. Arabyani Neyshaburi
sequence g is called an alternate dual or simply a dual for the Bessel
i i∈I
{ } ⊆ H
sequence f if
i i∈I
{ }
f = f,g f , (f ). (1.2)
i i
h i ∈H
i∈I
X
The synthesis operator T : l2 of a Bessel sequence f is defined
i i∈I
→ H { }
by T c = c f . By (1.2) two Bessel sequences f and g are
{ i}i∈I i∈I i i { i}i∈I { i}i∈I
dual of each other if and only if T T∗ = I , where T and T are the synthesis
P G F H F G
operators f and g , respectively. For more details on the frame theory
i i∈I i i∈I
{ } { }
we refer to [4, 8].
Nowwereviewbasicdefinitionsandprimaryresultsoffusionframes.Through-
out this paper, π denotes the orthogonal projection from Hilbert space onto
V
H
a closed subspace V.
Definition 1.1. Let W be a family of closed subspaces of and ω a
i i∈I i i∈I
{ } H { }
family of weights, i.e. ω > 0, i I. Then (W ,ω ) is called a fusion frame
i i i i∈I
∈ { }
for if there exist the constants 0<A B < such that
H ≤ ∞
A f 2 ω2 π f 2 B f 2, (f ). (1.3)
k k ≤ ik Wi k ≤ k k ∈H
i∈I
X
The constants A and B are called the fusion frame bounds. If we only have
the upper bound in (1.3) we call (W ,ω ) a Bessel fusion sequence. A fusion
i i i∈I
{ }
frame is called A-tight, if A = B, and Parseval if A = B = 1. If ω = ω for
i
all i I, the collection (W ,ω ) is called ω-uniform and we abbreviate 1-
i i i∈I
∈ { }
uniform fusion frames as W . A family of closed subspaces W is called
i i∈I i i∈I
{ } { }
an orthonormal basis for when W = and it is a Riesz decomposition of
i∈I i
H ⊕ H
, if for every f there is a unique choice of f W such that f = f .
H ∈ H i ∈ i i∈I i
Also a family of closed subspaces W is called a Riesz fusion basis whenever
i i∈I
{ } P
it is complete for and there exist positive constants A, B such that for every
H
finite subset J I and arbitrary vector f W , we have
i i
⊂ ∈
A f 2 f 2 B f 2.
i i i
k k ≤k k ≤ k k
i∈J i∈J i∈J
X X X
It is clear that every Riesz fusion basis is a 1- uniform fusion frame for ,
H
and also a fusion frame is a Riesz basis if and only if it is a Riesz decomposition
for , see [3, 5].
H
For every fusion frame a useful local frame is proposed in the following the-
orem.
Theorem1.2. [5] Let W be a family of subspaces in and ω a family
i i∈I i i∈I
{ } H { }
ofweights.Then (W ,ω ) isafusionframefor withboundsAandB,ifand
i i i∈I
{ } H
only if ω π e is a frame for , with the same bounds, where e
{ i Wi j}i∈I,j∈J H { j}j∈J
is an orthonormal basis for .
H
Also, a connection between local and global properties is given in the next
result, see [5].
Some properties of dual and approximate dual of fusion frames 3
Theorem 1.3. For each i I, let W be a closed subspace of and ω > 0. Also
i i
∈ H
let f be a frame for W with frame bounds α and β such that
{ i,j}j∈Ji i i i
0<α=inf α β =sup β < . (1.4)
i∈I i i∈I i
≤ ∞
Then the following conditions are equivalent.
(i) (W ,ω ) is a fusion frame of with bounds C and D.
i i i∈I
{ } H
(ii) ω f is a frame of with bounds αC and βD, .
{ i i,j}i∈I,j∈Ji H
Recall that for each sequence W of closed subspaces in , the space
i i∈I
{ } H
W = f :f W , f 2 < ,
i i i∈I i i i
⊕ {{ } ∈ k k ∞}
i∈I i∈I
X X
with the inner product
f , g = f ,g ,
i i∈I i i∈I i i
h{ } { } i h i
i∈I
X
is a Hilbert space. For a Bessel fusion sequence (W ,ω ) of , the synthesis
i i i∈I
{ } H
operator T : W is defined by
W i∈I⊕ i →H
PT ( f )= ω f , ( f W ).
W i i∈I i i i i∈I i
{ } { } ∈ ⊕
i∈I i∈I
X X
Its adjoint operator T∗ : W , which is called the analysis operator,
W H → i∈I⊕ i
is given by
P
T∗ (f)= ω π (f) , (f ).
W { i Wi }i∈I ∈H
Let (W ,ω ) be a fusion frame, the fusion frame operator S : is
i i i∈I W
defin{edbyS }f = ω2π f isabounded,invertibleaswellaspositiHve.→HeHnce,
W i∈I i Wi
we have the following reconstruction formula [5]
P
f = ω2S−1π f, (f ).
i W Wi ∈H
i∈I
X
The family (S−1W ,ω ) , which is also a fusion frame, is called the canoni-
{ W i i }i∈I
cal dual of (W ,ω ) . Also, a Bessel fusion sequence (V ,ν ) is called an
i i i∈I i i i∈I
{ } { }
alternate dual of (W ,ω ) , [10] whenever
i i i∈I
{ }
f = ω ν π S−1π f, (f ). (1.5)
i i Vi W Wi ∈H
i∈I
X
In [10], it is proved that every alternate dual of a fusion frame is a fusion
frame. Also, easily we can see that a Bessel fusion sequence (V ,υ ) is an
i i i∈I
{ }
alternate dual fusion frame of (W ,ω ) if and only if T φ T∗ =I , where
{ i i }i∈I V vw W H
the bounded operator φ : W V is given by
vw i∈I i → i∈I i
φvwP({fi}Li∈I)={πPViSW−1Lfi}i∈I. (1.6)
Moreover, a Bessel fusion sequence V = (V ,ω ) given by V = S−1W U ,
{ i i }i∈I i W i⊕ i
is an alternate dual fusion frame of (W ,ω ) in which U is a closedsubspace
i i i∈I i
{ }
of for all i I, [15]. Recently, Heineken et al. introduced the other concept of
H ∈
dual fusion frames, [12]. For two fusion frames (W ,ω ) and (V ,υ ) if
i i i∈I i i i∈I
{ } { }
there exists a mapping Q B( W , V ), such that T QT∗ = I
∈ i∈I i i∈I i V W H
P L P L
4 A. A. Arefijamaal and F. Arabyani Neyshaburi
then (V ,υ ) is called a Q-dual of (W ,ω ) . Clearly every alternate dual
i i i∈I i i i∈I
{ } { }
fusionframeisaφ -dual.Q-dualsareusefultoolsforestablishingreconstruction
vw
formula. For more information on fusion frames we refer the reader to [3, 5, 6].
2. Approximate duals
Alternate dual fusion frames play a key role in fusion frame theory, however,
theirexplicitcomputationsseemratherintricate.Inthis section,weintroducethe
notionofapproximatedualforfusionframesanddiscusstheexistenceofalternate
dual fusion frames from an approximate alternate dual. Moreover, we present a
complete characterization of alternate duals of Riesz fusion bases. The notion of
approximate dual for discrete frames has been already introduced by Christensen
and Laugesen in [7] and then for g-frames in [14], however many of theirs results
are invalid for fusion frames. Throughout this section we consider a Riesz fusion
basis as a 1-uniform fusion frame.
First, we recall the notion of approximate dual for discrete frames. Let F =
f and G = g be Bessel sequences for . Then F and G are called
i i∈I i i∈I
a{pp}roximate dual f{ram}es if I T T∗ <1.In thiHs case (T T∗)−1g is a dual
k H− G Fk { G F i}
of F, see [7].
Now we introduce approximate duality for fusion frames.
Definition 2.1. Let (W ,ω ) be a fusion frame. A Bessel fusion sequence
i i i∈I
{ }
(V ,υ ) is called an approximate alternate dual of (W ,ω ) if
i i i∈I i i i∈I
{ } { }
I T φ T∗ <1.
k H− V vw Wk
Putting
ψ =T φ T∗ . (2.1)
vw V vw W
Then, we have the following reconstruction formula
∞
f = (ψ )−1ω υ π S−1π f = (I ψ )nψ f, (f ).
vw i i Vi W Wi − vw vw ∈H
i∈I n=0
X X
Proposition2.2. LetV = (V ,υ ) beanapproximatealternatedualofafusion
i i i∈I
{ }
frame W = (W ,ω ) . Then V is a fusion frame.
i i i∈I
{ }
Proof. Let B and D be Bessel bounds of W and V, respectively. Then
ψ∗ f 2 = sup T φ∗ T∗f,g 2
k vw k kgk=1|h W vw V i|
= sup ω υ π S−1π f,g 2
kgk=1|h i i Wi W Vi i|
i∈I
X
sup υ2 π f 2 ω2 S−1π g 2
≤ kgk=1i∈I ik Vi k i∈I ik W Wi k
X X
S−1 2B υ2 π f 2,
≤ k W k ik Vi k
i∈I
X
Some properties of dual and approximate dual of fusion frames 5
for every f . This follows that
∈H
(ψ−1)∗ −2
f 2k vw k υ2 π f 2 D f 2.
k k S−1 2B ≤ ik Vi k ≤ k k
k W k Xi∈I
(cid:3)
Thefollowingpropositiondescribesapproximatedualityoffusionframeswith
respect to local frames.
Proposition 2.3. Let e be an orthonormal basis of . Then the Bessel se-
j j∈J
{ } H
quenceV = (V ,υ ) is an approximate alternate dual fusion frame of a fusion
i i i∈I
{ }
frame W = (W ,ω ) if and only if υ π e is an approximate dual
of ω π S−{1e i i }i∈.I { i Vi j}i∈I,j∈J
{ i Wi W j}i∈I,j∈J
Proof. For each f we have
∈H
f,ω π S−1e 2 = ω S−1π f,e 2
|h i Wi W ji| |h i W Wi ji|
i∈I,j∈J i∈I j∈J
X XX
= ω2 S−1π f 2
ik W Wi k
i∈I
X
S−1 2 ω2 π f 2.
≤ k W k ik Wi k
i∈I
X
This implies that F = ω π S−1e is a Bessel sequence for . Similarly,
{ i Wi W j}i∈I,j∈J H
G= υ π e is also Bessel sequence for . Moreover,
{ i Vi j}i∈I,j∈J H
T T∗f = f,ω π S−1e υ π e
G F h i Wi W ji i Vi j
i∈I,j∈J
X
= ω υ π S−1π f,e e
i i Vih W Wi ji j
i∈I,j∈J
X
= ω υ π S−1π f
i i Vi W Wi
i∈I
X
= T φ T∗ f =ψ .
V vw W vw
for all f . This completes the proof. (cid:3)
∈H
Theorem 2.4. Let W = (W ,ω ) be a fusion frame and V = (V ,υ ) be
i i i∈I i i i∈I
{ } { }
a Bessel fusion sequence, also g be a frame for V with bounds A and B
{ i,j}j∈Ji i i i
for every i I such that 0 < a = inf A sup B = b < . Then V is an
∈ i∈I i ≤ i∈I i ∞
approximate alternate dual fusion frame of W if and only if G = υ g
is an approximate dual of F = ω π S−1g where g{ i i,j}i∈iIs,j∈thJei
{ i Wi W i,j}i∈I,j∈Ji { i,j}j∈Ji
canonical dual of g .
{ i,j}j∈Ji
e e
6 A. A. Arefijamaal and F. Arabyani Neyshaburi
Proof. We first show that F is a Bessel sequence for . Indeed, for each f
H ∈H
f,ω π S−1g 2 = ω2 π S−1π f,g 2
|h i Wi W i,ji| i |h Vi W Wi i,ji|
i∈IX,j∈Ji Xi∈I jX∈Ji
e ωi2 π S−1π f 2 e
≤ A k Vi W Wi k
i
i∈I
X
S−1 2
k W k ω2 π f 2.
≤ a ik Wi k
i∈I
X
Moreover,by Theorem 1.3, G is a Bessel sequence for . On the other hand,
H
T φ T∗ f = ω υ π S−1π f
V vw W i i Vi W Wi
i∈I
X
= ω υ π S−1π f,g g
i i h Vi W Wi i,ji i,j
Xi∈I jX∈Ji
= f,ω π S−1g υeg =T T∗f.
h i Wi W i,ji i i,j G F
i∈IX,j∈Ji
This completes the proof. e (cid:3)
Thefollowingtheorem,givestheideathatwillleadtooneofthemainresults
of this section.
Theorem 2.5. Let W = (W ,ω ) be a Riesz fusion basis. For an approximate
i i i∈I
alternate dual fusion fra{me (V ,}ω ) of W, the sequence (ψ−1V ,ω ) is
{ i i }i∈I { vw i i }i∈I
an alternate dual fusion frame of W.
Proof. Supposethat e isanorthonormalbasisofW ,foreachi I.Then
{ i,j}j∈Ji i ∈
F := ω e is a frame for by Theorem 1.3. Now, for each f we
{ i i,j}i∈I,j∈Ji H ∈ H
obtain
f,ω π S−1e 2 = ω S−1π f,e 2
|h i Vi W i,ji| |h i W Vi i,ji|
i∈IX,j∈Ji Xi∈I jX∈Ji
ω2 S−1π f 2
≤ ik W Vi k
i∈I
X
S−1 2 ω2 π f 2.
≤ k W k ik Vi k
i∈I
X
Thus, G:= ω π S−1e is a Bessel sequence for . Moreover,
{ i Vi W i,j}i∈I,j∈Ji H
ψ f = ω2π S−1π f = ω2π S−1 f,e e
vw i Vi W Wi i Vi W h i,ji i,j
Xi∈I i∈IX,j∈Ji
= f,ω e ω π S−1e =T T∗f.
h i i,ji i Vi W i,j G F
i∈IX,j∈Ji
Hence, by assumption G is an approximate dual of F. This implies that the se-
quence (T T∗)−1ω π S−1e isadualfor ω e .Ontheother
{ G F i Vi W i,j}i∈I,j∈Ji { i i,j}i∈I,j∈Ji
Some properties of dual and approximate dual of fusion frames 7
hand the sequence ω e is a Riesz basis for by Theorem 3.6 of [3].
{ i i,j}i∈I,j∈Ji H
Using the fact that discrete Riesz bases have only one dual we obtain
(T T∗)−1ω π S−1e =S−1ω e (i I,j J ). (2.2)
G F i Vi W i,j F i i,j ∈ ∈ i
Furthermore,itisnotdifficulttoseethatS =S .Indeed,forallf wehave
F W
∈H
S f = f,ω e ω e
F i i,j i i,j
h i
i∈IX,j∈Ji
= ω2 π f,e e
i h Wi i,ji i,j
Xi∈I jX∈Ji
= ω2π f =S f.
i Wi W
i∈I
X
Now,sinceT T∗ =ψ .BySubstitutingψ andS in(2.2),wefinallyconclude
G F vw vw W
that
ψ−1π S−1e =S−1e , (i I,j J ).
vw Vi W i,j W i,j ∈ ∈ i
In particular,
ψ−1V S−1W , (i I). (2.3)
vw i ⊇ W i ∈
It immediately follows that (ψ−1V ,ω ) is an alternate dual fusion frame of
(W ,ω ) . { vw i i }i∈I (cid:3)
i i i∈I
{ }
By the above theorem we obtain the following characterization of alternate
duals of Riesz fusion bases.
Corollary 2.6. Let W = (W ,ω ) be a Riesz fusion basis. A Bessel sequence
i i i∈I
{ }
V = (V ,ω ) is an alternate dual fusion frame of W if and only if
i i i∈I
{ }
V S−1W , (i I). (2.4)
i ⊇ W i ∈
Proof. If V satisfies in (2.4), clearly V is an alternate dual of W. On the other
hand, since every dual fusion frame is an approximate dual with ψ =I , so by
vw H
(2.3) the result follows. (cid:3)
Corollary2.6,alsoshowsthat,unlikediscreteframes,Rieszfusionbasesmay
have more than one dual. Moreover, in the next proposition, we show that every
fusion frame has at least an alternate dual.
Proposition 2.7. Every fusion frame has an alternate dual fusion frame different
from the canonical dual fusion frame.
Proof. Let (W ,ω ) be afusionframewithframeoperatorS .Firstsuppose
i i i∈I W
tVhat=thSer−e1W{exists iU0 }∈wIhesurechUthat W(Si0−16=WH).⊥TaiskeanVia=rbiStrW−a1ryWiclofosredi s6=ubis0paacned.
i0 W i0 ⊕ i0 i0 ⊆ W i0
8 A. A. Arefijamaal and F. Arabyani Neyshaburi
Obviously (V ,ω ) is an alternate dual fusion frame of (W ,ω ) . Indeed
i i i∈I i i i∈I
{ } { }
T φ T∗ f = ω2π S−1π f +ω2π S−1π f
V vw W i Vi W Wi i0 Vi0 W Wi0
i∈XI,i6=i0
= ωi2πSW−1WiSW−1πWif +ωi20πSW−1Wi0⊕Ui0SW−1πWi0f
i∈XI,i6=i0
= ωi2πSW−1WiSW−1πWif =f,
i∈I
X
for every f . On the other hand, assume that W = for all i I. It
i
immediately∈follHows that ωi i∈I l2. Take V1 = and Vi =H 0 for i >∈1, and
assume that ν1 = Piω∈1Iωi2{an}d νi∈= ωi for i > 1. THhen SWf ={ } i∈Iωi2 f and
for every f we have
∈H (cid:0)P (cid:1)
ωiνiπViSW−1πWif = ωiνiπViSW−1f =ω1ν1πV1SW−1f
i∈I i∈I
X X
= ω2 S−1f =f.
i W
!
i∈I
X
Thisshowsthat (V ,ν ) isanalternatedualfusionframeof (W ,ω ) . (cid:3)
i i i∈I i i i∈I
{ } { }
Example 2.8. Let W = W be a Riesz fusion basis. Consider
i i∈I
{ }
V =(span W )⊥, (i I)
i j6=i{ j} ∈
weclaimthatV = (V ,υ ) isanalternatedualof W ,forall υ l2.
i i i∈I i i∈I i i∈I
Take f W , sinc{e S−1/}2W is an orthogonal f{ami}ly of subspa{ces}in ∈ so
S−1f i ∈V .iHence V{ W S−1iW}i∈,Ifor every i I and so V is a dual of WHby
W i ∈ i i ⊇ W i ∈
Corollary 2.6. In fact, this dual is the unique maximal biorthogonal sequence for
W , see also Proposition 4.3 in [5].
i i∈I
{ }
Suppose that W is a Riesz fusion basis, by Theorem 3.9 in [3], there
i i∈I
{ }
existsanorthonormalfusionbasis U andaboundedbijectivelinearoperator
i i∈I
{ }
T : forwhichTU =W foralli I.ThereforethecanonicaldualofaRiesz
i i
H→H ∈
fusion basis is also a Riesz fusion basis. The following theorem, shows that other
alternatedualsof W arenotRieszfusionbasis.Thisresultisageneralization
i i∈I
{ }
of Proposition 3.7 (2) in [11] for infinite case.
Theorem 2.9. Let W = W be a Riesz fusion basis. The only dual V of
i i∈I i i∈I
{ } { }
W that is Riesz basis is the canonical dual.
Proof. Suppose the Rieszbasis V isanalternatedualfusionframeofW.By
i i∈I
Corollary 2.6, S−1W V for a{ll i} I. Assume that there exists j I such that
S−1W V , piWck upi ⊆a noin-zero 0 =∈ f V (S−1W )⊥. Since S−∈1W is a
W j ⊂ j 6 ∈ j ∩ W j { W i}i∈I
Riesz fusion basis we can choose a unique sequence g such that f = g
where g S−1W for all i I. Therefore, the vec{tori}fi∈Ihas two representait∈iIonsi
i ∈ W i ∈ P
of the elements in the Riesz fusion basis V which is a contradiction. Hence
i i∈I
V =S−1W for every i I. { } (cid:3)
i W i ∈
Some properties of dual and approximate dual of fusion frames 9
Suppose L B( ) is invertible and (V ,υ ) is an alternate dual (ap-
i i i∈I
∈ H { }
proximate alternate dual) fusion frame of W = (W ,ω ) . It is natural to ask
i i i∈I
{ }
whether (LV ,υ ) is an alternate dual (approximate alternate dual) fusion
i i i∈I
{ }
frame of (LW ,ω ) .
i i i∈I
{ }
Theorem2.10. Let W = (W ,ω ) be a fusion frame and L B( ) be invert-
i i i∈I
{ } ∈ H
ible such that L∗LW W for every i I. The following statements hold:
i i
⊆ ∈
(i) If V = (V ,υ ) is an alternate dual fusion frame of W. Then the sequence
i i i∈I
{ }
LV = (LV ,υ ) is an alternate dual fusion frame of LW = (LW ,ω ) .
i i i∈I i i i∈I
{ } { }
(ii) If V is an approximate alternate dual fusion frame of W such that,
I ψ < L −1 L−1 −1.
H vw
k − k k k k k
Then LV is an approximate alternate dual fusion frame of LW.
Proof. The sequence (LW ,ω ) is a fusion frame with the frame operator
i i i∈I
LS L−1 and π ={Lπ L−1,}see [6]. Therefore, for each f we obtain
W LWi Wi ∈H
ω υ π S−1 π f = ω υ Lπ S−1π L−1f
i i LVi LW LWi i i Vi W Wi
i∈I i∈I
X X
= LL−1f =f.
This proves (i). To show (ii) first note that ψ =Lψ L−1, hence
Lv,Lw vw
(I ψ )f = (I Lψ L−1)f
H Lv,Lw H v,w
k − k k − k
= L(I ψ )L−1f
H v,w
k − k
< f
k k
for all f . This follows the result. (cid:3)
∈H
3. Stability of approximate duals
In frame theory, every f is represented by the collection of coefficients
∈ H
f,f . From these coefficients, f can be recovered using a reconstruction
i i∈I
{h i}
formula by dual frames. In realapplications,in these transmissionsusually a part
of the data vectors change or reshape in the other words, disturbances affect on
the information.In this respect,stability of frames anddual frames under pertur-
bations have a key role in practice. The stability of approximate dual of discrete
frames and g-frames can be found in [7, 14]. In the following, we discuss on the
stability of approximate alternate dual fusion frames under some perturbations.
First, we fix definition of perturbation.
Definition 3.1. Let W and W be closed subspaces in . Also let
i i∈I i i∈I
{ } { } H
ω be positive numbers and ǫ > 0. We call (W ,ω ) a ǫ-perturbation
i i∈I i i i∈I
{ } { }
of (W ,ω ) whenever for every ff ,
i i i∈I
{ } ∈H
ω2 (π π )f 2 <ǫ f f2.
ik Wfi − Wi k k k
i∈I
X
10 A. A. Arefijamaal and F. Arabyani Neyshaburi
Theorem3.2. Let V = (V ,υ ) be an approximate alternate dual fusion frame
i i i∈I
{ }
of a fusion frame W = (W ,ω ) . Also let (U ,υ ) be a ǫ-perturbation of
i i i∈I i i i∈I
{ } { }
V , such that
2
1 I ψ
ǫ< −k H− vwk , (3.1)
√B S−1 !
k W k
where B is the upper bound of W. Then (U ,υ ) is also an approximate alter-
i i i∈I
{ }
nate dual fusion frame of W. In particular, if W is a Parseval fusion frame and
choose V =W, then the result holds for ǫ<1.
Proof. Notice that (U ,υ ) is a Bessel fusion sequence, in fact
i i i∈I
{ }
υ2 π f 2 = υ2 π f +(π π )f 2
ik Ui k ik Vi Ui − Vi k
i∈I i∈I
X X
1/2 1/2 2
υ2 π f 2 + υ2 (π π )f 2
≤  ik Vi k ! ik Ui − Vi k ! 
i∈I i∈I
X X
(√D+√ǫ)2 f 2, 
≤ k k
where D is the upper bound of V. On the other hand,
(I ψ )f (I ψ )f + (ψ ψ )f
H uw H vw vw uw
k − k ≤ k − k k − k
(I ψ )f + sup ω υ (π π )S−1π f,g
≤ k H− vw k kgk=1(cid:12)* i i Vi − Ui W Wi +(cid:12)
(cid:12) Xi∈I (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= (I ψ )f + sup (cid:12) ω S−1π f,υ (π π )g (cid:12)
k H− vw k kgk=1(cid:12) i W Wi i Vi − Ui (cid:12)
(cid:12)(cid:12)(cid:12)Xi∈I (cid:10) 1/2 (cid:11)(cid:12)(cid:12)(cid:12)
(I ψ )f +√ǫ (cid:12) ω2 S−1π f 2 (cid:12)
≤ k H− vw k ik W Wi k !
i∈I
X
(I ψ )f +√ǫB S−1 f < f ,
≤ k H− vw k k W kk k k k
where the last inequality is implied from (3.1). The rest follows by the fact that
each Parsevalfusion frame is a dual of itself. (cid:3)
Example 3.3. Consider
W1 =R2 0 , W2 = 0 R2, W3 =span (1,0,0) ,
×{ } { }× { }
V1 =span (0,1,0) , V2 = 0 R2, V3 =span (1,0,0) .
{ } { }× { }
ThenW = W 3 isafusionframeand S−1 =1Also,wehave I ψ = 1
and so, the{Beis}sei=l1sequence V = V 3 kisWankapproximate alternkatHe−duavlwfkusion2
{ i}i=1
frame of W. Now, if we take
U1 =V1, U2 =V2, U3 =span (α,β,0) ,
{ }