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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 © 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 {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡 − 𝑛), 𝑚, 𝑛 ∈ Z},which is proposed by Fu et al. and associated with nonlinear Fourier atoms. To this end, we establish the relationships betweenspaces 𝐿2(R, 𝑑𝜃) and 𝐿2(R). We also introduce the concept of frame associated with nonlinear Fourier atoms for 𝐿2(R, 𝑑𝜃) andobtain many subsidiary results for this kind of (Gabor) frames.

1. Introduction

Note that the classical Fourier atoms 𝑒2𝜋𝑖𝑛𝑡 cannot expose thetime-varying property of nonstationary signals [1]. Recently,a kind of specific nonlinear phase function 𝜃

𝑎(2𝜋𝑡) is intro-

duced in [2–6]. Note that, for different 𝑎, the shapes ofcos 𝜃𝑎(2𝜋𝑡) (also those of sin 𝜃

𝑎(2𝜋𝑡)) are different. It is

observed that the closer 𝑎 gets to 1, the sharper the graphof cos 𝜃

𝑎(2𝜋𝑡) is. This means that the nontrivial Harmonic

waves 𝑒𝑖𝑚𝜃𝑎(2𝜋𝑡) can represent a conformal rescaling of classicFourier atoms. Thus, the nontrivial Harmonic waves areexpected to be better suitable and adaptable, along withdifferent choices of 𝑎, 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 {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡 − 𝑛), 𝑚, 𝑛 ∈ Z} generated by afunction𝑔, where 𝜃(𝑡) satisfies certain assumptions. Note thattheywere not restricted to the conformal phase functions 𝜃(𝑡)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 {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡 − 𝑛),𝑚, 𝑛 ∈ Z} proposed by [8] can be related to already existing

cases. A particular case of this kind of Gabor system is thenonlinear Fourier atoms 𝑒𝑖𝑚𝜃𝑎(2𝜋𝑡) which was discussed in [2–6]. Using the nonlinear Fourier atoms in [2–6], we have thatthe frequency modulation 𝑒𝑖𝑚𝜃𝑎(2𝜋𝑡) represents a conformaldilation of the classical modulation 𝑒𝑖𝑚2𝜋𝑡 on the unit circle.Taking the proposedGabor systemswith different parameters𝑎, 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 𝐿2(R, 𝑑𝜃) and 𝐿2(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 𝐿2(R, 𝑑𝜃) and 𝐿2(R). Basing on this relation-ship, we obtain many properties for general frame system{𝑒𝑖𝑚𝜃(2𝜋𝑡)

𝑔𝑛(𝑡), 𝑚, 𝑛 ∈ Z} and its special case {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡 −

𝑛), 𝑚, 𝑛 ∈ Z}, where 𝜃 is a nonlinear function. With theseresults for general Gabor system {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡−𝑛), 𝑚, 𝑛 ∈ 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 𝐿2(R, 𝑑𝜃) and𝐿2(R); we also depict some properties of general frame

2 Journal of Function Spaces and Applications

{𝑒𝑖𝑚𝜃(2𝜋𝑡)

𝑔𝑛(𝑡), 𝑚, 𝑛 ∈ Z} for 𝐿2(R, 𝑑𝜃). In Section 4, we

establish the relationship betweenGabor frame {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡−𝑛), 𝑚, 𝑛 ∈ Z} for 𝐿2(R, 𝑑𝜃) and classical one {𝑒𝑖2𝑚𝜋𝑡𝑔𝜃(𝑡 −𝑛), 𝑚, 𝑛 ∈ Z} for 𝐿2(R) under some assumptions on 𝜃; 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 𝜃 in R, consider the space𝐿2(R, 𝑑𝜃) of square integrable functions in R with respect to

𝜃 and the finite norm:

𝑓𝜃 = (

1

𝜃 (2𝜋) − 𝜃 (0)∫∞

−∞

𝑓 (𝑥)2𝑑𝜃 (2𝜋𝑥))

1/2

(1)

induced by the inner product

⟨𝑓, 𝑔⟩𝜃:=

1

𝜃 (2𝜋) − 𝜃 (0)∫∞

−∞

𝑓 (𝑥) 𝑔 (𝑥)𝑑𝜃 (2𝜋𝑥) . (2)

To obtain the Balian-Low theorem for Gabor system{𝑒𝑖𝑚𝜃(2𝜋𝑡)

𝑔(𝑡 − 𝑛), 𝑚, 𝑛 ∈ Z}, Fu et al. introduced someassumptions including the Assumptions 2.1 and 2.2 in [8]for a nonlinear phase function 𝜃. Combining these twoassumptions together, we obtain the following Assumption 1.

Assumption 1. Let function 𝜃 : R → R be a measure on Rand satisfy

𝜃 (𝑥 + 2𝑘𝜋) = 𝜃 (𝑥) + 2𝑘𝜋, (3)

for any 𝑥 ∈ R and 𝑘 ∈ Z. Further, 𝜃(𝑥) > 0 for all 𝑥 ∈ R.

Note that 𝜃(𝑥) > 0 for all 𝑥 ∈ R; one obtains that theinverse of 𝜃 (denote by 𝜃−1) exists. Moreover, it is obvious tocheck that 𝜃 satisfies

𝜃−1(𝑥 + 2𝑘𝜋𝛾) = 𝜃

−1(𝑥) + 2𝑘𝜋, (4)

for any 𝑥 ∈ R and 𝑘 ∈ Z. In fact, we obtain from (3) that

𝜃−1(𝜃 (𝑥 + 2𝑘𝜋)) = 𝜃

−1(𝜃 (𝑥) + 2𝑘𝜋) , (5)

or

𝑥 + 2𝑘𝜋 = 𝜃−1(𝜃 (𝑥) + 2𝑘𝜋) . (6)

Replacing 𝜃(𝑥) and 𝑥 by 𝑡 and 𝜃−1(𝑡) in (6), respectively, weobtain (4).

For a function 𝑓 defined in R, denote by

𝑓𝜃(𝑡) := 𝑓 (

1

2𝜋𝜃−1(2𝜋𝑡)) (7)

through the rest of paper.

For 𝑎, 𝑏 ∈ R, consider the translation operator (𝑇𝑎𝑔)(𝑥) =

𝑔(𝑥 − 𝑎) and the modulation operator (𝐸𝜃𝑏𝑔)(𝑥) = 𝑒

𝑖𝑏𝜃(2𝜋𝑥)

𝑔(𝑥), both acting on 𝐿2(R, 𝑑𝜃). In [8], Fu et al. proposed ageneral Gabor frame for 𝐿2(R, 𝑑𝜃). We say that the system{𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} is a general Gabor frame for 𝐿2(R, 𝑑𝜃) if

there exist two constants 𝐴, 𝐵 > 0 such that

𝐴𝑓2

𝜃⩽ ∑𝑚, 𝑛∈Z

⟨𝑓, 𝐸𝜃

𝑚𝑏𝑇𝑛𝑎𝑔⟩𝜃

2

⩽ 𝐵𝑓2

𝜃 (8)

holds for all 𝑓 ∈ 𝐿2(R, 𝑑𝜃). To further study the generalGabor frame as defined in [8], we introduce a general frameconcept as follows.

Definition 2. Let 𝑔𝑛∈ 𝐿2(R, 𝑑𝜃), 𝑛 ∈ Z. One says that the

system {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} is a general frame for 𝐿2(R, 𝑑𝜃) if

there exist two constants 𝐴, 𝐵 > 0 such that

𝐴𝑓2


⟨𝑓, 𝐸𝜃

𝑚𝑔𝑛⟩𝜃

2

⩽ 𝐵𝑓2

𝜃 (9)

holds for all 𝑓 ∈ 𝐿2(R, 𝑑𝜃); moreover, one says that the frame{𝐸𝜃

𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} is tight if 𝐴 = 𝐵; in particular, the frame

{𝐸𝜃

𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} is Parseval if 𝐴 = 𝐵 = 1.

Given a frame {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} for 𝐿2(R, 𝑑𝜃), a dual

frame is a frame {𝐸𝜃𝑚ℎ𝑛, 𝑚, 𝑛 ∈ Z} of 𝐿2(R, 𝑑𝜃) which

satisfies the reconstruction property

𝑓 = ∑𝑛,𝑚∈Z

⟨𝑓, 𝐸𝜃

𝑚𝑔𝑛⟩𝜃𝐸𝜃

𝑚ℎ𝑛, ∀𝑓 ∈ 𝐿

2(R, 𝑑𝜃) , (10)

and we say that the systems {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚ℎ𝑛, 𝑚,

𝑛 ∈ Z} constitute a pair of dual frames for 𝐿2(R, 𝑑𝜃), wherethe convergence is in the 𝐿2 sense. Note that if 𝜃(2𝜋𝑥) = 2𝜋𝑥for all 𝑥 ∈ R, then the frame {𝐸𝜃

𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} for 𝐿2(R, 𝑑𝜃)

constitutes a frame for 𝐿2(R).For fixed 𝑓, 𝑔 ∈ 𝐿2(R, 𝑑𝜃) and 𝑏 > 0, we introduce the

𝜃-bracket product as follows:

[𝑓, 𝑔]𝜃

𝑏(𝑥) := ∑

𝑘∈Z

𝑓𝜃(𝑥 +

𝑘

𝑏)𝑔𝜃 (𝑥 +

𝑘

𝑏), a.e. 𝑥 ∈ R.

(11)

If 𝜃(2𝜋𝑥) = 2𝜋𝑥 for 𝑥 ∈ R, then [𝑓, 𝑔]𝜃𝑏is bracket product

(denote by [𝑓, 𝑔]𝑏) introduced by Ron and Shen in [11]. Thus,

[𝑓, 𝑔]𝜃

𝑏= [𝑓𝜃, 𝑔𝜃]𝑏. (12)

Note that [𝑓, 𝑔]𝜃𝑏is 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 𝑔𝑛, ℎ𝑛∈ 𝐿2(R), 𝑛 ∈ Z, and 𝑏 > 0. Let the

systems {𝐸𝑚𝑏𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸

𝑚𝑏ℎ𝑛, 𝑚, 𝑛 ∈ Z} be Bessel

sequences in 𝐿2(R). Define

𝑆𝑓 = ∑𝑚, 𝑛∈Z

⟨𝑓, 𝐸𝑚𝑏𝑔𝑛⟩ 𝐸𝑚𝑏ℎ𝑛, ∀𝑓 ∈ 𝐿

2(R) . (13)

Journal of Function Spaces and Applications 3

Then, the following holds:

(𝑆𝑓) (𝑥) =1

𝑏∑𝑛∈Z

[𝑓, 𝑔𝑛]𝑏(𝑥) ℎ𝑛 (𝑥)

=1

𝑏∑𝑛∈Z

∑𝑘∈Z

𝑓(𝑥 +𝑘

𝑏)𝑔𝑛(𝑥 +

𝑘

𝑏)ℎ𝑛 (𝑥) ,

∀𝑓 ∈ 𝐿2(R) ,

(14)

where the convergence is in the 𝐿2 sense. Moreover, {𝐸𝑚𝑏𝑔𝑛, 𝑚,

𝑛 ∈ Z} and {𝐸𝑚𝑏ℎ𝑛, 𝑚, 𝑛 ∈ Z} are a pair of dual frames for

𝐿2(R) if and only if

∑𝑘∈Z

𝑔𝑛(𝑥 +

𝑘

𝑏) ℎn (𝑥 +

𝑘

𝑏) = 𝑏𝛿

𝑛,0, a.e. 𝑥 ∈ 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 𝑔𝑛∈ 𝐿2(R), 𝑛 ∈ Z, 𝑏 > 0, and

𝐵 :=1

𝑏sup𝑥∈R

∑𝑘∈Z

∑𝑛∈Z

𝑔𝑛 (𝑥) 𝑔𝑛 (𝑥 −

𝑘

𝑏)

< ∞. (16)

Then, {𝐸𝑚𝑏𝑔𝑛, 𝑚, 𝑛 ∈ Z} is a Bessel sequence with upper frame

bound 𝐵. If also

𝐴 :=1

𝑏inf𝑥∈R

(∑𝑛∈Z

𝑔𝑛 (𝑥)2− ∑𝑘 ̸= 0

∑𝑛∈Z

𝑔𝑛 (𝑥) 𝑔𝑛 (𝑥 −

𝑘

𝑏)

) > 0,

(17)

then {𝐸𝑚𝑏𝑔𝑛, 𝑚, 𝑛 ∈ Z} constitutes a frame𝐿2(R)with bounds

𝐴 and 𝐵.

3. Frame for 𝐿2(R, 𝑑𝜃)

In this section, we discuss frames for 𝐿2(R, 𝑑𝜃). Here, we willestablish the relationship between frames for 𝐿2(R, 𝑑𝜃) and𝐿2(R). We will also give necessary conditions for frames and

characterize a pair of dual frames in 𝐿2(R, 𝑑𝜃). Above all,we establish the relationship between 𝐿2(R, 𝑑𝜃) and 𝐿2(R) asfollows.

Theorem 5. Let 𝑓, 𝑔 be functions defined on R. Then,⟨𝑓, 𝑔⟩

𝜃= ⟨𝑓𝜃, 𝑔𝜃⟩; in particular, ‖𝑓‖

𝜃= ‖𝑓

𝜃‖, which means

that 𝑓 ∈ 𝐿2(R, 𝑑𝜃) if and only if 𝑓𝜃 ∈ 𝐿2(R).

Proof. Denote 𝑡 := (1/2𝜋)𝜃(2𝜋𝑥). Then,

∫∞

−∞

𝑓 (𝑥) 𝑔 (𝑥)𝑑𝜃 (2𝜋𝑥) = ∫∞

−∞

𝑓(1

2𝜋𝜃−1(2𝜋𝑡))

× 𝑔 (1

2𝜋𝜃−1(2𝜋𝑡))𝑑 (2𝜋𝑡)

= 2𝜋∫∞

−∞

𝑓𝜃(𝑡) 𝑔𝜃(𝑡)𝑑𝑡.

(18)

This means that

⟨𝑓, 𝑔⟩𝜃= ⟨𝑓𝜃, 𝑔𝜃⟩ . (19)

Thus, we can obtain the desired result.

Theorem 6. Let 𝑔𝑛, 𝑛 ∈ Z be functions defined on R. Then,

the system {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} constitutes a frame for 𝐿2(R, 𝑑𝜃)

if and only if the system {𝐸𝑚𝑔𝜃

𝑛, 𝑚, 𝑛 ∈ Z} constitutes a frame

for 𝐿2(R) and these two systems have the same bounds.

Proof. From Theorem 5, one obtains that 𝑔𝑛∈ 𝐿2(R, 𝑑𝜃) if

and only if 𝑔𝜃𝑛∈ 𝐿2(R) for 𝑛 ∈ Z. Note that

⟨𝑓, 𝐸𝜃

𝑚𝑔𝑛⟩𝜃=1

2𝜋∫∞

−∞

𝑓 (𝑥) 𝑒𝑖𝑚𝜃(2𝜋𝑥)𝑔𝑛(𝑥)𝑑𝜃 (2𝜋𝑥)

=1

2𝜋∫∞

−∞

𝑓(1

2𝜋𝜃−1(2𝜋𝑥))

× 𝑒𝑖2𝜋𝑚𝑥𝑔𝑛(1

2𝜋𝜃−1 (2𝜋𝑥))𝑑 (2𝜋𝑥)

= ∫∞

−∞

𝑓𝜃(𝑥) 𝑒𝑖2𝜋𝑚𝑥𝑔𝜃

𝑛(𝑥)𝑑𝑥 = ⟨𝑓

𝜃, 𝐸𝑚𝑔𝜃

𝑛⟩ .

(20)

Then,

𝐴𝑓2


⟨𝑓, 𝐸𝜃

𝑚𝑔𝑛⟩𝜃

2

⩽ 𝐵𝑓2

𝜃, ∀𝑓 ∈ 𝐿

2(R, 𝑑𝜃)

(21)

is equivalent to

𝐴𝑓𝜃

2

⩽ ∑𝑚, 𝑛∈Z

⟨𝑓𝜃, 𝐸𝑚𝑔𝜃

𝑛⟩

2

⩽ 𝐵𝑓𝜃

2

, ∀𝑓𝜃∈ 𝐿2(R) .

(22)

Now, we can obtain the desired results.

Theorem7. Let 𝑔𝑛, 𝑛 ∈ Z be functions defined onR.Then, the

systems {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚ℎ𝑛, 𝑚, 𝑛 ∈ Z} constitute

a pair of dual frames for 𝐿2(R, 𝑑𝜃) if and only if the systems{𝐸𝑚𝑔𝜃

𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸

𝑚ℎ𝜃

𝑛, 𝑚, 𝑛 ∈ Z} constitute a pair of

dual frames for 𝐿2(R).

Proof. “if ” part. If the systems {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚ℎ𝑛,

𝑚, 𝑛 ∈ Z} constitute a pair of dual frames for 𝐿2(R, 𝑑𝜃).Then, by Theorem 6, these two systems {𝐸𝜃

𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z}

and {𝐸𝜃𝑚ℎ𝑛, 𝑚, 𝑛 ∈ Z} are frames for 𝐿2(R, 𝑑𝜃). Moreover,

we obtain from (20) that

𝑓 (𝑥) = ∑𝑛,𝑚∈Z


𝑛⟩𝐸𝜃

𝑚ℎ𝑛 (𝑥) , 𝑥 ∈ R, (23)

for any 𝑓 ∈ 𝐿2(R, 𝑑𝜃), where the convergence is in the 𝐿2sense. Replacing 𝑥 by (1/2𝜋)𝜃−1(2𝜋𝑥) in the above equation,we obtain

𝑓𝜃(𝑥) = ∑

𝑛,𝑚∈Z


𝑛⟩𝐸𝑚ℎ𝜃

𝑛(𝑥) , 𝑥 ∈ R, (24)


for any 𝑓𝜃 ∈ 𝐿2(R), where the convergence is in the 𝐿2 sense.Therefore, the systems {𝐸

𝑚𝑔𝜃


𝑚ℎ𝜃

𝑛, 𝑚, 𝑛 ∈

Z} constitute a pair of dual frames for 𝐿2(R).The proof of “only if ” part is similar to the “if ” part, and

we omit it.

With the 𝜃-bracket product proposed in the above sec-tion, we can prove the following theorem.

Theorem 8. Let 𝑔𝑛, ℎ𝑛∈ 𝐿2(R, 𝑑𝜃) for 𝑛 ∈ Z. Let {𝐸𝜃

𝑚𝑔𝑛, 𝑚,

𝑛 ∈ Z} and {𝐸𝜃𝑚ℎ𝑛, 𝑚, 𝑛 ∈ Z} be Bessel sequences in𝐿2(R, 𝑑𝜃).

Define


⟨𝑓, 𝐸𝜃


𝑚ℎ𝑛, (25)

for 𝑓 ∈ 𝐿2(R, 𝑑𝜃). Then,

(𝑆𝑓𝜃) (𝑥) = ∑

𝑛∈Z

[𝑓, 𝑔𝑛]𝜃(𝑥) ℎ𝜃

𝑛(𝑥)

× ∑𝑛∈Z

∑𝑘∈Z

𝑓𝜃(𝑥 + 𝑘) 𝑔𝜃

𝑛(𝑥 + 𝑘)ℎ

𝜃

𝑛(𝑥)

(26)

holds for 𝑓 ∈ 𝐿2(R, 𝑑𝜃), where the convergence is in the 𝐿2

sense. Moreover, the systems {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚ℎ𝑛, 𝑚,

𝑛 ∈ Z} constitute a pair of dual frames for 𝐿2(R, 𝑑𝜃) if and onlyif

∑𝑘∈Z

𝑔𝜃

𝑛(𝑥 + 𝑘) ℎ𝜃

𝑛(𝑥 + 𝑘) = 𝛿𝑛,0, a.e. 𝑥 ∈ R. (27)

Proof. For fixed 𝑓 ∈ 𝐿2(R, 𝑑𝜃), one obtains from (20) that

𝑆𝑓 (𝑥) = ∑𝑚, 𝑛∈Z

⟨𝑓, 𝐸𝜃


𝑚ℎ𝑛 (𝑥)

= ∑𝑚, 𝑛∈Z


𝑛⟩𝐸𝜃

𝑚ℎ𝑛 (𝑥) ,

(28)

where the convergence is in the 𝐿2 sense. Replacing 𝑥 by(1/2𝜋)𝜃

−1(2𝜋𝑥) in the above equation, we obtain

𝑆𝑓𝜃(𝑥) = ∑

𝑚, 𝑛∈Z


𝑛⟩𝐸𝑚ℎ𝜃

𝑛(𝑥) , ∀𝑓

𝜃∈ 𝐿2(R) .

(29)

Note that the systems {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚ℎ𝑛, 𝑚, 𝑛 ∈

Z} are Bessel sequences in 𝐿2(R, 𝑑𝜃). We can deduce that thesystems {𝐸

𝑚𝑔𝜃


𝑚ℎ𝜃

𝑛, 𝑚, 𝑛 ∈ Z} are Bessel

sequences in 𝐿2(R). Therefore, one obtains (26) from (14).From Theorem 7, we know that the systems

{𝐸𝜃

𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚ℎ𝑛, 𝑚, 𝑛 ∈ Z} constitute a

pair of dual frames for 𝐿2(R, 𝑑𝜃) if and only if the systems{𝐸𝑚𝑔𝜃


𝑚ℎ𝜃

𝑛, 𝑚, 𝑛 ∈ Z} constitute a pair of

dual frames for 𝐿2(R). Thus, by (15) in Lemma 3, one obtainsthe desired result.

Theorem 9. Let 𝑔𝑛∈ 𝐿2(R, 𝑑𝜃), 𝑛 ∈ Z, and suppose that

𝐵 := sup𝑥∈R

∑𝑘∈Z

∑𝑛∈Z

𝑔𝜃

𝑛(𝑥) 𝑔𝜃𝑛(𝑥 + 𝑘)

< ∞. (30)

Then, the system {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} is a Bessel sequence with

upper frame bound 𝐵 for 𝐿2(R, 𝑑𝜃). If also

𝐴 := inf𝑥∈R

(∑𝑛∈Z

𝑔𝜃

𝑛(𝑥)

2

− ∑𝑘 ̸= 0

∑𝑛∈Z

𝑔𝜃

𝑛(𝑥) 𝑔𝜃𝑛(𝑥 + 𝑘)

) > 0,

(31)

then the system {𝐸𝜃𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} constitutes a frame for𝐿2(R,

𝑑𝜃) with bounds 𝐴 and 𝐵.

Proof. Since 𝑔𝑛∈ 𝐿2(R, 𝑑𝜃), 𝑛 ∈ Z, then 𝑔𝜃

𝑛∈ 𝐿2(R), 𝑛 ∈

Z. If 0 < 𝐴, 𝐵 < ∞, then by Lemma 4, the system {𝐸𝑚𝑔𝜃

𝑛,

𝑚, 𝑛 ∈ Z} constitutes a frame for 𝐿2(R)with frame bounds𝐴and 𝐵. Therefore, by Theorem 6, one obtains that the system{𝐸𝜃

𝑚𝑔𝑛, 𝑚, 𝑛 ∈ Z} constitutes a frame for 𝐿2(R, 𝑑𝜃) with the

same frame bounds 𝐴 and 𝐵.

4. Gabor Frame for 𝐿2(R, 𝑑𝜃)

In this section, Gabor frames for 𝐿2(R, 𝑑𝜃) are discussed.We establish the relationship between the generalized Gaborframe {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡 − 𝑛), 𝑚, 𝑛 ∈ Z} for 𝐿2(R, 𝑑𝜃) and theclassical one {𝑒𝑖2𝑚𝜋𝑡𝑔𝜃(𝑡 − 𝑛), 𝑚, 𝑛 ∈ Z} for 𝐿2(R); further,we prove the Balian-Low theorem for Gabor system {𝑒𝑖𝑚𝜃(2𝜋𝑡)𝑔(𝑡−𝑛), 𝑚, 𝑛 ∈ Z} proposed by Fu et al. in [8] from a differentviewpoint.

Theorem 10. Let 𝑔 ∈ 𝐿2(R, 𝑑𝜃). Then,

(𝑇𝑛𝑔)𝜃(𝑥) = 𝑇𝑛𝑔

𝜃(𝑥) . (32)

Proof. Since

𝜃−1(𝑥 + 2𝑘𝜋) = 𝜃

−1(𝑥) + 2𝑘𝜋, ∀𝑥 ∈ R, 𝑘 ∈ Z, (33)

then

𝜃−1(2𝜋 (𝑥 + 𝑘)) = 𝜃

−1(2𝜋𝑥 + 2𝑘𝜋)

= 𝜃−1(2𝜋𝑥) + 2𝑘𝜋, ∀𝑥 ∈ R, 𝑘 ∈ Z.

(34)

Hence,

(𝑇𝑛𝑔)𝜃(𝑥) = 𝑔 (

1

2𝜋𝜃−1(2𝜋𝑥) − 𝑛)

= 𝑔(1

2𝜋𝜃−1(2𝜋 (𝑥 − 𝑛))) = 𝑇𝑛𝑔

𝜃(𝑥) .

(35)

We complete the proof.

Theorem 11. Let 𝑔 be a function defined on R. Then, thegeneral Gabor system {𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a frame

for 𝐿2(R, 𝑑𝜃) if and only if the classical Gabor system{𝐸𝑚𝑇𝑛𝑔𝜃, 𝑚, 𝑛 ∈ Z} constitutes a frame for 𝐿2(R) with the

same bounds.


Proof. Define 𝑔𝑛(𝑥) := 𝑇

𝑛𝑔(𝑥), 𝑛 ∈ Z. FromTheorem 6, one

obtains that the general Gabor system {𝐸𝜃𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z}

constitutes a frame for 𝐿2(R, 𝑑𝜃) if and only if the system{𝐸𝑚(𝑇𝑛𝑔)𝜃, 𝑚, 𝑛 ∈ Z} constitutes a frame for 𝐿2(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 𝑔, ℎ ∈ 𝐿2(R, 𝑑𝜃). Let the systems{𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚𝑇𝑛ℎ, 𝑚, 𝑛 ∈ Z} be Bessel

sequences in 𝐿2(R, 𝑑𝜃). Define


⟨𝑓, 𝐸𝜃

𝑚𝑇𝑛𝑔⟩𝐸𝜃

𝑚𝑇𝑛ℎ, ∀𝑓 ∈ 𝐿

2(R, 𝑑𝜃) . (36)

Then, for any 𝑓 ∈ 𝐿2(R, 𝑑𝜃),

(𝑆𝑓𝜃) (𝑥) = ∑

𝑛∈Z

[𝑓𝜃, 𝑇𝑛𝑔𝜃] (𝑥) 𝑇𝑛ℎ

𝜃(𝑥)

= ∑𝑛∈Z

∑𝑘∈Z

𝑓𝜃(𝑥 + 𝑘) 𝑇𝑛𝑔

𝜃 (𝑥 + 𝑘) 𝑇𝑛ℎ𝜃(𝑥) ,

(37)

where the convergence is in the 𝐿2 sense. Moreover, the systems{𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} and {𝐸𝜃

𝑚𝑇𝑛ℎ, 𝑚, 𝑛 ∈ Z} constitute a pair

of dual frames for 𝐿2(R, 𝑑𝜃) if and only if

∑𝑘∈Z

𝑇𝑛𝑔𝜃(𝑥 + 𝑘) 𝑇𝑛ℎ

𝜃 (𝑥 + 𝑘) = 𝛿𝑛,0, a.e. 𝑥 ∈ R. (38)

Proof. Replacing 𝑔𝑛and ℎ

𝑛by 𝑇𝑛𝑔 and 𝑇

𝑛ℎ in (26) and (27),

respectively, we have

(𝑆𝑓𝜃) (𝑥) = ∑

𝑛∈Z

[𝑓, 𝑇𝑛𝑔]𝜃(𝑥) (𝑇𝑛ℎ)

𝜃(𝑥)

= ∑𝑛∈Z

∑𝑘∈Z

𝑓𝜃(𝑥 + 𝑘) (𝑇𝑛𝑔)

𝜃(𝑥 + 𝑘)(𝑇𝑛ℎ)

𝜃(𝑥) ,

(39)

∑𝑘∈Z

(𝑇𝑛𝑔)𝜃(𝑥 + 𝑘) (𝑇𝑛ℎ)

𝜃(𝑥 + 𝑘) = 𝛿𝑛, 0, a.e. 𝑥 ∈ R.

(40)

Equations (37) and (38) follow from (39) and (40), respec-tively. Here, we used the facts that (𝑇

𝑛𝑔)𝜃(𝑥) = 𝑇

𝑛𝑔𝜃(𝑥) and

(𝑇𝑛ℎ)𝜃(𝑥) = 𝑇

𝑛ℎ𝜃(𝑥).

By Theorems 9 and 10, we obtainTheorem 13.

Theorem 13. Consider 𝑔 ∈ 𝐿2(R, 𝑑𝜃), 𝑛 ∈ Z, and supposethat

𝐵 := sup𝑥∈[0,1]

∑𝑘∈Z

∑𝑛∈Z

𝑇𝑛𝑔𝜃(𝑥) 𝑇𝑛𝑔

𝜃 (𝑥 + 𝑘)

< ∞. (41)

Then, the system {𝐸𝜃𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} is a Bessel sequence for

𝐿2(R, 𝑑𝜃) with upper frame bound 𝐵. If also

𝐴 := inf𝑥∈[0,1]

(∑𝑛∈Z

𝑇𝑛𝑔𝜃(𝑥)

2

− ∑𝑘 ̸= 0

∑𝑛∈Z

𝑇𝑛𝑔𝜃(𝑥) 𝑇𝑛𝑔

𝜃 (𝑥 + 𝑘)

) > 0,

(42)

then the system {𝐸𝜃𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a frame for

𝐿2(R, 𝑑𝜃) with bounds 𝐴 and 𝐵.

Proof. Since 𝑇𝑛𝑔𝜃(𝑥) = (𝑇

𝑛𝑔)𝜃(𝑥) and 𝑇

𝑛ℎ𝜃(𝑥) = (𝑇

𝑛ℎ)𝜃(𝑥),

then

𝐵 = sup𝑥∈[0,1]

∑𝑘∈Z

∑𝑛∈Z

(𝑇𝑛𝑔)𝜃(𝑥) (𝑇𝑛𝑔)

𝜃(𝑥 + 𝑘)

,

𝐴 = inf𝑥∈[0,1]

(∑𝑛∈Z

(𝑇𝑛𝑔)𝜃(𝑥)

2

−∑𝑘 ̸= 0

∑𝑛∈Z


𝜃(𝑥 + 𝑘)

) .

(43)

Define

𝐻1 (𝑥) := ∑

𝑘∈Z

∑𝑛∈Z


𝜃(𝑥 + 𝑘)

,

𝐻2 (𝑥) := ∑

𝑛∈Z


2

− ∑𝑘 ̸= 0

∑𝑛∈Z


𝜃(𝑥 + 𝑘)

,

(44)

then𝐻1and𝐻

2are 1-periodic functions. Thus,

𝐵 = sup𝑥∈R

∑𝑘∈Z

∑𝑛∈Z


𝜃(𝑥 + 𝑘)

,

𝐴 = inf𝑥∈R

(∑𝑛∈Z


2

− ∑𝑘 ̸= 0

∑𝑛∈Z


𝜃(𝑥 + 𝑘)

) .

(45)

ByTheorem 9, one obtains the results.

Theorem 14. Let 𝑔 ∈ 𝐿2(R, 𝑑𝜃). Assume that the system{𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a generalized Gabor frame for

𝐿2(R, 𝑑𝜃) with bounds 𝐴 and 𝐵. Then,

𝐴 ⩽ ∑𝑛∈Z

𝑔𝜃(𝑥 − 𝑛)

2

⩽ 𝐵, a.e. 𝑥 ∈ R. (46)


Proof. If the system {𝐸𝜃𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a

generalized Gabor frame for 𝐿2(R, 𝑑𝜃) with bounds 𝐴 and𝐵. Then, by Theorem 7, {𝐸

𝑚(𝑇𝑛𝑔)𝜃, 𝑚, 𝑛 ∈ Z} constitutes a

frame for 𝐿2(R) with the same bounds 𝐴 and 𝐵. Note that(𝑇𝑛𝑔)𝜃= 𝑇𝑛𝑔𝜃. We can say that the system {𝐸

𝑚𝑇𝑛𝑔𝜃, 𝑚, 𝑛 ∈

Z} constitutes a Gabor frame for 𝐿2(R)with the same bounds𝐴 and 𝐵. Thus, one obtains from [14, Proposition 8.3.2] thedesired result.

Theorem 15. Let 𝑔 ∈ 𝐿2(R, 𝑑𝜃). Suppose that the system{𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a general Gabor frame for 𝐿2

(R, 𝑑𝜃). If the derivative 𝜃 of function 𝜃 is continuous on R,then either

∫∞

−∞

𝑥2𝑔 (𝑥)

2𝑑𝜃 (2𝜋𝑥) = ∞ (47)

or

∫∞

−∞

𝜉2𝑔 (𝜉)

2𝑑𝜃 (2𝜋𝜉) = ∞. (48)

Proof. Since {𝐸𝜃𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a general

Gabor frame for 𝐿2(R, 𝑑𝜃), then, by Theorem 11, the system{𝐸𝑚𝑇𝑛𝑔𝜃, 𝑚, 𝑛 ∈ Z} constitutes a classical Gabor frame for

𝐿2(R). Therefore, by the classical Balian-Low theorem, we

have either

∫∞

−∞

𝑥2𝑔𝜃(𝑥)

2

𝑑𝑥 = ∞ (49)

or

∫∞

−∞

𝜉2𝑔𝜃 (𝜉)

2

𝑑𝜉 = ∞. (50)

That is either

∫∞

−∞

𝜃2(2𝜋𝑥)

𝑔 (𝑥)2𝑑𝜃 (2𝜋𝑥) = ∞ (51)

or

∫∞

−∞

𝜃2(2𝜋𝜉)

𝑔 (𝜉)2𝑑𝜃 (2𝜋𝜉) = ∞. (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

∫∞

0

𝜃2(2𝜋𝑥)

𝑔 (𝑥)2𝑑𝜃 (2𝜋𝑥) = ∞. (53)

Since the derivative 𝜃 of function 𝜃 is continuous on R and𝜃 satisfies Assumption 1, then

0 < min𝑥∈[0,2𝜋]

𝜃(𝑥) ≤ 𝜃

(𝑡) ≤ max𝑥∈[0,2𝜋]

𝜃(𝑥) , 𝑡 ∈ R. (54)

By the Lagrange mean-valued theorem, there exists 𝜁 ∈ [0, 𝑥]such that

𝜃 (𝑥) = 𝜃 (0) + 𝜃(𝜁) 𝑥. (55)

Therefore, for any fixed 𝑥0> 0, there exists a constant 𝐶 > 0

such that

|𝜃 (𝑥)| ≤ 𝐶𝑥, ∀𝑥 ≥ 𝑥0. (56)

Note that

∫𝑥0

0

𝜃2(2𝜋𝑥)

𝑔 (𝑥)2𝑑𝜃 (2𝜋𝑥) < ∞. (57)

Therefore,

∫∞

𝑥0

𝐶2𝑥2𝑔 (𝑥)

2𝑑𝜃 (2𝜋𝑥)

≥ ∫∞

𝑥0

𝜃2(2𝜋𝑥)

𝑔 (𝑥)2𝑑𝜃 (2𝜋𝑥) = ∞.

(58)

That is,

∫∞

𝑥0

𝑥2𝑔 (𝑥)

2𝑑𝜃 (2𝜋𝑥) = ∞, (59)

or

∫∞

−∞

𝑥2𝑔 (𝑥)

2𝑑𝜃 (2𝜋𝑥) = ∞. (60)

In the proof of Theorem 15, the main technique is theinequality

|𝜃 (𝑥)| ≤ 𝐶𝑥, ∀𝑥 ≥ 𝑥0, (61)

for some positive constant 𝐶. Note that

𝜃 (𝑥 + 2𝑘𝜋) = 𝜃 (𝑥) + 2𝑘𝜋 (62)

is equivalent to

𝜃 (𝑥) = 𝑥 + 𝛽 (𝑥) , (63)

where 𝛽 is a 2𝜋-periodic function. We obtain the followingBalian-Low theorem which weakens the conditions imposedon 𝜃 in Theorem 15.

Theorem 16. Let 𝑔 ∈ 𝐿2(R, 𝑑𝜃). Suppose that the system{𝐸𝜃

𝑚𝑇𝑛𝑔, 𝑚, 𝑛 ∈ Z} constitutes a general Gabor frame for

𝐿2(R, 𝑑𝜃). Let 𝛽 be a 2𝜋-periodic function such that

𝜃 (𝑥) = 𝑥 + 𝛽 (𝑥) , ∀𝑥 ∈ R,

𝛽 (𝑥) ≤ 𝐶𝑥, ∀𝑥 ≥ 𝑥0,

(64)

where 𝐶 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 𝐿2(R, 𝑑𝜃).


Theorem 17. Let 𝑔 ∈ 𝐿2(R, 𝑑𝜃). Then, the system {𝐸𝜃𝑚𝑇𝑛𝑔,

𝑚, 𝑛 ∈ Z} constitutes a tight frame for 𝐿2(R, 𝑑𝜃) with 𝐴 = 1 ifand only if

∑𝑛∈Z

𝑔𝜃(𝑥 − 𝑛)

2

= 1,

∑𝑛∈Z

𝑔𝜃(𝑥 − 𝑛) 𝑔𝜃 (𝑥 − 𝑛 − 𝑘) = 0, for 𝑘 ̸= 0

(65)

holds a.e. in R.

Proof. By Theorem 11, one obtains that the system {𝐸𝜃𝑚𝑇𝑛𝑔,

𝑚, 𝑛 ∈ Z} constitutes a tight frame for 𝐿2(R, 𝑑𝜃) with 𝐴 = 1if and only if {𝐸

𝑚𝑇𝑛𝑔𝜃, 𝑚, 𝑛 ∈ Z} constitutes a tight frame

for 𝐿2(R) with 𝐴 = 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).

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