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Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013, Article ID 530172, 7 pageshttp://dx.doi.org/10.1155/2013/530172
Research ArticleNew Proof for Balian-Low Theorem of Nonlinear Gabor System
D. H. Yuan,1 S. Z. Yang,2 X. W. Zheng,2 and Y. F. Shen2
1 Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong 521041, China2Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
Correspondence should be addressed to D. H. Yuan; [email protected]
Received 27 July 2013; Revised 24 September 2013; Accepted 25 September 2013
Academic Editor: T. S. S. R. K. Rao
Copyright Β© 2013 D. H. Yuan et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The main purpose of this paper is to give a new proof of the Balian-Low theorem for Gabor system {ππππ(2ππ‘)π(π‘ β π), π, π β Z},which is proposed by Fu et al. and associated with nonlinear Fourier atoms. To this end, we establish the relationships betweenspaces πΏ2(R, ππ) and πΏ2(R). We also introduce the concept of frame associated with nonlinear Fourier atoms for πΏ2(R, ππ) andobtain many subsidiary results for this kind of (Gabor) frames.
1. Introduction
Note that the classical Fourier atoms π2ππππ‘ cannot expose thetime-varying property of nonstationary signals [1]. Recently,a kind of specific nonlinear phase function π
π(2ππ‘) is intro-
duced in [2β6]. Note that, for different π, the shapes ofcos ππ(2ππ‘) (also those of sin π
π(2ππ‘)) are different. It is
observed that the closer π gets to 1, the sharper the graphof cos π
π(2ππ‘) is. This means that the nontrivial Harmonic
waves πππππ(2ππ‘) can represent a conformal rescaling of classicFourier atoms. Thus, the nontrivial Harmonic waves areexpected to be better suitable and adaptable, along withdifferent choices of π, to capture nonstationary features ofband-limited signals. In fact, Ren et al. [7] obtained some newphenomena on the Shannon sampling theorem by dealingwith sampling points which are nonequally distributed.
Motivated by these points, Fu et al. [8] considered a newlyGabor system {ππππ(2ππ‘)π(π‘ β π), π, π β Z} generated by afunctionπ, where π(π‘) satisfies certain assumptions. Note thattheywere not restricted to the conformal phase functions π(π‘)in their discussion. This freedom allows us to choose phasefunctions adequate to the necessary nonuniform samplingof the signal [7]. Using the Zak transform technique, theyestablished the Balian-Low theorem for this newly Gaborsystem.
We point out that the Gabor system {ππππ(2ππ‘)π(π‘ β π),π, π β Z} proposed by [8] can be related to already existing
cases. A particular case of this kind of Gabor system is thenonlinear Fourier atoms πππππ(2ππ‘) which was discussed in [2β6]. Using the nonlinear Fourier atoms in [2β6], we have thatthe frequency modulation πππππ(2ππ‘) represents a conformaldilation of the classical modulation πππ2ππ‘ on the unit circle.Taking the proposedGabor systemswith different parametersπ, we can obtain a dictionary of Gabor frames with differentdilation parameters in the modulation part. A simple changeof variables can establish a clear relation between this systemand the system generated by the affine Weyl-Heisenberggroup with dilation on the window function [9, 10].
Basing on these points, we can say that establishingrelationships between frames for πΏ2(R, ππ) and πΏ2(R) is aninteresting issue. In this paper, our main purpose is to givea different proof for the Balian-Low theorem proposed in[8]. For this purpose, we firstly establish the relationshipsbetween spaces πΏ2(R, ππ) and πΏ2(R). Basing on this relation-ship, we obtain many properties for general frame system{ππππ(2ππ‘)
ππ(π‘), π, π β Z} and its special case {ππππ(2ππ‘)π(π‘ β
π), π, π β Z}, where π is a nonlinear function. With theseresults for general Gabor system {ππππ(2ππ‘)π(π‘βπ), π, π β Z},we give a new and simple proof for the Balian-Low theoremproposed in [8].
The rest of the paper is organized as follows. Section 2 isdevoted to giving some notations and lemmas. In Section 3,we establish the relationship between spaces πΏ2(R, ππ) andπΏ2(R); we also depict some properties of general frame
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2 Journal of Function Spaces and Applications
{ππππ(2ππ‘)
ππ(π‘), π, π β Z} for πΏ2(R, ππ). In Section 4, we
establish the relationship betweenGabor frame {ππππ(2ππ‘)π(π‘βπ), π, π β Z} for πΏ2(R, ππ) and classical one {ππ2πππ‘ππ(π‘ βπ), π, π β Z} for πΏ2(R) under some assumptions on π; fur-ther, a new and simple proof is presented for the Balian-Lowtheorem which was proposed by Fu et al. [8].
2. Notations
In this section, we present some notations and lemmas, whichwill be needed in the rest of the paper.
For an arbitrary measure π in R, consider the spaceπΏ2(R, ππ) of square integrable functions in R with respect to
π and the finite norm:
ππ = (
1
π (2π) β π (0)β«β
ββ
π (π₯)2ππ (2ππ₯))
1/2
(1)
induced by the inner product
β¨π, πβ©π:=
1
π (2π) β π (0)β«β
ββ
π (π₯) π (π₯)ππ (2ππ₯) . (2)
To obtain the Balian-Low theorem for Gabor system{ππππ(2ππ‘)
π(π‘ β π), π, π β Z}, Fu et al. introduced someassumptions including the Assumptions 2.1 and 2.2 in [8]for a nonlinear phase function π. Combining these twoassumptions together, we obtain the following Assumption 1.
Assumption 1. Let function π : R β R be a measure on Rand satisfy
π (π₯ + 2ππ) = π (π₯) + 2ππ, (3)
for any π₯ β R and π β Z. Further, π(π₯) > 0 for all π₯ β R.
Note that π(π₯) > 0 for all π₯ β R; one obtains that theinverse of π (denote by πβ1) exists. Moreover, it is obvious tocheck that π satisfies
πβ1(π₯ + 2πππΎ) = π
β1(π₯) + 2ππ, (4)
for any π₯ β R and π β Z. In fact, we obtain from (3) that
πβ1(π (π₯ + 2ππ)) = π
β1(π (π₯) + 2ππ) , (5)
or
π₯ + 2ππ = πβ1(π (π₯) + 2ππ) . (6)
Replacing π(π₯) and π₯ by π‘ and πβ1(π‘) in (6), respectively, weobtain (4).
For a function π defined in R, denote by
ππ(π‘) := π (
1
2ππβ1(2ππ‘)) (7)
through the rest of paper.
For π, π β R, consider the translation operator (πππ)(π₯) =
π(π₯ β π) and the modulation operator (πΈπππ)(π₯) = π
πππ(2ππ₯)
π(π₯), both acting on πΏ2(R, ππ). In [8], Fu et al. proposed ageneral Gabor frame for πΏ2(R, ππ). We say that the system{πΈπ
ππππ, π, π β Z} is a general Gabor frame for πΏ2(R, ππ) if
there exist two constants π΄, π΅ > 0 such that
π΄π2
πβ©½ βπ, πβZ
β¨π, πΈπ
ππππππβ©π
2
β©½ π΅π2
π (8)
holds for all π β πΏ2(R, ππ). To further study the generalGabor frame as defined in [8], we introduce a general frameconcept as follows.
Definition 2. Let ππβ πΏ2(R, ππ), π β Z. One says that the
system {πΈππππ, π, π β Z} is a general frame for πΏ2(R, ππ) if
there exist two constants π΄, π΅ > 0 such that
π΄π2
πβ©½ βπ, πβZ
β¨π, πΈπ
πππβ©π
2
β©½ π΅π2
π (9)
holds for all π β πΏ2(R, ππ); moreover, one says that the frame{πΈπ
πππ, π, π β Z} is tight if π΄ = π΅; in particular, the frame
{πΈπ
πππ, π, π β Z} is Parseval if π΄ = π΅ = 1.
Given a frame {πΈππππ, π, π β Z} for πΏ2(R, ππ), a dual
frame is a frame {πΈππβπ, π, π β Z} of πΏ2(R, ππ) which
satisfies the reconstruction property
π = βπ,πβZ
β¨π, πΈπ
πππβ©ππΈπ
πβπ, βπ β πΏ
2(R, ππ) , (10)
and we say that the systems {πΈππππ, π, π β Z} and {πΈπ
πβπ, π,
π β Z} constitute a pair of dual frames for πΏ2(R, ππ), wherethe convergence is in the πΏ2 sense. Note that if π(2ππ₯) = 2ππ₯for all π₯ β R, then the frame {πΈπ
πππ, π, π β Z} for πΏ2(R, ππ)
constitutes a frame for πΏ2(R).For fixed π, π β πΏ2(R, ππ) and π > 0, we introduce the
π-bracket product as follows:
[π, π]π
π(π₯) := β
πβZ
ππ(π₯ +
π
π)ππ (π₯ +
π
π), a.e. π₯ β R.
(11)
If π(2ππ₯) = 2ππ₯ for π₯ β R, then [π, π]ππis bracket product
(denote by [π, π]π) introduced by Ron and Shen in [11]. Thus,
[π, π]π
π= [ππ, ππ]π. (12)
Note that [π, π]ππis a 1-periodic function.
With the classical bracket product, Christensen and Sun[12] proved the following Lemma 3, which is [13, Lemma 2.3].
Lemma 3. Let ππ, βπβ πΏ2(R), π β Z, and π > 0. Let the
systems {πΈππππ, π, π β Z} and {πΈ
ππβπ, π, π β Z} be Bessel
sequences in πΏ2(R). Define
ππ = βπ, πβZ
β¨π, πΈππππβ© πΈππβπ, βπ β πΏ
2(R) . (13)
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Journal of Function Spaces and Applications 3
Then, the following holds:
(ππ) (π₯) =1
πβπβZ
[π, ππ]π(π₯) βπ (π₯)
=1
πβπβZ
βπβZ
π(π₯ +π
π)ππ(π₯ +
π
π)βπ (π₯) ,
βπ β πΏ2(R) ,
(14)
where the convergence is in the πΏ2 sense. Moreover, {πΈππππ, π,
π β Z} and {πΈππβπ, π, π β Z} are a pair of dual frames for
πΏ2(R) if and only if
βπβZ
ππ(π₯ +
π
π) βn (π₯ +
π
π) = ππΏ
π,0, a.e. π₯ β R. (15)
The following lemma follows from general properties ofshift-invariant frames; see [11, Corollary 1.6.2]. Alternatively,it can be proved similarly to [14, Theorem 8.4.4].
Lemma 4. Let ππβ πΏ2(R), π β Z, π > 0, and
π΅ :=1
πsupπ₯βR
βπβZ
βπβZ
ππ (π₯) ππ (π₯ β
π
π)
< β. (16)
Then, {πΈππππ, π, π β Z} is a Bessel sequence with upper frame
bound π΅. If also
π΄ :=1
πinfπ₯βR
(βπβZ
ππ (π₯)2β βπ ΜΈ= 0
βπβZ
ππ (π₯) ππ (π₯ β
π
π)
) > 0,
(17)
then {πΈππππ, π, π β Z} constitutes a frameπΏ2(R)with bounds
π΄ and π΅.
3. Frame for πΏ2(R, ππ)
In this section, we discuss frames for πΏ2(R, ππ). Here, we willestablish the relationship between frames for πΏ2(R, ππ) andπΏ2(R). We will also give necessary conditions for frames and
characterize a pair of dual frames in πΏ2(R, ππ). Above all,we establish the relationship between πΏ2(R, ππ) and πΏ2(R) asfollows.
Theorem 5. Let π, π be functions defined on R. Then,β¨π, πβ©
π= β¨ππ, ππβ©; in particular, βπβ
π= βπ
πβ, which means
that π β πΏ2(R, ππ) if and only if ππ β πΏ2(R).
Proof. Denote π‘ := (1/2π)π(2ππ₯). Then,
β«β
ββ
π (π₯) π (π₯)ππ (2ππ₯) = β«β
ββ
π(1
2ππβ1(2ππ‘))
Γ π (1
2ππβ1(2ππ‘))π (2ππ‘)
= 2πβ«β
ββ
ππ(π‘) ππ(π‘)ππ‘.
(18)
This means that
β¨π, πβ©π= β¨ππ, ππβ© . (19)
Thus, we can obtain the desired result.
Theorem 6. Let ππ, π β Z be functions defined on R. Then,
the system {πΈππππ, π, π β Z} constitutes a frame for πΏ2(R, ππ)
if and only if the system {πΈπππ
π, π, π β Z} constitutes a frame
for πΏ2(R) and these two systems have the same bounds.
Proof. From Theorem 5, one obtains that ππβ πΏ2(R, ππ) if
and only if πππβ πΏ2(R) for π β Z. Note that
β¨π, πΈπ
πππβ©π=1
2πβ«β
ββ
π (π₯) ππππ(2ππ₯)ππ(π₯)ππ (2ππ₯)
=1
2πβ«β
ββ
π(1
2ππβ1(2ππ₯))
Γ ππ2πππ₯ππ(1
2ππβ1 (2ππ₯))π (2ππ₯)
= β«β
ββ
ππ(π₯) ππ2πππ₯ππ
π(π₯)ππ₯ = β¨π
π, πΈπππ
πβ© .
(20)
Then,
π΄π2
πβ©½ βπ, πβZ
β¨π, πΈπ
πππβ©π
2
β©½ π΅π2
π, βπ β πΏ
2(R, ππ)
(21)
is equivalent to
π΄ππ
2
β©½ βπ, πβZ
β¨ππ, πΈπππ
πβ©
2
β©½ π΅ππ
2
, βππβ πΏ2(R) .
(22)
Now, we can obtain the desired results.
Theorem7. Let ππ, π β Z be functions defined onR.Then, the
systems {πΈππππ, π, π β Z} and {πΈπ
πβπ, π, π β Z} constitute
a pair of dual frames for πΏ2(R, ππ) if and only if the systems{πΈπππ
π, π, π β Z} and {πΈ
πβπ
π, π, π β Z} constitute a pair of
dual frames for πΏ2(R).
Proof. βif β part. If the systems {πΈππππ, π, π β Z} and {πΈπ
πβπ,
π, π β Z} constitute a pair of dual frames for πΏ2(R, ππ).Then, by Theorem 6, these two systems {πΈπ
πππ, π, π β Z}
and {πΈππβπ, π, π β Z} are frames for πΏ2(R, ππ). Moreover,
we obtain from (20) that
π (π₯) = βπ,πβZ
β¨ππ, πΈπππ
πβ©πΈπ
πβπ (π₯) , π₯ β R, (23)
for any π β πΏ2(R, ππ), where the convergence is in the πΏ2sense. Replacing π₯ by (1/2π)πβ1(2ππ₯) in the above equation,we obtain
ππ(π₯) = β
π,πβZ
β¨ππ, πΈπππ
πβ©πΈπβπ
π(π₯) , π₯ β R, (24)
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4 Journal of Function Spaces and Applications
for any ππ β πΏ2(R), where the convergence is in the πΏ2 sense.Therefore, the systems {πΈ
πππ
π, π, π β Z} and {πΈ
πβπ
π, π, π β
Z} constitute a pair of dual frames for πΏ2(R).The proof of βonly if β part is similar to the βif β part, and
we omit it.
With the π-bracket product proposed in the above sec-tion, we can prove the following theorem.
Theorem 8. Let ππ, βπβ πΏ2(R, ππ) for π β Z. Let {πΈπ
πππ, π,
π β Z} and {πΈππβπ, π, π β Z} be Bessel sequences inπΏ2(R, ππ).
Define
ππ = βπ, πβZ
β¨π, πΈπ
πππβ©ππΈπ
πβπ, (25)
for π β πΏ2(R, ππ). Then,
(πππ) (π₯) = β
πβZ
[π, ππ]π(π₯) βπ
π(π₯)
Γ βπβZ
βπβZ
ππ(π₯ + π) ππ
π(π₯ + π)β
π
π(π₯)
(26)
holds for π β πΏ2(R, ππ), where the convergence is in the πΏ2
sense. Moreover, the systems {πΈππππ, π, π β Z} and {πΈπ
πβπ, π,
π β Z} constitute a pair of dual frames for πΏ2(R, ππ) if and onlyif
βπβZ
ππ
π(π₯ + π) βπ
π(π₯ + π) = πΏπ,0, a.e. π₯ β R. (27)
Proof. For fixed π β πΏ2(R, ππ), one obtains from (20) that
ππ (π₯) = βπ, πβZ
β¨π, πΈπ
πππβ©ππΈπ
πβπ (π₯)
= βπ, πβZ
β¨ππ, πΈπππ
πβ©πΈπ
πβπ (π₯) ,
(28)
where the convergence is in the πΏ2 sense. Replacing π₯ by(1/2π)π
β1(2ππ₯) in the above equation, we obtain
πππ(π₯) = β
π, πβZ
β¨ππ, πΈπππ
πβ©πΈπβπ
π(π₯) , βπ
πβ πΏ2(R) .
(29)
Note that the systems {πΈππππ, π, π β Z} and {πΈπ
πβπ, π, π β
Z} are Bessel sequences in πΏ2(R, ππ). We can deduce that thesystems {πΈ
πππ
π, π, π β Z} and {πΈ
πβπ
π, π, π β Z} are Bessel
sequences in πΏ2(R). Therefore, one obtains (26) from (14).From Theorem 7, we know that the systems
{πΈπ
πππ, π, π β Z} and {πΈπ
πβπ, π, π β Z} constitute a
pair of dual frames for πΏ2(R, ππ) if and only if the systems{πΈπππ
π, π, π β Z} and {πΈ
πβπ
π, π, π β Z} constitute a pair of
dual frames for πΏ2(R). Thus, by (15) in Lemma 3, one obtainsthe desired result.
Theorem 9. Let ππβ πΏ2(R, ππ), π β Z, and suppose that
π΅ := supπ₯βR
βπβZ
βπβZ
ππ
π(π₯) πππ(π₯ + π)
< β. (30)
Then, the system {πΈππππ, π, π β Z} is a Bessel sequence with
upper frame bound π΅ for πΏ2(R, ππ). If also
π΄ := infπ₯βR
(βπβZ
ππ
π(π₯)
2
β βπ ΜΈ= 0
βπβZ
ππ
π(π₯) πππ(π₯ + π)
) > 0,
(31)
then the system {πΈππππ, π, π β Z} constitutes a frame forπΏ2(R,
ππ) with bounds π΄ and π΅.
Proof. Since ππβ πΏ2(R, ππ), π β Z, then ππ
πβ πΏ2(R), π β
Z. If 0 < π΄, π΅ < β, then by Lemma 4, the system {πΈπππ
π,
π, π β Z} constitutes a frame for πΏ2(R)with frame boundsπ΄and π΅. Therefore, by Theorem 6, one obtains that the system{πΈπ
πππ, π, π β Z} constitutes a frame for πΏ2(R, ππ) with the
same frame bounds π΄ and π΅.
4. Gabor Frame for πΏ2(R, ππ)
In this section, Gabor frames for πΏ2(R, ππ) are discussed.We establish the relationship between the generalized Gaborframe {ππππ(2ππ‘)π(π‘ β π), π, π β Z} for πΏ2(R, ππ) and theclassical one {ππ2πππ‘ππ(π‘ β π), π, π β Z} for πΏ2(R); further,we prove the Balian-Low theorem for Gabor system {ππππ(2ππ‘)π(π‘βπ), π, π β Z} proposed by Fu et al. in [8] from a differentviewpoint.
Theorem 10. Let π β πΏ2(R, ππ). Then,
(πππ)π(π₯) = πππ
π(π₯) . (32)
Proof. Since
πβ1(π₯ + 2ππ) = π
β1(π₯) + 2ππ, βπ₯ β R, π β Z, (33)
then
πβ1(2π (π₯ + π)) = π
β1(2ππ₯ + 2ππ)
= πβ1(2ππ₯) + 2ππ, βπ₯ β R, π β Z.
(34)
Hence,
(πππ)π(π₯) = π (
1
2ππβ1(2ππ₯) β π)
= π(1
2ππβ1(2π (π₯ β π))) = πππ
π(π₯) .
(35)
We complete the proof.
Theorem 11. Let π be a function defined on R. Then, thegeneral Gabor system {πΈπ
ππππ, π, π β Z} constitutes a frame
for πΏ2(R, ππ) if and only if the classical Gabor system{πΈπππππ, π, π β Z} constitutes a frame for πΏ2(R) with the
same bounds.
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Journal of Function Spaces and Applications 5
Proof. Define ππ(π₯) := π
ππ(π₯), π β Z. FromTheorem 6, one
obtains that the general Gabor system {πΈπππππ, π, π β Z}
constitutes a frame for πΏ2(R, ππ) if and only if the system{πΈπ(πππ)π, π, π β Z} constitutes a frame for πΏ2(R) with
the same bounds. So, one obtains the desired result fromTheorem 10.
Combining Theorems 8 and 10 together, we obtain thefollowingTheorem 12.
Theorem 12. Consider π, β β πΏ2(R, ππ). Let the systems{πΈπ
ππππ, π, π β Z} and {πΈπ
πππβ, π, π β Z} be Bessel
sequences in πΏ2(R, ππ). Define
ππ = βπ, πβZ
β¨π, πΈπ
ππππβ©πΈπ
πππβ, βπ β πΏ
2(R, ππ) . (36)
Then, for any π β πΏ2(R, ππ),
(πππ) (π₯) = β
πβZ
[ππ, ππππ] (π₯) ππβ
π(π₯)
= βπβZ
βπβZ
ππ(π₯ + π) πππ
π (π₯ + π) ππβπ(π₯) ,
(37)
where the convergence is in the πΏ2 sense. Moreover, the systems{πΈπ
ππππ, π, π β Z} and {πΈπ
πππβ, π, π β Z} constitute a pair
of dual frames for πΏ2(R, ππ) if and only if
βπβZ
ππππ(π₯ + π) ππβ
π (π₯ + π) = πΏπ,0, a.e. π₯ β R. (38)
Proof. Replacing ππand β
πby πππ and π
πβ in (26) and (27),
respectively, we have
(πππ) (π₯) = β
πβZ
[π, πππ]π(π₯) (ππβ)
π(π₯)
= βπβZ
βπβZ
ππ(π₯ + π) (πππ)
π(π₯ + π)(ππβ)
π(π₯) ,
(39)
βπβZ
(πππ)π(π₯ + π) (ππβ)
π(π₯ + π) = πΏπ, 0, a.e. π₯ β R.
(40)
Equations (37) and (38) follow from (39) and (40), respec-tively. Here, we used the facts that (π
ππ)π(π₯) = π
πππ(π₯) and
(ππβ)π(π₯) = π
πβπ(π₯).
By Theorems 9 and 10, we obtainTheorem 13.
Theorem 13. Consider π β πΏ2(R, ππ), π β Z, and supposethat
π΅ := supπ₯β[0,1]
βπβZ
βπβZ
ππππ(π₯) πππ
π (π₯ + π)
< β. (41)
Then, the system {πΈπππππ, π, π β Z} is a Bessel sequence for
πΏ2(R, ππ) with upper frame bound π΅. If also
π΄ := infπ₯β[0,1]
(βπβZ
ππππ(π₯)
2
β βπ ΜΈ= 0
βπβZ
ππππ(π₯) πππ
π (π₯ + π)
) > 0,
(42)
then the system {πΈπππππ, π, π β Z} constitutes a frame for
πΏ2(R, ππ) with bounds π΄ and π΅.
Proof. Since ππππ(π₯) = (π
ππ)π(π₯) and π
πβπ(π₯) = (π
πβ)π(π₯),
then
π΅ = supπ₯β[0,1]
βπβZ
βπβZ
(πππ)π(π₯) (πππ)
π(π₯ + π)
,
π΄ = infπ₯β[0,1]
(βπβZ
(πππ)π(π₯)
2
ββπ ΜΈ= 0
βπβZ
(πππ)π(π₯) (πππ)
π(π₯ + π)
) .
(43)
Define
π»1 (π₯) := β
πβZ
βπβZ
(πππ)π(π₯) (πππ)
π(π₯ + π)
,
π»2 (π₯) := β
πβZ
(πππ)π(π₯)
2
β βπ ΜΈ= 0
βπβZ
(πππ)π(π₯) (πππ)
π(π₯ + π)
,
(44)
thenπ»1andπ»
2are 1-periodic functions. Thus,
π΅ = supπ₯βR
βπβZ
βπβZ
(πππ)π(π₯) (πππ)
π(π₯ + π)
,
π΄ = infπ₯βR
(βπβZ
(πππ)π(π₯)
2
β βπ ΜΈ= 0
βπβZ
(πππ)π(π₯) (πππ)
π(π₯ + π)
) .
(45)
ByTheorem 9, one obtains the results.
Theorem 14. Let π β πΏ2(R, ππ). Assume that the system{πΈπ
ππππ, π, π β Z} constitutes a generalized Gabor frame for
πΏ2(R, ππ) with bounds π΄ and π΅. Then,
π΄ β©½ βπβZ
ππ(π₯ β π)
2
β©½ π΅, a.e. π₯ β R. (46)
-
6 Journal of Function Spaces and Applications
Proof. If the system {πΈπππππ, π, π β Z} constitutes a
generalized Gabor frame for πΏ2(R, ππ) with bounds π΄ andπ΅. Then, by Theorem 7, {πΈ
π(πππ)π, π, π β Z} constitutes a
frame for πΏ2(R) with the same bounds π΄ and π΅. Note that(πππ)π= ππππ. We can say that the system {πΈ
πππππ, π, π β
Z} constitutes a Gabor frame for πΏ2(R)with the same boundsπ΄ and π΅. Thus, one obtains from [14, Proposition 8.3.2] thedesired result.
Theorem 15. Let π β πΏ2(R, ππ). Suppose that the system{πΈπ
ππππ, π, π β Z} constitutes a general Gabor frame for πΏ2
(R, ππ). If the derivative π of function π is continuous on R,then either
β«β
ββ
π₯2π (π₯)
2ππ (2ππ₯) = β (47)
or
β«β
ββ
π2π (π)
2ππ (2ππ) = β. (48)
Proof. Since {πΈπππππ, π, π β Z} constitutes a general
Gabor frame for πΏ2(R, ππ), then, by Theorem 11, the system{πΈπππππ, π, π β Z} constitutes a classical Gabor frame for
πΏ2(R). Therefore, by the classical Balian-Low theorem, we
have either
β«β
ββ
π₯2ππ(π₯)
2
ππ₯ = β (49)
or
β«β
ββ
π2ππ (π)
2
ππ = β. (50)
That is either
β«β
ββ
π2(2ππ₯)
π (π₯)2ππ (2ππ₯) = β (51)
or
β«β
ββ
π2(2ππ)
π (π)2ππ (2ππ) = β. (52)
We need to prove that (51) implies (47) or (52) implies(48). Next, we only prove that (51) implies (47) (the case(52) implies (48) can be obtained similarly). Without loss ofgenerality, let
β«β
0
π2(2ππ₯)
π (π₯)2ππ (2ππ₯) = β. (53)
Since the derivative π of function π is continuous on R andπ satisfies Assumption 1, then
0 < minπ₯β[0,2π]
π(π₯) β€ π
(π‘) β€ maxπ₯β[0,2π]
π(π₯) , π‘ β R. (54)
By the Lagrange mean-valued theorem, there exists π β [0, π₯]such that
π (π₯) = π (0) + π(π) π₯. (55)
Therefore, for any fixed π₯0> 0, there exists a constant πΆ > 0
such that
|π (π₯)| β€ πΆπ₯, βπ₯ β₯ π₯0. (56)
Note that
β«π₯0
0
π2(2ππ₯)
π (π₯)2ππ (2ππ₯) < β. (57)
Therefore,
β«β
π₯0
πΆ2π₯2π (π₯)
2ππ (2ππ₯)
β₯ β«β
π₯0
π2(2ππ₯)
π (π₯)2ππ (2ππ₯) = β.
(58)
That is,
β«β
π₯0
π₯2π (π₯)
2ππ (2ππ₯) = β, (59)
or
β«β
ββ
π₯2π (π₯)
2ππ (2ππ₯) = β. (60)
In the proof of Theorem 15, the main technique is theinequality
|π (π₯)| β€ πΆπ₯, βπ₯ β₯ π₯0, (61)
for some positive constant πΆ. Note that
π (π₯ + 2ππ) = π (π₯) + 2ππ (62)
is equivalent to
π (π₯) = π₯ + π½ (π₯) , (63)
where π½ is a 2π-periodic function. We obtain the followingBalian-Low theorem which weakens the conditions imposedon π in Theorem 15.
Theorem 16. Let π β πΏ2(R, ππ). Suppose that the system{πΈπ
ππππ, π, π β Z} constitutes a general Gabor frame for
πΏ2(R, ππ). Let π½ be a 2π-periodic function such that
π (π₯) = π₯ + π½ (π₯) , βπ₯ β R,
π½ (π₯) β€ πΆπ₯, βπ₯ β₯ π₯0,
(64)
where πΆ is a positive constant. Then, one and only one of theinequalities (47) and (48) holds.
In applications of frames, it is inconvenient that the framedecomposition, as stated in [15, Theorem 5.1.7], requires theinverse of a frame operator. As we have seen in the discussionof general frame theory, one way of avoiding the problem isto consider tight frames. Hence, we give characterization fortight Gabor frames in πΏ2(R, ππ).
-
Journal of Function Spaces and Applications 7
Theorem 17. Let π β πΏ2(R, ππ). Then, the system {πΈπππππ,
π, π β Z} constitutes a tight frame for πΏ2(R, ππ) with π΄ = 1 ifand only if
βπβZ
ππ(π₯ β π)
2
= 1,
βπβZ
ππ(π₯ β π) ππ (π₯ β π β π) = 0, for π ΜΈ= 0
(65)
holds a.e. in R.
Proof. By Theorem 11, one obtains that the system {πΈπππππ,
π, π β Z} constitutes a tight frame for πΏ2(R, ππ) with π΄ = 1if and only if {πΈ
πππππ, π, π β Z} constitutes a tight frame
for πΏ2(R) with π΄ = 1. From [14, Theorem 9.5.2], one obtainsthe desired result.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
The authors are grateful to the referees for their valuablesuggestions that helped to improve the paper in its presentform. This research is supported by the National NaturalScience Foundation of China (Grant no. 11071152), the Nat-ural Science Foundation of Guangdong Province (Grant nos.S2013010013101 and S2011010004511), and the Foundation ofHanshan Normal University (Grant nos. QD20131101 andLQ200905) This work was also partially supported by theOpening Project of Guangdong Province Key Laboratory ofComputational Science at the Sun Yat-sen University (Grantno. 201206012).
References
[1] B. Picinbono, βOn instantaneous amplitude and phase of sig-nals,β IEEE Transactions on Signal Processing, vol. 45, no. 3, pp.552β560, 1997.
[2] Q.Chen, L. Li, andT.Qian, βTwo families of unit analytic signalswith nonlinear phase,β Physica D, vol. 221, no. 1, pp. 1β12, 2006.
[3] Q. Chen, L. Li, and T. Qian, βStability of frames generatedby nonlinear Fourier atoms,β International Journal of Wavelets,Multiresolution and Information Processing, vol. 3, no. 4, pp.465β476, 2005.
[4] Y. Fu and L. Li, βNontrivial harmonic waves with positiveinstantaneous frequency,β Nonlinear Analysis: Theory, Methodsand Applications, vol. 68, no. 8, pp. 2431β2444, 2008.
[5] T. Qian, Q. Chen, and L. Li, βAnalytic unit quadrature signalswith nonlinear phase,β Physica D, vol. 203, no. 1-2, pp. 80β87,2005.
[6] T. Qian, βCharacterization of boundary values of functions inHardy spaces with applications in signal analysis,β Journal ofIntegral Equations and Applications, vol. 17, no. 2, pp. 159β198,2005.
[7] G. Ren,Q.Chen, P. Cerejeiras, andU.Kaehle, βChirp transformsand chirp series,β Journal of Mathematical Analysis and Applica-tions, vol. 373, no. 2, pp. 356β369, 2011.
[8] Y. Fu, U. Kaehler, and P. Cerejeiras, βThe Balian-Low theoremfor a new kind of Gabor system,βApplicable Analysis, vol. 92, no.4, pp. 799β813, 2013.
[9] S. Dahlke, M. Fornasier, H. Rauhut, G. Steidl, and G. Teschke,βGeneralized coorbit theory, Banach frames, and the relation toπΌ-modulation spaces,β Proceedings of the London MathematicalSociety III, vol. 96, no. 2, pp. 464β506, 2008.
[10] G. Teschke, S. Dahlke, and K. Stingl, βCoorbit theory, multi-πΌ-modulation frames, and the concept of joint sparsity formedicalmultichannel data analysis,β Eurasip Journal on Advances inSignal Processing, vol. 2008, Article ID 471601, 19 pages, 2008.
[11] A. Ron and Z. Shen, βFrames and stable bases for shift-invariantsubspaces of πΏ2(Rπ),β Canadian Journal of Mathematics, vol. 47,no. 5, pp. 1051β1094, 1995.
[12] O. Christensen and W. Sun, βExplicitly given pairs of dualframes with compactly supported generators and applicationsto irregular B-splines,β Journal of ApproximationTheory, vol. 151,no. 2, pp. 155β163, 2008.
[13] O. Christensen, βPairs of dual Gabor frame generators withcompact support and desired frequency localization,β Appliedand Computational Harmonic Analysis, vol. 20, no. 3, pp. 403β410, 2006.
[14] O. Christensen, An Introduction to Frames and Riesz Bases,Applied and Numerical Harmonic Analysis, BirkhaΜuser,Boston, Mass, USA, 2003.
[15] O. Christensen, Frames and Bases an Introductory Course,BirkhaΜuser, Boston, Mass, USA, 2008.
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