M. M. SHAMOOSHAKY
Received 13 January 2005 and in revised form 17 September 2005
The concepts of basis and frame are studied in the classical literature of functional anal-ysis, Fourier analanal-ysis, and wavelet theory in a wide range. In this paper, we consider an operator-theoretic approach to discrete frame theory on a separable Hilbert space. For this purpose, we define a special type of frames and bases, called wavelet-type frames and wavelet-type bases, obtained by acting with a family of bounded linear operators on some vectors, and then investigate the elementary properties of these concepts.
1. Introduction
The idea of this paper comes from wavelets theory. In classical functional analysis, a basis of a separable Hilbert space is a countable subset of H with certain properties. In the wavelet theory, a basis is made by a countable family of unitary operators and a single number (or finite number) of vectors inH. In fact, ifUis a unitary system onHandΨis the mother wavelet corresponding to it, thenUΨis an orthonormal basis forH. In other words, the bases which one studies in the wavelet theory are of this form.
Hence generally, a frame{xn:n∈J}forHis called awavelet-type frameif there exist
a family{fi:i∈J}of bounded operators onHand a vectorh∈Hsuch thatxi= fihfor
eachi∈J. The vectorhis called a framer vector.
Similarly, a normalized tight frame (Riesz basis, orthonormal basis) {xj:j∈J} is
called a wavelet-type normalized tight frame (wavelet-type Riesz basis, wavelet-type or-thonormal basis)if there exist a family{fi:i∈J}of bounded linear invertible operators
onHand a vectorh∈Hsuch thatxi= fihfor eachi∈J.
Our idea in this paper is based on the wavelet-type frames, which we organize as fol-lows.
InSection 2, definitions and some elementary properties of framers and wavelet-type frames, basers and wavelet-type bases are given. Then, we will explain the relations be-tween, framers and normal framers, normal framers and orthonormal basers, framers and Riesz basers, and finally, framers and the canonical dual of it. These results will be needed inSection 3.
Copyright©2005 Hindawi Publishing Corporation
Section 3is devoted to the study of the structure of framers and basers, by defining the concepts of complement, strong complement, strongly disjoint complement, and similar-ity. We will show that the complement of each wavelet-type frame is of wavelet-type. The same result will be proven for the strong complement, the strongly disjoint complement, and similarity. Also, the problem of construction of a baser from a framer on a Hilbert space will be considered in this section. Indeed by using a complement framer, we can find a (Riesz or orthonormal) baser.
2. Elementary properties of framers and basers
LetHbe a separable Hilbert space, and letB(H) denote the algebra of all bounded linear operators onH. LetJdenote a generic countable (or finite) index set such asN,Z, and so forth.
In the notations, we follow [8].
Definition 2.1. LetH be a separable complex Hilbert space andB(H) the algebra of all bounded linear operators on H. If U= {fj: j∈J} is a countable subset of bounded
operators onH, andλ∈H, then the following hold.
(i) ({fj},λ) is aframeronHifUλ= {fjλ: fj∈U}is a frame with frame boundsA,
B, forH. AlsoA,Bare called framer bounds.
(ii) ({fj},λ) is atight frameronH if the fj’s are invertible, for each j∈J, andUλ=
{fjλ:fj∈U}is a tight frame forH.
(iii) ({fj},λ) is anormal framer onH if the fj’s are invertible, for each j∈J, and
Uλ= {fjλ:fj∈U}is a normalized tight frame forH.
(iv) Finally, ({fj},λ) is called (an orthonormal-Riesz) baseron H if the fj’s are
in-vertible, for each j∈J, andUλ= {fjλ:fj∈U}is (an orthonormal-Riesz) basis
forH.
2.1. Examples.
(a) LetH=l2(N∪ {0}) andT:l2(N∪ {0})→l2(N∪ {0}) is defined byT(x
1,x2,. . .)=
(0,x1,x2,. . .), thenTis a bounded invertible operator withT =1. SetTn=Tn
with
Tnx1,x2,. . .=0, 0, . . ., 0 ntimes
,x1,x2,. . ., T0=T, λ=(1, 0, 0,. . .), (2.1)
then ({Tn:n∈N∪ {0}},λ) is an orthonormal baser onH.
(b) Let H =L2[0, 1], define T(x(t))=t
0x(q)dq and by induction, Tn(x(t))=
(1/n)Tn−1(x(t)), ifx(t)=1, thenT(x(t))=t,T2(x(t))=(1/2)t2,. . .,Tn(x(t))=
(1/n!)tn, and{1,t, (1/2)t2,. . ., (1/n!)tn,. . .}is a tight frame (see [8]), so if we define
Tn=Tn,n∈N, then ({Tn:n∈N},x(t)) is a tight framer onH.
(c) LetΨbe the mother wavelet corresponding to unitary systemUfor Hilbert space H, then (U,Ψ) is an orthonormal baser onH.
Theorem2.2. Let{fn:n∈J}be a normal framer onH. Then there exist a Hilbert spaceK
n∈J, whereθ:H→θ(H)is a unitary operator andP:K→θ(H)is the orthogonal projec-tion ofHontoθ(H).
Proof. Since {fn:n∈J} is a normal framer, there exists h∈H such that{fnh:n∈
J}is a normalized tight frame for H. Set K=l2(J) and defineθ
0:H→K byθ0(x)=
(x,fnh)n∈Jas the usual frame transform which is an isometry. Thus,H can be
embed-ded intoK by identifyingH withθ0(H). Sinceθ:H→θ0(H)⊂K withθ(x)=θ0(x) is
onto, so it is unitary. DefineT1:K→Kas identity and
T2
xn
n∈J
=0,x1,x2,. . .
,. . .,Tm
xn
n∈J
=0, 0, . . ., 0
m−1 times
,x1,x2,. . .
, (2.2)
then{Tn}n∈Jwithe1=(1, 0, 0,. . .) is an orthonormal baser onK. SincePis a projection,
soP=P∗, and we haveθ f
m(x),PTn(θ(x))=Pθ fm(x),Tn(θ(x))=θ fm(x),Tn(θ(x))=
fm(x),fn(θ(x)) = θ(fm(x)),θ fn(θ(x)), hence θ fm(x),PTn(θ(x))−θ fn(θ(x)) =0.
But by definition of framer, we haveθ(y),Tn(e1) = y,fn(h). On the other hand, since
fm(h)’s spanH, thus vectorsθ fm(h) spanθ(H), so (PTne1−θ(fnh))⊥θ(H), and hence
(PTnθ(x)−θ(fn(x)))⊥θ(H). However, the range ofPisθ(H), and thereforePTnθ(x)−
θ(fn(x))=0 andPTnθ(x)=θ(fn(x)).
The relation between normal framers and orthonormal basers will be shown in the following theorem.
Theorem2.3. LetHbe a Hilbert space, then({fn:n∈J},h1)is a normal framer onH if
and only if there exist a Hilbert spaceMand a normal framer({gn:n∈J},h2)onMsuch
that({fn⊕gn:n∈J},h1⊕h2)is an orthonormal baser onH⊕M.
Proof. ByTheorem 2.2, there exist a Hilbert spaceKand an orthonormal baser ({un:n∈
J},e1) onK such thatθ0(fn(x))=Pun(θ0(x)), whereθ0:H→K is the frame transform
corresponding to normalized tight frame{fnh1:n∈J}, which is an isometry. Now set
M=(I−P)K andgn=(I−P)un, whereI is the identity operator onK, then the set
{gne1:n∈J}is a normalized tight frame forM, hence ({gn:n∈J},e1) is a normal framer
on M and ({θ0(fn)⊕gn:n∈J},θ0(h1)⊕h2) is an orthonormal baser onθ0(H)⊕M.
Sinceθ0:H→K is an isometry, if we defineθ:H →θ0(H) by θ(x)=θ0(x), then θ is
unitary, soθ⊕Iandθ−1⊕Iare unitaries. Since ({θ
0(fn)⊕gn:n∈J},θ0(h1)⊕h2) is an
orthonormal baser onθ0(H)⊕M, hence
θ−1⊕Iθ 0
fn⊕gn:n∈J ,θ−1⊕Iθ0
h1
⊕h2
=fn⊕gn:n∈J ,h1⊕h2
(2.3)
is an orthonormal baser on (θ−1⊕I)(θ
0(H)⊕M)=H⊕Mas required.
The next theorem show that a Riesz baser is the image of an orthonormal baser under a bounded invertible operator.
Theorem2.4. Let({vn:n∈J},λ1)be a Riesz baser on Hilbert spaceH, then there exist a
un(θ(x)). Likewise, a framer is the image of a normal framer under a bounded invertible
operator.
Proof. Let ({vn:n∈J},λ1) be a Riesz baser on Hilbert space H, then by the proof of
Theorem 2.2, we haveθ0(H)=Kandθ0−1(K)=H, hence if ({un:n∈J},λ2) is the normal
baser as mentioned inTheorem 2.2, thenθ0−1(un(θ0(x)))=vn(x), orθ0vn(x)=un(θ0(x)).
Let ({un:n∈J},h) be a normal framer onHandTa bounded linear invertible
opera-tor. Definefn(x)=Tun(x), then forx0∈H, we havej|x0,fj(h)|2=j|x0,Tuj(h)|2=
jT∗x0,uj(h)|2= T∗x02.
But
T∗x
0≤T∗x0= Tx0. (2.4)
Since T−1 exists, so T∗ is invertible and T∗−1
=(T−1)∗, thus T∗−1
= (T−1)∗ = T−1, hence
x0=T∗−1
T∗x0≤T−1T∗
x0. (2.5)
Now (2.4) and (2.5) imply that T−1−1x
0 ≤ T∗(x0) ≤ Tx0. Therefore,
T−1−2x 02≤
j|x0,fj(h)|2≤ T2x02, hence{fn(Th) :n∈J}is a frame with
frame boundsA≥ T−1−2andB≤ T2, and consequently ({f
n:n∈J},h) is a framer.
InTheorem 2.3, we showed that each normal framer is the direct summand of a nor-mal baser. We will prove the same result for Riesz basers as follows.
Theorem 2.5. Let ({fn:n∈J},h) be a framer on Hilbert space H. Then there exist a
Hilbert space M and a normal framer ({un:n∈J},λ)on M such that ({fn⊕un:n∈
J},h⊕λ)is a Riesz baser onH⊕M.
Proof. Since by Theorem 2.4 each framer is the image of a normal framer under a bounded invertible operator, there exist a normal framer ({vn:n∈J},λ1) onH and
in-vertible operatorT:H→H such thatTvn(x)=fn(x) for alln∈J. So byTheorem 2.3,
there are a Hilbert space M and a normal framer ({un:n∈J},λ2) on M such that
({θ(vn)⊕un:n∈J},θ(λ1)⊕λ2) is an orthonormal baser onθ(H)⊕M. Since T and
θ−1are invertible, soTθ−1is invertible, (Tθ−1⊕I)(θ(λ
1)⊕λ2)=Tλ1⊕λ2, and (Tθ−1⊕
I)(θ(vn)⊕un)=T(vn)⊕un= fn⊕un. Hence,Theorem 2.4implies that ({fn⊕un:n∈
J},Tλ1⊕λ2) is a Riesz baser onH⊕M.
Lemma2.6. Let({un:n∈J},λ2)be a Riesz baser onHand letP be a projection fromH
onto a subspacePH ofH, then{Pun:n∈J}is a framer onPH.
Proof. LetT:H→H be a bounded invertible operator onH, and letQbe a selfadjoint projection, setP=TQT−1 and letv
n=T−1un, another Riesz baser onH. Forx∈QH,
we haven|x,Qvn(T−1λ2)|2=n|Qx,vn(T−1λ2)|2=n|x,vn(T−1λ2)|2, so by the
definition of framer,{Qvn:n∈J}is a framer onH. Now, sinceT|QHis a bounded
Theorem2.7. A pair({fn:n∈J},λ1)is a framer on a Hilbert spaceHif and only if there
exist a Hilbert spaceMand a normal framer({un:n∈J},λ2)onM such that({fn⊕un:
n∈J},λ1⊕λ2)is a Riesz baser onH⊕M. In short , framers are precisely the inner direct
summands of Riesz basers.
The proof is obvious.
Remark 2.8. The last theorem allows us to generalize the concept of framer in Banach spaces to be simply an inner direct summand of a Riesz baser. In this case, the index set should beN. For instance, a Schauder framer would be an inner direct summand of a Schauder baser. Similarly, the bounded unconditional framer would be an inner direct summand of a bounded unconditional baser.
In the following theorems, we will prove that coisometries preserve the structure of framers, that is, the image of framers under coisometries are framers.
Theorem2.9. Let{Aj}j∈Jbe a family of coisometries onH and let({fj:j∈J},λ)be a
framer for which{fjλ}j∈Jis a frame with frame bounds{B1,B2}. SetCj=Ajfj,then({Cj:
j∈J},λ)is a framer with framer bounds{B1,B2}. If({fj:j∈J},λ)is tight (normal), then
({Cj:j∈J},λ)is tight (normal), respectively.
Proof. Letx∈H be given, then j|x,Cjλ|2=j|x,Ajfj(λ)|2=j|A∗jx,fjλ|2.
Since ({fj:j∈J},λ) is a framer for which{fjλ}j∈Jis a frame with frame bounds{B1,B2},
so B1A∗jx2 <
j|A∗jx,fjλ|2 < B2A∗jx2. But Aj is coisometry, for all j, hence
A∗
jx=x, and thenB1x2<j|A∗jx,fjλ|2< B2x2, orB1x2<j|x,Cjλ|2<
B2x2. So ({Cj:j∈J},λ) is a framer with bounds{B1,B2}.
If ({fj: j∈J},λ) is a tight framer, then a similar calculation shows that j|x,
Cjλ|2=j|A∗jx,fjλ|2=B1A∗jx2=B1x2, that is, ({Cj:j∈J},λ) is a tight framer
onH.
And finally, if ({fj:j∈J},λ) is normal, then ({Cj:j∈J},λ) is a normal framer onH.
We can generalize this theorem as follows.
Theorem2.10. Let{A1
j}j∈J,{A2j}j∈J,. . .,{Anj}j∈Jbe families of coisometries, and let({fj:
j∈J},λ)be a framer with framer bounds{B1,B2}. IfCj=A1jA2j···Anjfj, then({Cj:j∈
J},λ)is a framer with framer bounds{B1,B2}. If({fj:j∈J},λ)is tight (normal), then
({Cj:j∈J}λ)is tight (normal), respectively.
Proof. The proof is an easy consequence byTheorem 2.9and induction. Remark 2.11. For a tight framer ({fj:j∈J},λ) with framer boundA, the reconstruction
formula isx=(1/A)jx,fjλfjλ. If framer is not tight, then there is a similar
recon-struction formula in terms of dual framers.
Let ({fj:j∈J},λ) be given. A framer ({gj:j∈J},λ) is called a dual framer of ({fj:
j∈J},λ) if for eachx∈H,x=jx,gjλfjλ.
Theorem2.12. Let ({fj: j∈J},λ)be a framer on a Hilbert space H. Then there exists
a unique operatorS:H→H such thatx=j∈Nx,S fjλfjλ, for allx∈H. IfA:H→K
is any invertible operator for some Hilbert spaceK such that({A fj:j∈J},λ)is a normal
Proof. Since ({fj:j∈J},λ) is a framer, so{fjλ:j∈J}is a frame forH, now the theorem
is trivial.
We call ({S fj:j∈J},λ) the canonical dual of ({fj:j∈J},λ), and for eachx∈H, one
can writex=j∈Nx,S fjλfjλ=
j∈Nx,fjλS fjλ.
The last theorem proves that the canonical dual of each wavelet-type frame can be of wavelet-type as the following theorem proves the same results for the dual of a frame. Theorem2.13. Let({fj:j∈J},h)and({gj:j∈J},λ)be framers on a Hilbert spaceH,
such thatx=jx,gjλfjhfor allx∈H. Thenx=
jx,fjhgjλ, for allx∈H.
The proof is obvious.
3. Complement in framers and basers
This section investigates some additional properties of framers and basers. First, we give some definitions.
Definition 3.1. Let ({fj:j∈J},λ1) be a framer on a Hilbert spaceH. A framer ({gj:j∈
J},λ2) on a Hilbert spaceMis called a complement to ({fj:j∈J},λ1) if ({fj⊕gj:j∈
J},λ1⊕λ2) is a Riesz baser onH⊕M. Note that byTheorem 2.7, the complement to each
framer ({fj:j∈J},λ1) exists.
If ({fj: j∈J},λ1) is a normal framer, then a normal framer ({gj:j∈J},λ2) on a
Hilbert spaceMis called a strong complement to ({fj:j∈J},λ1) if ({fj⊕gj:j∈J},λ1⊕
λ2) is an orthonormal baser onH⊕M.
Two framers ({fj:j∈J},λ1) and ({gj:j∈J},λ2) on Hilbert spacesH,K,
respec-tively, are similar if there exists a linear bounded invertible operator T:H→K such that T fjT∗=gj for each j∈J. In this case, ifT is unitary, then ({fj:j∈J},λ1) and
({gj:j∈J},λ2) are called unitarily equivalent.
A framer ({gj: j∈J},λ2) is called strong complement to ({fj: j∈J},λ1) if there
exist a pair of strong complement normal framers ({fj:j∈J},λ1) and ({gj:j∈J},λ2)
and linear bounded invertible operatorsT1andT2such that fj=T1fjT1∗andgj=T2gjT2∗
for allj∈J.
Theorem3.2. Let({fj:j∈J},λ1)be a framer onH, and let({gj:j∈J},λ2)and({ej:
j∈J},λ3)be strong complements to({fj:j∈J},λ1)in Hilbert spacesM,N, respectively.
Then there exists an invertible operator∆:M→Nsuch thatej=∆gj∆∗for allj∈J.
The proof is immediate.
In the next theorem, we will show that the strong complementary of normal framers are preserved under coisometries.
Theorem3.3. Let ({fj:j∈J},λ1)be a normal framer onH, and let {Aj:j∈J} be a
family of coisometries operators onH. Then there exist a Hilbert spaceMwith normal framer ({gj:j∈J},λ2), and a family of coisometries operators{Bj:j∈J}onMsuch that({gj:
j∈J},λ2)is a strong complement to({fj:j∈J},λ1)and({Bjgj:j∈J},λ2)is a strong
complement to({Ajfj:j∈J},λ1).
Proof. Let ({fj:j∈J},λ1) be a normal framer and{Aj:j∈J}is a family of
are unitaries. Hence byTheorems 2.3and2.9, there exist a Hilbert spaceM and a nor-mal framer ({ej: j∈J},λ3) such that ({Ajfj⊕ej: j∈J},λ1+λ3) is an orthonormal
baser onH⊕M. On the other hand, since ({fj:j∈J},λ1) is normal framer, so there
exists a normal framer ({gj:j∈J},λ2) which is a strong complement to ({fj:j∈J},λ1),
that is, ({fj⊕gj:j∈J},λ1⊕λ2) is an orthonormal baser onH⊕M. SetBj=ejg−j1since
B∗ j y2=
j∈J|B∗j y,gjλ3|2=
j∈J|y,Bjgjλ3|2=
j∈J|y,ejλ3|2= y2. SoBj, for
each j∈J, is a bounded invertible coisometry. Now,Bjgj=ej and ({ej:j∈J},λ3) is a
strong complement to ({Ajfj:j∈J},λ1), so ({Bjgj:j∈J},λ3) is a strong complement
to ({Ajfj:j∈J},λ1).
A similar conclusion holds for framers as follows.
Theorem3.4. Let({fj:j∈J},λ1)be a framer onH, and let{Aj:j∈J}be a family of
coisometry operators onH. Then there exist a Hilbert spaceMwith normal framer({gj:j∈
J},λ2), and a family of coisometry operators{Cj:j∈J}onM such that({gj:j∈J},λ2)
is complement to({fj:j∈J},λ1)and({Cjgj:j∈J},λ3)is complement to({Ajfj:j∈
J},λ1).
Proof. Let ({fj:j∈J},λ1) be a framer and{Aj:j∈J}are coisometries, so ({Ajfj:j∈
J},λ1) is a framer onH. On the other hand, there are normal framers ({gj:j∈J},λ2) and
({ej:j∈J},λ3) onMsuch that ({fj⊕gj:j∈J},λ1⊕λ2) and ({Ajfj⊕ej:j∈J},λ1⊕
λ3) are Riesz basers onH⊕M, that is, ({gj:j∈J},λ2) is complement to ({fj:j∈J},λ1).
For the other part of theorem, setCj=ejg−j1, then a calculation such asTheorem 3.3
shows that{Cj:j∈J} are coisometries and since ({Ajfj⊕Cjgj: j∈J},λ1⊕λ3) is a
Riesz baser onH⊕M, hence ({Cjgj:j∈J},λ3) is a complement to ({Ajfj:j∈J},λ1).
In the following, we will prove that the strong complement of a wavelet-type normal-ized tight frame is of wavelet-type.
Theorem3.5. Let{xj:j∈J}be a normalized tight wavelet-type frame for a Hilbert space
H, and let{yj:j∈J}be a strong complement to{xj:j∈J}in the frame sense, then{yj:
j∈J}is a wavelet-type frame.
Proof. Let{xj:j∈J}be a normalized tight wavelet-type frame for a Hilbert spaceH, so
there exists a normal framer ({fj:j∈J},λ1) onHsuch thatxj=fjλ1for allj∈J. Since
({fj:j∈J},λ1) is normal framer, so there exists a normal framer ({gj:j∈J},λ2) such
that ({fj⊕gj:j∈J},λ1⊕λ2) is an orthonormal baser onH⊕M. So{gjλ2:j∈J}is a
strong complement to{fjλ1:j∈J}in the sense of frames. But {yj:j∈J}is another
strong complement to{fjλ1:j∈J}, so byTheorem 3.2, there exists a unitary operator
U:M→Nsuch thatyj=Ugj(λ2) andyj=UgjU∗(λ3), forλ3∈N. Setej=UgjU∗, then
({ej:j∈J},λ3) is a normal framer andejλ3=yj, that is,{yj:j∈J}is wavelet-type.
For the case that{xj:j∈J}is a general wavelet-type frame, first we prove the
follow-ing lemma, stated that the similar of a wavelet-type frame is of wavelet-type.
Lemma3.6. Let{xj:j∈J}be a wavelet-type frame for a Hilbert spaceH, and let{yj:j∈
Proof. Since{xj:j∈J}is similar to{yj:j∈J}, there exists a bounded invertible
oper-atorT:H→Hsuch thatTxj=yj, since{xj:j∈J}is a wavelet-type frame, there exists
a framer ({fj:j∈J},λ1) such thatxj= fjλ1, henceT fjλ1=yj. Now setgj=T fj, then
({gj: j∈J},λ1) is a framer andgjλ1=yj, for all j∈J. Thus,{yj: j∈J}is
wavelet-type.
The following theorem shows that the strong complement of each wavelet-type frame is of wavelet-type.
Theorem3.7. Let{xj:j∈J}be a wavelet-type frame for a Hilbert spaceH, let{yj:j∈J}
be a strong complement to{xj:j∈J}, then{yj:j∈J}is wavelet-type.
Proof. Since{xj:j∈J}and{yj:j∈J}are strong complements, so by the definition,
there exists normalized tight frames{xj:j∈J}and{yj:j∈J}such that{xj:j∈J}
is similar to{xj:j∈J},{yj:j∈J}is similar to{yj:j∈J}, and{xj⊕yj:j∈J}is an
orthonormal basis. Since{xj:j∈J}is a wavelet-type frame and{xj:j∈J}is similar
to it, so byLemma 3.6, the frame{xj:j∈J}is wavelet-type. On the other hand, since {y
j:j∈J}is a strong complement to {xj:j∈J}and{xj:j∈J} is wavelet-type, so
byTheorem 3.5, the frame{yj:j∈J}is wavelet-type. Therefore, there exists a normal
framer ({gj:j∈J},λ2) such thatgjλ2=yj. Since{yj:j∈J}and{yj:j∈J}are similar,
so there is a bounded invertible operatorTsuch thatT yj=yj=gjλ2. Thusyj=T−1gjλ2.
Now setej=T−1gj, thenyj=ejλ2. That is,{yj:j∈J}is wavelet-type.
Definition 3.8. Let ({fj:j∈J},λ1) and ({gj:j∈J},λ2) be normal framers on Hilbert
spacesHandK, respectively. ({fj:j∈J},λ1) and ({gj:j∈J},λ2) are strongly disjoint
(strongly completely disjoint) if ({fj⊕gj:j∈J},λ1⊕λ2) is a normal framer
(orthonor-mal baser) onH⊕M.
If ({fj:j∈J},λ1) and ({gj:j∈J},λ2) are framers (not necessarily tight or normal),
({fj:j∈J},λ1) and ({gj:j∈J},λ2) are disjoint if ({fj⊕gj:j∈J},λ1⊕λ2) is a framer
onH⊕M.
Two framers ({fj:j∈J},λ1) and ({gj:j∈J},λ2) are called strongly disjoint (strongly
completely disjoint) if they are similar to a pair of strongly disjoint (strongly completely disjoint) normal framers.
Finally our last task in this paper is to prove that the strongly disjointness of each wavelet-type normalized tight frame is of wavelet-type. This will be shown in the follow-ing theorem.
Theorem3.9. Let{xj:j∈J}be a wavelet-type normalized tight frame for a Hilbert space
H, and let{yj:j∈J}be a strongly disjoint frame to it. Then there exists a normalized tight
frame{zj:j∈J}such that{yj⊕zj:j∈J}is of wavelet-type.
Proof. Let{xj:j∈J}be a wavelet-type normalized tight frame. So there exists a normal
framer ({fj:j∈J},λ1) such thatxj= fjλ1 for all j∈J. Since {yj: j∈J}is strongly
disjoint to{xj:j∈J}, thus{xj⊕yj:j∈J}is a normalized tight frame and there exists
a normalized tight frame{zj:j∈J}such that{(xj⊕yj)⊕zj:j∈J}is an orthonormal
But{(xj⊕yj)⊕zj:j∈J}and{xj⊕(yj⊕zj) :j∈J}are unitarily equivalent, hence
{xj⊕(yj⊕zj) :j∈J}is an orthonormal basis. Since{xj:j∈J}is a normalized tight
frame, so {yj⊕zj:j∈J} is a normalized tight frame, which is a strong complement
to{xj:j∈J}. Now{xj:j∈J}is a wavelet-type normalized tight frame, so byTheorem
3.7, the frame{yj⊕zj:j∈J}is wavelet-type as required.
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M. M. Shamooshaky: Department of Mathematics, Imam Hossein University, P.O. Box 16895-198, Tehran, Iran