R E S E A R C H
Open Access
Dilation-and-modulation systems on the
half real line
Yun-Zhang Li
*and Wei Zhang
*Correspondence:
College of Applied Sciences, Beijing University of Technology, Beijing, 100124, P.R. China
Abstract
Translation, dilation, and modulation are fundamental operations in wavelet analysis. Affine frames based on translation-and-dilation operation and Gabor frames based on translation-and-modulation operation have been extensively studied and seen great achievements. But dilation-and-modulation frames have not. This paper addresses a class of dilation-and-modulation systems inL2(R
+). We characterize
frames, dual frames, and Parseval frames inL2(R
+) generated by such systems.
Interestingly, it turns out that, for such systems, Parseval frames, orthonormal bases, and orthonormal systems are mutually equivalent to each other, while this is not the case for affine systems and Gabor systems.
MSC: 42C15; 42C40
Keywords: dilation-and-modulation system; frame; dual frame
1 Introduction
Before proceeding, we recall some notions and notations. An at most countable sequence {ei}i∈Iin a separable Hilbert spaceHis called aframeforHif there exist <C≤C<∞
such that
Cf≤
i∈I
f,ei≤Cf forf ∈H, (.)
whereC,Care calledframe bounds; it is called aBessel sequenceinHif the right-hand
side inequality in (.) holds, whereCis called aBessel bound. In particular,{ei}i∈Iis called
aParseval frameifC=C= in (.). Given a frame{ei}i∈IforH, a sequence{˜ei}i∈Iis called adualof{ei}i∈Iif it is a frame such that
f = i∈I
f,e˜iei forf∈H. (.)
It is easy to check that{ei}i∈Iis also a dual of{˜ei}i∈Iif{˜ei}i∈Iis a dual of{ei}i∈I. So, in this case, we say{ei}i∈Iand{˜ei}i∈Iform a pair of dual frames forH. It is well known that{ei}i∈I and{˜ei}i∈Iform a pair of dual frames forHif they are Bessel sequences and satisfy (.). The fundamentals of frames can be found in [–]. TheFourier transformofc∈l(Z) is
is defined by
c∗d(k) = m∈Z
c(k–m)d(m) fork∈Z
if it is well defined. TheKronecker deltais defined byδn,m=
ifn=m;
ifn=m. l(Z) denotes the set of finitely supported sequences onZ. We denote byIthe identity operator onl(Z),
and byχEits characteristic function for a setE. WriteR+= (,∞). For a positive number a> , a functionhdefined onR+is said to bea-dilation periodicifh(a·) =h(·) onR+. For
a functionf defined on [,a), we define the functionf˜onR+by
˜
f(·) =fa–l· onal,al+forl∈Z,
which is called thea-dilation periodizationoff. Obviously, it isa-dilation periodic. The translation operatorTx, the modulation operatorMxwithx∈R, and the dilation Dcwithc> are, respectively, defined by
Txf(·) =f(·–x),
Mxf(·) =e
πix·f(·),
and
Dcf(·) = √
cf(c·)
forf ∈L(R). They are the basis of wavelet analysis. Affine systems of the form{D
ajTbkψ:
j,k∈Z}withψ∈L(R) anda,b> , and Gabor systems of the form{E
mbTnag:m,n∈ Z} with g∈L(R) anda, b> have been extensively studied. However,
dilation-and-modulation systems of the form
{MmbDajψ:m,j∈Z} witha,b> (.)
have not been extensively studied. This paper focuses on the following systems that are like (.):
{ψmDajψ:m,j∈Z} witha> , (.)
where
ψm(·) =√
a– e
πiam–· on [,a) form∈Z. (.)
We will investigate the theory ofL(R
+)-frames of the form (.). It is obvious thatL(R+)
can be considered as the Fourier transform of the Hardy spaceH(R) defined by
where the Fourier transform is defined by
ˆ
f(·) =
Rf(x)e
–πix·dx forf ∈L(R)∩L(R)
and extended toL(R) by the Plancherel theorem. Wavelet frames inH(R) of the form
{DajTmϕ :j,m∈Z} were studied in [, ], and some variations can be found in [– ]. By the Plancherel theorem, anH(R)-frame {D
jTmϕ:j,m∈Z}leads to aL(R+
)-frame
e–πi–jm·ϕˆ–j·:j,m∈Z. (.)
In (.),e–πi–jm·
is jZ-periodic with respect to additive operation, and the period de-pends on the dilation factor j. However,ψ
min (.) isa-dilation periodic and unrelated toj. Therefore, frames of the form (.) are different from ones of the form (.) forL(R+)
and of independent interest. They are related to a kind of function-valued frames in []. In [], numerical experiments were made to establish that the nonnegative integer shifts of the Gaussian function formed a Riesz sequence inL(R
+). In [], a sufficient condition
was obtained to determine whether the nonnegative translates form a Riesz sequence on
L(R +).
The rest of this paper is organized as follows. Section is devoted to characterizing frames and dual frames forL(R
+) with the structure of (.). Section is devoted to
Par-seval frames and orthonormal bases forL(R
+) of the form (.). It turns out that Parseval
frames, orthonormal bases, and orthonormal systems inL(R
+) of the form (.) are
mu-tually equivalent to each other. It is worth noting that neither affine systems nor Gabor systems have such a property.
2 Frame and dual frame characterization
This section characterizesL(R
+)-frames and dual frames of the form (.). For this
pur-pose, we need some notations and lemmas. Forf ∈L(R
+), we define
G(f,·) =Daj+lf(·)
j,l∈Z (.)
and
F(·) =Dalf(·)
l∈Z. (.)
Lemma . For m,j∈Z,defineψmas in(.),then
(i) {ψm:m∈Z}is an orthonormal bases forL([,a));
(ii)
[,a)
f(x)dx= m∈Z
[,a)f(x)ψm(x)dx
(.)
Proof (i) By a simple computation, we see that TD(a–)– is a unitary operator from
L([, )) ontoL([,a)), and
ψm(·) =eaπ–imTD
(a–)–eπim·
form∈Z. Also observing that{eπim·:m∈Z}is an orthonormal basis forL([, )), we
have (i).
(ii) By (i), (.) holds iff ∈L([,a)). Whenf ∈L([,a))\L([,a)), the left-hand side
of (.) is infinity. Now we prove by contradiction that the right-hand side of (.) is also infinity. Suppose it is finite. Since{ψm:m∈Z}is an orthonormal basis forL([,a)), we see that
g= m∈Z
[,a)
f(x)ψm(x)dx
ψm
belongs toL([,a)), and thus toL([,a)). It has the same Fourier coefficients asf. Sof =g
by the uniqueness of Fourier coefficients, and thusf ∈L([,a)). This is a contradiction.
The proof is completed.
Lemma . Let c,d∈l(Z).Suppose c∗d∈l(Z),then
(c∗d)∧(·) =ˆc(·)dˆ(·) a.e. onR.
Proof Sincec∗d∈l(Z), we have (c∗d)∧∈L([, )), and thus (c∗d)∧∈L([, )). Also observing thatˆc· ˆd∈L([, )), to finish the proof we only need to prove that
[,)ˆ
c(ξ)dˆ(ξ)eπikξdξ=c∗d(k) fork∈Z. (.)
Definecnby
cn(k) =
c(k) if|k| ≤n; if|k|>n
forn∈N. Thencn∈l(Z), and
cn–cl(Z)=ˆcn–cˆL([,))→ asn→ ∞. (.)
Sincecn∈l(Z) andd∈l(Z), we have (cn∗d)∧=cˆndˆ∈L([, )), and thus
[,)
ˆ
cn(ξ)dˆ(ξ)eπikξdξ=
l∈Z
cn(k–l)d(l) fork∈Z. (.)
Arbitrarily fixk∈Z. Observe that
l∈Z
cn(k–l)d(l) – l∈Z
and
[,)ˆcn(ξ)dˆ(ξ)eπikξdξ–
[,)
ˆ
c(ξ)dˆ(ξ)eπikξdξ≤ ˆcn–cˆL([,))ˆdL([,))→
asn→ ∞by (.). So, lettingn→ ∞in (.) we have (.). The proof is completed.
Lemma . For f,ψ∈L(R
+),we have
j∈Z
m∈Z
f,ψmDajψL(R+)= a
G(ψ,x)F(x)l(Z)dx. (.)
Proof Sincef(·)ψm(·)Dajψ(·)∈L(R+), we have
f,ψmDajψL(R+)= a
l∈Z
alfalx ψm
alx(D ajψ)
alxdx
= a
ψm(x)
l∈Z
(Dalf)(x)Daj+lψ(x)dx.
It follows that
j∈Z
m∈Z
f,ψmDajψL(R +)
= j∈Z
a
l∈Z
(Dalf)(x)(Daj+lψ)(x) dx
by Lemma .(ii), and thus (.) holds. The proof is completed.
Theorem . Letψ∈L(R
+).The following are equivalent:
(i) {ψmDajψ:m,j∈Z}is a Bessel sequence inL(R+)with Bessel boundB.
(ii) G∗G(ψ,·)≤BIa.e.on[,a).
(iii) |l∈Zalψ(alx)e–πilξ| ≤√Bfor a.e.(x,ξ)∈[,a)×[, ).
Proof Since
a
j∈Z
Dajψ(x)dx= ∞
ψ(x)dx<∞,
we havej∈Z|Dajψ(·)|<∞a.e. on [,a). So all rows and columns ofG(ψ,x) belong to
l(Z) for a.e.x∈[,a). It follows thatG∗G(ψ,·) is well defined a.e. on [,a) by the
Cauchy-Schwartz inequality. ThusG∗G(ψ,·)c,cl(Z)andG(ψ,·)c
l(Z)are well defined, and
G∗G(ψ,·)c,c
l(Z)=G(ψ,·)c
l(Z) (.)
a.e. on [,a) for eachc∈l(Z). First we show the equivalence between (i) and (ii). Suppose
G∗G(ψ,·)≤BIa.e. on [,a). Arbitrarily fixf ∈L(R
+). It is easy to check that
a
SoF(x)∈l(Z), and thus
G(ψ,x)F(x)l(Z)=
G∗G(ψ,x)F(x),F(x)
l(Z)≤BF(x)
l(Z), for a.e.x∈[,a). It follows that
a
G(ψ,x)F(x)l(Z)dx≤B a
F(x)l(Z)dx=BfL(R +).
Therefore,
j∈Z
m∈Z
f,ψmDajψL(R+)≤Bf L(R
+),
by Lemma ., and thus{ψmDajψ :m,j∈Z}is a Bessel sequence inL(R+) with Bessel
boundB.
Now suppose (i) holds. We prove (ii) by contradiction. SupposeG∗G(ψ,·) >BIon some
E⊂[,a) with|E|> . Take =c∈l(Z) and definef∈L(R+) by
F(·) =χE(·)c on [,a).
Thenf is well defined since
fL(R+)= a
F(x)l(Z)dx=|E|cl(Z).
Let us checkaG(ψ,x)F(x)
l(Z)dx: a
G(ψ,x)F(x)l(Z)dx= a
G∗G(ψ,x)F(x),F(x)
l(Z)dx =
E
G∗G(ψ,x)F(x),F(x)
l(Z)dx >B
a
F(x)l(Z)dx
=BfL(R +).
Soj∈Zm∈Z|f,ψmDajψL(R +)|
>Bf
L(R
+)by Lemma ., contradicting the fact that {ψmDajψ:m,j∈Z}is a Bessel sequence inL(R+) with Bessel boundB.
Next we prove the equivalence between (ii) and (iii). By (.) and the density ofl(Z) in l(Z), (ii) is equivalent to
G(ψ,x)cl(Z)≤Bcl(Z) forc∈l(Z) and a.e.x∈[,a),
which is equivalent to
Forxsatisfying (.), defineαx∈l(Z) by
αx(l) =a l ψalx.
Then (.) can be rewritten as
αx∗dl(Z)≤Bdl(Z) ford∈l(Z) and a.e.x∈[,a),
equivalently,
αˆx(ξ)dˆ(ξ)
dξ≤B
dˆ(ξ)dξ ford∈l(Z) and a.e.x∈[,a)
by Lemma ., this is also equivalent to
αˆx(ξ)ˆd(ξ)dξ≤B
ˆd(ξ)dξ ford∈l(Z) and a.e.x∈[,a), (.)
by the density of trigonometric polynomials inL([, )). It is obvious that (.) is equiv-alent to (iii). The proof is completed.
By the same procedure as in the proof of Theorem ., we have the following theorem.
Theorem . Letψ∈L(R
+).The following are equivalent:
(i) {ψmDajψ:m,j∈Z}is a frame forL(R+)with frames boundsAandB; (ii) AI≤G∗G(ψ,·)≤BIa.e.on[,a);
(iii) √A≤ |l∈Zalψ(alx)e–πilξ| ≤√Bfor a.e.(x,ξ)∈[,a)×[, ).
Theorem . Letψ,ϕ∈L(R
+).Suppose{ψmDajψ:m,j∈Z}and{ψmDajϕ:m,j∈Z}are
Bessel sequences in L(R
+).The following are equivalent:
(i) {ψmDajψ:m,j∈Z}and{ψmDajϕ:m,j∈Z}form a pair of dual frames forL(R+). (ii) j∈Zajϕ(aj·)ψ(aj+l·) =δ
l,a.e.on[,a)forl∈Z. (iii) (l∈Zalϕ(alx)e–πilξ)(
l∈Za l
ψ(alx)eπilξ) = for a.e.(x,ξ)∈[,a)×[, ).
Proof Since both {ψmDajψ :m,j∈Z} and{ψmDajϕ:m,j∈Z}are Bessel sequences in
L(R +),
(m,j)∈Z
f,ψmDajψL(R+)ψmDajϕ,gL(R+)
is well defined for eachf,g∈L(R
+). By Theorem .,G(ψ,x) andG(ϕ,x) are bounded
linear operators onl(Z) for a.e.x∈[,a). Let us check the (k,l)-entry ofG∗(ϕ,·)G(ψ,·):
G∗(ϕ,·)G(ψ,·)
k,l=
j∈Z
Daj+kϕ(·)Daj+lψ(·) =
j∈Z
aj+kϕaj+k·a j+l
ψaj+l·
=al–k j∈Z
So (ii) is equivalent to
G∗(ϕ,·)G(ψ,·) =I a.e. on [,a). (.)
By Lemma . and the polarization identity, we have
(m,j)∈Z
f,ψmDajψL(R+)ψmDajϕ,gL(R+)= a
G∗(ϕ,x)G(ψ,x)F(x),G(x)
l(Z)dx
forf,g∈L(R+). By a simply computation, we also have
f,gL(R+)= a
F(x),G(x)l(Z)dx
forf,g∈L(R+). So (i) holds if and only if
a
G∗(ϕ,x)G(ψ,x)F(x),G(x)
l(Z)dx =
a
F(x),G(x)l(Z)dx forf,g∈L(R+). (.)
So, to show the equivalence between (i) and (ii), next we only need to prove the equiva-lence between (.) and (.). Obviously, (.) implies (.). Now we prove the converse implication. Suppose (.) holds. Letenbe the vector inl(Z) with thenth component being and the others being zero for eachn∈Z. Since all entries ofG∗(ϕ,·)G(ψ,·) are in
L[,a), there existsE⊂[,a) with|E|= such that every point of (,a)\Eis a Lebesgue
point of all entries ofG∗(ϕ,·)G(ψ,·). Arbitrarily fixx∈(,a)\Eandn,m∈Z. Take
ε> so small that (x–ε,x+ε)⊂(,a), and takef andgsuch that
F(·) =χ(x–ε,x+ε)(·)en and G(·) =χ(x–ε,x+ε)(·)em
a.e. on [,a). Then from (.) we have
ε
x+ε
x–ε
G∗(ϕ,x)G(ψ,x)e
n,em
l(Z)dx=δn,m. This leads to
G∗(ϕ,x
)G(ψ,x)en,em
l(Z)=δn,m,
by lettingε→. By the arbitrariness ofx,n, andm, we have
G∗(ϕ,·)G(ψ,·)e
n,em
l(Z)=δn,m on [,a)\Eforn,m∈Z. This implies that
G∗(ϕ,·)G(ψ,·) =I on (,a)\E .
Now we prove the equivalence between (ii) and (iii). Obviously, (ii) is equivalent to
j∈Z
ajϕaj·a j+l
ψaj+l·=δl, a.e. on [,a) forl∈Z,
that is,
j∈Z
a–jϕa–j·a l–j
ψal–j·=δl, a.e. on [,a) forl∈Z. (.)
Forxsatisfying (.), defineαx,βx∈l(Z) by
αx(l) =alψalx and βx(l) =a–lϕa–lx.
Then (.) can be rewritten as
αx∗βx=δl, for a.e.x∈[,a) andl∈Z.
This is equivalent to
l∈Z
alψalxe–πilξ l∈Z
alϕalxeπilξ
= a.e. (x,ξ)∈[,a)×[, )
by Lemma ., and thus equivalent to (iii) by itsZ-periodicity with respect toξ. The proof
is completed.
3 Parseval frames and orthonormal bases
The following theorem characterizes Parseval frames of the form (.). It is worth noting that the theorem shows all such Parseval frames must be orthonormal bases. This is not the case for Parseval wavelet (Gabor) frames.
Theorem . Letψ∈L(R
+).The following are equivalent: (i) {ψmDajψ:m,j∈Z}is a Parseval frames forL(R+). (ii) j∈Zajψ(aj·)ψ(aj+l·) =δ
l,a.e.on[,a)forl∈Z. (iii) |l∈Zalψ(alx)eπilξ|= for a.e.(x,ξ)∈[,a)×[, ).
(iv) {ψmDajψ:m,j∈Z}is an orthonormal basis forL(R+).
Proof Since (ii) and (iii) are equivalent, and (iii) implies that{ψmDajψ:m,j∈Z}is a Bessel sequence by Theorem ., we can obtain the equivalence among (i), (ii), and (iii) by tak-ingϕ=ψ in Theorem .. It is obvious that (iv) implies (i). Now we prove the converse implication to finish the proof. Suppose (i) holds. Then (iv) holds if and only if
ψmDajψL(R+)= form,j∈Z,
equivalently,ψ
L(R
+)=a– . By the equivalence between (i) and (ii), we have
j∈Z
Integrating this equation over [,a) leads to
ψL(R+)=a– .
So (iv) holds. The proof is completed.
Theorem . Let =ψ∈L(R+).Then we have
(i) {ψmDajψ:m,j∈Z}is an orthogonal system inL(R+)if and only if
j∈Z
ajψaj·ψaj+l·=ψ
L(R+)
a– δl, a.e. on[,a)forl∈Z; (.)
(ii) {ψmDajψ:m,j∈Z}is an orthonormal system forL(R+)if and only if
j∈Z
ajψaj·ψaj+l·=δ
l, a.e. on[,a)forl∈Z. (.)
Proof (i) By a simple computation, we have
ψmDajψ,ψmDajψL(R+)= √
a– a j–j
a
ψm–m(x)
j∈Z
ajψajxψaj+j–jxdx
for (m,j), (m,j)∈Z. It follows that{ψmDajψ :m,j∈Z} is an orthogonal system in
L(R
+) if and only if
a
ψm(x) j∈Z
ajψajxψaj+lxdx= for (, ) = (m,l)∈Z. (.)
Also observing that
ψL(R +)=
a
j∈Z
ajψajxdx, (.)
we have (i) by Lemma .(i) and the uniqueness theorem of Fourier coefficients.
(ii){ψmDajψ:m,j∈Z}is an orthonormal system forL(R+) if and only if it is an orthog-onal system andψ
L(R+)=a– . So we have (ii) by (i) and (.). The proof is completed.
As an immediate consequence of Theorems . and ., we have the following corollary.
Corollary . Let =ψ∈L(R
+).Then the following are equivalent: (i) {ψmDajψ:m,j∈Z}is a Parseval frame forL(R+).
(ii) {ψmDajψ:m,j∈Z}is an orthonormal basis forL(R+). (iii) {ψmDajψ:m,j∈Z}is an orthonormal system inL(R+). (iv) j∈Zajψ(aj·)ψ(aj+l·) =δ
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
Supported by the National Natural Science Foundation of China (Grant No. 11271037).
Received: 30 April 2016 Accepted: 6 July 2016 References
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