#### Abstract

The main purpose of this paper is to give a new proof of the Balian-Low theorem for Gabor system , which is proposed by Fu et al. and associated with nonlinear Fourier atoms. To this end, we establish the relationships between spaces and . We also introduce the concept of frame associated with nonlinear Fourier atoms for and obtain many subsidiary results for this kind of (Gabor) frames.

#### 1. Introduction

Note that the classical Fourier atoms cannot expose the time-varying property of nonstationary signals [1]. Recently, a kind of specific nonlinear phase function is introduced in [2–6]. Note that, for different , the shapes of (also those of ) are different. It is observed that the closer gets to 1, the sharper the graph of is. This means that the nontrivial Harmonic waves can represent a conformal rescaling of classic Fourier atoms. Thus, the nontrivial Harmonic waves are expected to be better suitable and adaptable, along with different choices of , to capture nonstationary features of band-limited signals. In fact, Ren et al. [7] obtained some new phenomena on the Shannon sampling theorem by dealing with sampling points which are nonequally distributed.

Motivated by these points, Fu et al. [8] considered a newly Gabor system generated by a function , where satisfies certain assumptions. Note that they were not restricted to the conformal phase functions in their discussion. This freedom allows us to choose phase functions adequate to the necessary nonuniform sampling of the signal [7]. Using the Zak transform technique, they established the Balian-Low theorem for this newly Gabor system.

We point out that the Gabor system proposed by [8] can be related to already existing cases. A particular case of this kind of Gabor system is the nonlinear Fourier atoms which was discussed in [2–6]. Using the nonlinear Fourier atoms in [2–6], we have that the frequency modulation represents a conformal dilation of the classical modulation on the unit circle. Taking the proposed Gabor systems with different parameters , we can obtain a dictionary of Gabor frames with different dilation parameters in the modulation part. A simple change of variables can establish a clear relation between this system and the system generated by the affine Weyl-Heisenberg group with dilation on the window function [9, 10].

Basing on these points, we can say that establishing relationships between frames for and is an interesting issue. In this paper, our main purpose is to give a different proof for the Balian-Low theorem proposed in [8]. For this purpose, we firstly establish the relationships between spaces and . Basing on this relationship, we obtain many properties for general frame system and its special case , where is a nonlinear function. With these results for general Gabor system , we give a new and simple proof for the Balian-Low theorem proposed in [8].

The rest of the paper is organized as follows. Section 2 is devoted to giving some notations and lemmas. In Section 3, we establish the relationship between spaces and ; we also depict some properties of general frame for . In Section 4, we establish the relationship between Gabor frame for and classical one for under some assumptions on ; further, a new and simple proof is presented for the Balian-Low theorem which was proposed by Fu et al. [8].

#### 2. Notations

In this section, we present some notations and lemmas, which will be needed in the rest of the paper.

For an arbitrary measure in , consider the space of square integrable functions in with respect to and the finite norm: induced by the inner product To obtain the Balian-Low theorem for Gabor system , Fu et al. introduced some assumptions including the Assumptions 2.1 and 2.2 in [8] for a nonlinear phase function . Combining these two assumptions together, we obtain the following Assumption 1.

*Assumption 1. *Let function be a measure on and satisfy
for any and . Further, for all .

Note that for all ; one obtains that the inverse of (denote by ) exists. Moreover, it is obvious to check that satisfies for any and . In fact, we obtain from (3) that or Replacing and by and in (6), respectively, we obtain (4).

For a function defined in , denote by through the rest of paper.

For , consider the translation operator and the modulation operator , both acting on . In [8], Fu et al. proposed a general Gabor frame for . We say that the system is a general Gabor frame for if there exist two constants such that holds for all . To further study the general Gabor frame as defined in [8], we introduce a general frame concept as follows.

*Definition 2. *Let , . One says that the system is a general frame for if there exist two constants such that
holds for all ; moreover, one says that the frame is tight if ; in particular, the frame is Parseval if .

Given a frame for , a dual frame is a frame of which satisfies the reconstruction property and we say that the systems and constitute a pair of dual frames for , where the convergence is in the sense. Note that if for all , then the frame for constitutes a frame for .

For fixed and , we introduce the -bracket product as follows: If for , then is bracket product (denote by ) introduced by Ron and Shen in [11]. Thus, Note that is 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 , , and . Let the systems and be Bessel sequences in . Define
**
Then, the following holds:
**
where the convergence is in the sense. Moreover, and , are a pair of dual frames for if and only if
*

The following lemma follows from general properties of shift-invariant frames; see [11, Corollary ]. Alternatively, it can be proved similarly to [14, Theorem ].

Lemma 4. *Let , , , and
**
Then, is a Bessel sequence with upper frame bound . If also
**
then constitutes a frame with bounds and .*

#### 3. Frame for

In this section, we discuss frames for . Here, we will establish the relationship between frames for and . We will also give necessary conditions for frames and characterize a pair of dual frames in . Above all, we establish the relationship between and as follows.

Theorem 5. *Let be functions defined on . Then, ; in particular, , which means that if and only if .*

*Proof. *Denote . Then,
This means that
Thus, we can obtain the desired result.

Theorem 6. *Let be functions defined on . Then, the system constitutes a frame for if and only if the system constitutes a frame for and these two systems have the same bounds.*

*Proof. *From Theorem 5, one obtains that if and only if for . Note that
Then,
is equivalent to
Now, we can obtain the desired results.

Theorem 7. *Let be functions defined on . Then, the systems and constitute a pair of dual frames for if and only if the systems and constitute a pair of dual frames for .*

*Proof. *“if” part. If the systems and constitute a pair of dual frames for . Then, by Theorem 6, these two systems and are frames for . Moreover, we obtain from (20) that
for any , where the convergence is in the sense. Replacing by in the above equation, we obtain
for any , where the convergence is in the sense. Therefore, the systems and constitute a pair of dual frames for .

The proof of “only if” part is similar to the “if” part, and we omit it.

With the -bracket product proposed in the above section, we can prove the following theorem.

Theorem 8. *Let for . Let and be Bessel sequences in . Define
**
for . Then,
**
holds for , where the convergence is in the sense. Moreover, the systems and constitute a pair of dual frames for if and only if
*

*Proof. *For fixed , one obtains from (20) that
where the convergence is in the sense. Replacing by in the above equation, we obtain
Note that the systems and are Bessel sequences in . We can deduce that the systems and are Bessel sequences in . Therefore, one obtains (26) from (14).

From Theorem 7, we know that the systems and constitute a pair of dual frames for if and only if the systems and constitute a pair of dual frames for . Thus, by (15) in Lemma 3, one obtains the desired result.

Theorem 9. *Let , , and suppose that
**
Then, the system is a Bessel sequence with upper frame bound for . If also
**
then the system constitutes a frame for with bounds and .*

*Proof. *Since , , then , . If , then by Lemma 4, the system constitutes a frame for with frame bounds and . Therefore, by Theorem 6, one obtains that the system constitutes a frame for with the same frame bounds and .

#### 4. Gabor Frame for

In this section, Gabor frames for are discussed. We establish the relationship between the generalized Gabor frame for and the classical one for ; further, we prove the Balian-Low theorem for Gabor system proposed by Fu et al. in [8] from a different viewpoint.

Theorem 10. *Let . Then,
*

*Proof. *Since
then
Hence,
We complete the proof.

Theorem 11. *Let be a function defined on . Then, the general Gabor system constitutes a frame for if and only if the classical Gabor system constitutes a frame for with the same bounds.*

*Proof. *Define . From Theorem 6, one obtains that the general Gabor system constitutes a frame for if and only if the system constitutes a frame for with the same bounds. So, one obtains the desired result from Theorem 10.

Combining Theorems 8 and 10 together, we obtain the following Theorem 12.

Theorem 12. *Consider . Let the systems and be Bessel sequences in . Define
**
Then, for any ,
**
where the convergence is in the sense. Moreover, the systems and constitute a pair of dual frames for if and only if
*

*Proof. *Replacing and by and in (26) and (27), respectively, we have
Equations (37) and (38) follow from (39) and (40), respectively. Here, we used the facts that and .

By Theorems 9 and 10, we obtain Theorem 13.

Theorem 13. *Consider , , and suppose that
**
Then, the system is a Bessel sequence for with upper frame bound . If also
**
then the system constitutes a frame for with bounds and .*

*Proof. *Since and , then
Define
then and are 1-periodic functions. Thus,
By Theorem 9, one obtains the results.

Theorem 14. *Let . Assume that the system constitutes a generalized Gabor frame for with bounds and . Then,
*

*Proof. *If the system constitutes a generalized Gabor frame for with bounds and . Then, by Theorem 7, constitutes a frame for with the same bounds and . Note that . We can say that the system constitutes a Gabor frame for with the same bounds and . Thus, one obtains from [14, Proposition ] the desired result.

Theorem 15. *Let . Suppose that the system constitutes a general Gabor frame for . If the derivative of function is continuous on , then either
**
or
*

*Proof. *Since constitutes a general Gabor frame for , then, by Theorem 11, the system constitutes a classical Gabor frame for . Therefore, by the classical Balian-Low theorem, we have either
or
That is either
or
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 of generality, let
Since the derivative of function is continuous on and satisfies Assumption 1, then
By the Lagrange mean-valued theorem, there exists such that
Therefore, for any fixed , there exists a constant such that
Note that
Therefore,
That is,
or

In the proof of Theorem 15, the main technique is the inequality for some positive constant . Note that is equivalent to where is a -periodic function. We obtain the following Balian-Low theorem which weakens the conditions imposed on in Theorem 15.

Theorem 16. *Let . Suppose that the system constitutes a general Gabor frame for . Let be a -periodic function such that
**
where is a positive constant. Then, one and only one of the inequalities (47) and (48) holds.*

In applications of frames, it is inconvenient that the frame decomposition, as stated in [15, Theorem ], requires the inverse of a frame operator. As we have seen in the discussion of general frame theory, one way of avoiding the problem is to consider tight frames. Hence, we give characterization for tight Gabor frames in .

Theorem 17. *Let . Then, the system constitutes a tight frame for with if and only if
**
holds a.e. in .*

*Proof. *By Theorem 11, one obtains that the system constitutes a tight frame for with if and only if constitutes a tight frame for with . From [14, Theorem ], one obtains the desired result.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are grateful to the referees for their valuable suggestions that helped to improve the paper in its present form. This research is supported by the National Natural Science Foundation of China (Grant no. 11071152), the Natural Science Foundation of Guangdong Province (Grant nos. S2013010013101 and S2011010004511), and the Foundation of Hanshan Normal University (Grant nos. QD20131101 and LQ200905) This work was also partially supported by the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (Grant no. 201206012).