JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 530172 10.1155/2013/530172 530172 Research Article New Proof for Balian-Low Theorem of Nonlinear Gabor System Yuan D. H. 1 Yang S. Z. 2 Zheng X. W. 2 Shen Y. F. 2 Rao T. S. S. R. K. 1 Department of Mathematics Hanshan Normal University Chaozhou Guangdong 521041 China 2 Department of Mathematics Shantou University Shantou Guangdong 515063 China stu.edu.cn 2013 6 11 2013 2013 27 07 2013 24 09 2013 25 09 2013 2013 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(tn),  m,n}, which is proposed by Fu et al. and associated with nonlinear Fourier atoms. To this end, we establish the relationships between spaces L2(,dθ) and L2(). We also introduce the concept of frame associated with nonlinear Fourier atoms for L2(,dθ) and obtain many subsidiary results for this kind of (Gabor) frames.

1. Introduction

Note that the classical Fourier atoms e2πint cannot expose the time-varying property of nonstationary signals . Recently, a kind of specific nonlinear phase function θa(2πt) is introduced in . Note that, for different a, the shapes of cosθa(2πt) (also those of sinθa(2πt)) are different. It is observed that the closer a gets to 1, the sharper the graph of cosθa(2πt) is. This means that the nontrivial Harmonic waves eimθa(2πt) 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 a, to capture nonstationary features of band-limited signals. In fact, Ren et al.  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.  considered a newly Gabor system {eimθ(2πt)g(t-n),m,n} generated by a function g, where θ(t) satisfies certain assumptions. Note that they were not restricted to the conformal phase functions θ(t) in their discussion. This freedom allows us to choose phase functions adequate to the necessary nonuniform sampling of the signal . Using the Zak transform technique, they established the Balian-Low theorem for this newly Gabor system.

We point out that the Gabor system {eimθ(2πt)g(t-n),m,n} proposed by  can be related to already existing cases. A particular case of this kind of Gabor system is the nonlinear Fourier atoms eimθa(2πt) which was discussed in . Using the nonlinear Fourier atoms in , we have that the frequency modulation eimθa(2πt) represents a conformal dilation of the classical modulation eim2πt on the unit circle. Taking the proposed Gabor systems with different parameters a, 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 L2(,dθ) and L2() is an interesting issue. In this paper, our main purpose is to give a different proof for the Balian-Low theorem proposed in . For this purpose, we firstly establish the relationships between spaces L2(,dθ) and L2(). Basing on this relationship, we obtain many properties for general frame system {eimθ(2πt)gn(t),m,n} and its special case {eimθ(2πt)g(t-n),m,n}, where θ is a nonlinear function. With these results for general Gabor system {eimθ(2πt)g(t-n),m,n}, we give a new and simple proof for the Balian-Low theorem proposed in .

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 L2(,dθ) and L2(); we also depict some properties of general frame {eimθ(2πt)gn(t),m,n} for L2(,dθ). In Section 4, we establish the relationship between Gabor frame {eimθ(2πt)g(t-n),m,n} for L2(,dθ) and classical one {ei2mπtgθ(t-n),m,n} for L2() under some assumptions on θ; further, a new and simple proof is presented for the Balian-Low theorem which was proposed by Fu et al. .

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 θ in , consider the space L2(,dθ) of square integrable functions in with respect to θ and the finite norm: (1)fθ=(1θ(2π)-θ(0)-|f(x)|2dθ(2πx))1/2 induced by the inner product (2)f,gθ:=1θ(2π)-θ(0)-f(x)g(x)¯dθ(2πx). To obtain the Balian-Low theorem for Gabor system {eimθ(2πt)g(t-n),m,n}, Fu et al. introduced some assumptions including the Assumptions 2.1 and 2.2 in  for a nonlinear phase function θ. Combining these two assumptions together, we obtain the following Assumption 1.

Assumption 1.

Let function θ: be a measure on and satisfy (3)θ(x+2kπ)=θ(x)+2kπ, for any x and k. Further, θ(x)>0 for all x.

Note that θ(x)>0 for all x; one obtains that the inverse of θ (denote by θ-1) exists. Moreover, it is obvious to check that θ satisfies (4)θ-1(x+2kπγ)=θ-1(x)+2kπ, for any x and k. In fact, we obtain from (3) that (5)θ-1(θ(x+2kπ))=θ-1(θ(x)+2kπ), or (6)x+2kπ=θ-1(θ(x)+2kπ). Replacing θ(x) and x by t and θ-1(t) in (6), respectively, we obtain (4).

For a function f defined in , denote by (7)fθ(t):=f(12πθ-1(2πt)) through the rest of paper.

For a,b, consider the translation operator (Tag)(x)=g(x-a) and the modulation operator (Ebθg)(x)=eibθ(2πx)g(x), both acting on L2(,dθ). In , Fu et al. proposed a general Gabor frame for L2(,dθ). We say that the system {EmθTng,  m,n} is a general Gabor frame for L2(,dθ) if there exist two constants A,B>0 such that (8)Afθ2m,n|f,EmbθTnagθ|2Bfθ2 holds for all fL2(,dθ). To further study the general Gabor frame as defined in , we introduce a general frame concept as follows.

Definition 2.

Let gnL2(,dθ), n. One says that the system {Emθgn,m,n} is a general frame for L2(,dθ) if there exist two constants A,B>0 such that (9)Afθ2m,n|f,Emθgnθ|2Bfθ2 holds for all fL2(,dθ); moreover, one says that the frame {Emθgn,m,n} is tight if A=B; in particular, the frame {Emθgn,m,n} is Parseval if A=B=1.

Given a frame {Emθgn,m,n} for L2(,dθ), a dual frame is a frame {Emθhn,m,n} of L2(,dθ) which satisfies the reconstruction property (10)f=n,mf,EmθgnθEmθhn,fL2(,dθ), and we say that the systems {Emθgn,m,n} and {Emθhn,m,n} constitute a pair of dual frames for L2(,dθ), where the convergence is in the L2 sense. Note that if θ(2πx)=2πx for all x, then the frame {Emθgn,m,n} for L2(,dθ) constitutes a frame for L2().

For fixed f,gL2(,dθ) and b>0, we introduce the θ-bracket product as follows: (11)[f,g]bθ(x):=kfθ(x+kb)gθ(x+kb)¯,a.e.x. If θ(2πx)=2πx for x, then [f,g]bθ is bracket product (denote by [f,g]b) introduced by Ron and Shen in . Thus, (12)[f,g]bθ=[fθ,gθ]b. Note that [f,g]bθ is a 1-periodic function.

With the classical bracket product, Christensen and Sun  proved the following Lemma 3, which is [13, Lemma  2.3].

Lemma 3.

Let gn,hnL2(), n, and b>0. Let the systems {Embgn,m,n} and {Embhn,m,n} be Bessel sequences in L2(). Define (13)Sf=m,nf,EmbgnEmbhn,fL2(). Then, the following holds: (14)(Sf)(x)=1bn[f,gn]b(x)hn(x)=1bnkf(x+kb)gn(x+kb)¯hn(x),cccccccccccccccccccccccicccfL2(), where the convergence is in the L2 sense. Moreover, {Embgn,  m,n} and {Embhn,    m,n} are a pair of dual frames for L2() if and only if (15)kgn(x+kb)hn(x+kb)¯=bδn,0,  a.e.x.

The following lemma follows from general properties of shift-invariant frames; see [11, Corollary 1.6.2]. Alternatively, it can be proved similarly to [14, Theorem 8.4.4].

Lemma 4.

Let gnL2(), n, b>0, and (16)B:=1bsupxk|ngn(x)gn(x-kb)¯|<. Then, {Embgn,m,n} is a Bessel sequence with upper frame bound B. If also (17)A:=1binfx(n|gn(x)|2-k0|ngn(x)gn(x-kb)¯|)>0, then {Embgn,m,n} constitutes a frame L2() with bounds A and B.

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In this section, we discuss frames for L2(,dθ). Here, we will establish the relationship between frames for L2(,dθ) and L2(). We will also give necessary conditions for frames and characterize a pair of dual frames in L2(,dθ). Above all, we establish the relationship between L2(,dθ) and L2() as follows.

Theorem 5.

Let f,g be functions defined on . Then, f,gθ=fθ,gθ; in particular, fθ=fθ, which means that fL2(,dθ) if and only if fθL2().

Proof.

Denote t:=(1/2π)θ(2πx). Then, (18)-f(x)g(x)¯dθ(2πx)=-f(12πθ-1(2πt))×g(12πθ-1(2πt))¯d(2πt)=2π-fθ(t)gθ(t)¯dt. This means that (19)f,gθ=fθ,gθ. Thus, we can obtain the desired result.

Theorem 6.

Let gn,n be functions defined on . Then, the system {Emθgn,m,n} constitutes a frame for L2(,dθ) if and only if the system {Emgnθ,m,n} constitutes a frame for L2() and these two systems have the same bounds.

Proof.

From Theorem 5, one obtains that gnL2(,dθ) if and only if gnθL2() for n. Note that (20)f,Emθgnθ=12π-f(x)eimθ(2πx)gn(x)¯dθ(2πx)=12π-f(12πθ-1(2πx))×ei2πmxgn(12πθ-1(2πx))¯d(2πx)=-fθ(x)ei2πmxgnθ(x)¯dx=fθ,Emgnθ. Then, (21)Afθ2m,n|f,Emθgnθ|2Bfθ2,fL2(,dθ) is equivalent to (22)Afθ2m,n|fθ,Emgnθ|2Bfθ2,fθL2(). Now, we can obtain the desired results.

Theorem 7.

Let gn,n be functions defined on . Then, the systems {Emθgn,m,n} and {Emθhn,m,n} constitute a pair of dual frames for L2(,dθ) if and only if the systems {Emgnθ,m,n} and {Emhnθ,m,n} constitute a pair of dual frames for L2().

Proof.

“if” part. If the systems {Emθgn,m,n} and {Emθhn,m,n} constitute a pair of dual frames for L2(,dθ). Then, by Theorem 6, these two systems {Emθgn,m,n} and {Emθhn,m,n} are frames for L2(,dθ). Moreover, we obtain from (20) that (23)f(x)=n,mfθ,EmgnθEmθhn(x),x, for any fL2(,dθ), where the convergence is in the L2 sense. Replacing x by (1/2π)θ-1(2πx) in the above equation, we obtain (24)fθ(x)=n,mfθ,EmgnθEmhnθ(x),x, for any fθL2(), where the convergence is in the L2 sense. Therefore, the systems {Emgnθ,m,n} and {Emhnθ,m,n} constitute a pair of dual frames for L2().

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 section, we can prove the following theorem.

Theorem 8.

Let gn,hnL2(,dθ) for n. Let {Emθgn,m,n} and {Emθhn,m,n} be Bessel sequences in L2(,dθ). Define (25)Sf=m,nf,EmθgnθEmθhn, for fL2(,dθ). Then, (26)(Sfθ)(x)=n[f,gn]θ(x)hnθ(x)×nkfθ(x+k)gnθ(x+k)¯hnθ(x) holds for fL2(,dθ), where the convergence is in the L2 sense. Moreover, the systems {Emθgn,m,n} and {Emθhn,m,n} constitute a pair of dual frames for L2(,dθ) if and only if (27)kgnθ(x+k)hnθ(x+k)¯=δn,0,a.e.x.

Proof.

For fixed fL2(,dθ), one obtains from (20) that (28)Sf(x)=m,nf,EmθgnθEmθhn(x)=m,nfθ,EmgnθEmθhn(x), where the convergence is in the L2 sense. Replacing x by (1/2π)θ-1(2πx) in the above equation, we obtain (29)Sfθ(x)=m,nfθ,EmgnθEmhnθ(x),fθL2(). Note that the systems {Emθgn,m,n} and {Emθhn,m,n} are Bessel sequences in L2(,dθ). We can deduce that the systems {Emgnθ,m,n} and {Emhnθ,m,n} are Bessel sequences in L2(). Therefore, one obtains (26) from (14).

From Theorem 7, we know that the systems {Emθgn,m,n} and {Emθhn,m,n} constitute a pair of dual frames for L2(,dθ) if and only if the systems {Emgnθ,m,n} and {Emhnθ,m,n} constitute a pair of dual frames for L2(). Thus, by (15) in Lemma 3, one obtains the desired result.

Theorem 9.

Let gnL2(,dθ), n, and suppose that (30)B:=supxk|ngnθ(x)gnθ(x+k)¯|<. Then, the system {Emθgn,m,n} is a Bessel sequence with upper frame bound B for L2(,dθ). If also (31)A:=infx(n|gnθ(x)|2-k0|ngnθ(x)gnθ(x+k)¯|)>0, then the system {Emθgn,m,n} constitutes a frame for L2(,dθ) with bounds A and B.

Proof.

Since gnL2(,dθ), n, then gnθL2(), n. If 0<A,B<, then by Lemma 4, the system {Emgnθ,m,n} constitutes a frame for L2() with frame bounds A and B. Therefore, by Theorem 6, one obtains that the system {Emθgn,m,n} constitutes a frame for L2(,dθ) with the same frame bounds A and B.

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In this section, Gabor frames for L2(,dθ) are discussed. We establish the relationship between the generalized Gabor frame {eimθ(2πt)g(t-n),m,n} for L2(,dθ) and the classical one {ei2mπtgθ(t-n),m,n} for L2(); further, we prove the Balian-Low theorem for Gabor system {eimθ(2πt)g(t-n),m,n} proposed by Fu et al. in  from a different viewpoint.

Theorem 10.

Let gL2(,dθ). Then, (32)(Tng)θ(x)=Tngθ(x).

Proof.

Since (33)θ-1(x+2kπ)=θ-1(x)+2kπ,x,k, then (34)θ-1(2π(x+k))=θ-1(2πx+2kπ)=θ-1(2πx)+2kπ,x,k. Hence, (35)(Tng)θ(x)=g(12πθ-1(2πx)-n)=g(12πθ-1(2π(x-n)))=Tngθ(x). We complete the proof.

Theorem 11.

Let g be a function defined on . Then, the general Gabor system {EmθTng,m,n} constitutes a frame for L2(,dθ) if and only if the classical Gabor system {EmTngθ,m,n} constitutes a frame for L2() with the same bounds.

Proof.

Define gn(x):=Tng(x),n. From Theorem 6, one obtains that the general Gabor system {EmθTng,m,n} constitutes a frame for L2(,dθ) if and only if the system {Em(Tng)θ,m,n} constitutes a frame for L2() 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 g,hL2(,dθ). Let the systems {EmθTng,m,n} and {EmθTnh,m,n} be Bessel sequences in L2(,dθ). Define (36)Sf=m,nf,EmθTngEmθTnh,fL2(,dθ). Then, for any fL2(,dθ), (37)(Sfθ)(x)=n[fθ,Tngθ](x)Tnhθ(x)=nkfθ(x+k)Tngθ(x+k)¯  Tnhθ(x), where the convergence is in the L2 sense. Moreover, the systems {EmθTng,m,n} and {EmθTnh,m,n} constitute a pair of dual frames for L2(,dθ) if and only if (38)kTngθ(x+k)Tnhθ(x+k)¯=δn,0,a.e.x.

Proof.

Replacing gn and hn by Tng and Tnh in (26) and (27), respectively, we have (39)(Sfθ)(x)=n[f,Tng]θ(x)(Tnh)θ(x)=nkfθ(x+k)(Tng)θ(x+k)¯(Tnh)θ(x),(40)k(Tng)θ(x+k)(Tnh)θ(x+k)¯=δn,0,a.e.x. Equations (37) and (38) follow from (39) and (40), respectively. Here, we used the facts that (Tng)θ(x)=Tngθ(x) and (Tnh)θ(x)=Tnhθ(x).

By Theorems 9 and 10, we obtain Theorem 13.

Theorem 13.

Consider gL2(,dθ), n, and suppose that (41)B:=supx[0,1]k|nTngθ(x)Tngθ(x+k)¯|<. Then, the system {EmθTng,m,n} is a Bessel sequence for L2(,dθ) with upper frame bound B. If also (42)A:=infx[0,1](n|Tngθ(x)|2infx[0,1]ccccccc-k0|nTngθ(x)Tngθ(x+k)¯|)>0, then the system {EmθTng,m,n} constitutes a frame for L2(,dθ) with bounds A and B.

Proof.

Since Tngθ(x)=(Tng)θ(x) and Tnhθ(x)=(Tnh)θ(x), then (43)B=supx[0,1]k|n(Tng)θ(x)(Tng)θ(x+k)¯|,A=infx[0,1](n|(Tng)θ(x)|2infx[0,1]ccccc-k0|n(Tng)θ(x)(Tng)θ(x+k)¯|). Define (44)H1(x):=k|n(Tng)θ(x)(Tng)θ(x+k)¯|,H2(x):=n|(Tng)θ(x)|2-k0|n(Tng)θ(x)(Tng)θ(x+k)¯|, then H1 and H2 are 1-periodic functions. Thus, (45)B=supxk|n(Tng)θ(x)(Tng)θ(x+k)¯|,A=infx(n|(Tng)θ(x)|2infxccccc-k0|n(Tng)θ(x)(Tng)θ(x+k)¯|). By Theorem 9, one obtains the results.

Theorem 14.

Let gL2(,dθ). Assume that the system {EmθTng,m,n} constitutes a generalized Gabor frame for L2(,dθ) with bounds A and B. Then, (46)An|gθ(x-n)|2B,a.e.x.

Proof.

If the system {EmθTng,m,n} constitutes a generalized Gabor frame for L2(,dθ) with bounds A and B. Then, by Theorem 7, {Em(Tng)θ,m,n} constitutes a frame for L2() with the same bounds A and B. Note that (Tng)θ=Tngθ. We can say that the system {EmTngθ,m,n} constitutes a Gabor frame for L2() with the same bounds A and B. Thus, one obtains from [14, Proposition 8.3.2] the desired result.

Theorem 15.

Let gL2(,dθ). Suppose that the system {EmθTng,m,n} constitutes a general Gabor frame for L2(,dθ). If the derivative θ of function θ is continuous on , then either (47)-x2|g(x)|2dθ(2πx)= or (48)-ξ2|g^(ξ)|2dθ(2πξ)=.

Proof.

Since {EmθTng,m,n} constitutes a general Gabor frame for L2(,dθ), then, by Theorem 11, the system {EmTngθ,m,n} constitutes a classical Gabor frame for L2(). Therefore, by the classical Balian-Low theorem, we have either (49)-x2|gθ(x)|2dx= or (50)-ξ2|gθ^(ξ)|2dξ=. That is either (51)-θ2(2πx)|g(x)|2dθ(2πx)= or (52)-θ2(2πξ)|g^(ξ)|2dθ(2πξ)=. 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 (53)0θ2(2πx)|g(x)|2dθ(2πx)=. Since the derivative θ of function θ is continuous on and θ satisfies Assumption 1, then (54)0<minx[0,2π]θ(x)θ(t)maxx[0,2π]θ(x),t. By the Lagrange mean-valued theorem, there exists ζ[0,x] such that (55)θ(x)=θ(0)+θ(ζ)x. Therefore, for any fixed x0>0, there exists a constant C>0 such that (56)|θ(x)|Cx,xx0. Note that (57)0x0θ2(2πx)|g(x)|2dθ(2πx)<. Therefore, (58)x0C2x2|g(x)|2dθ(2πx)x0θ2(2πx)|g(x)|2dθ(2πx)=. That is, (59)x0x2|g(x)|2dθ(2πx)=, or (60)-x2|g(x)|2dθ(2πx)=.

In the proof of Theorem 15, the main technique is the inequality (61)|θ(x)|Cx,xx0, for some positive constant C. Note that (62)θ(x+2kπ)=θ(x)+2kπ is equivalent to (63)θ(x)=x+β(x), where β is a 2π-periodic function. We obtain the following Balian-Low theorem which weakens the conditions imposed on θ in Theorem 15.

Theorem 16.

Let gL2(,dθ). Suppose that the system {EmθTng,m,n} constitutes a general Gabor frame for L2(,dθ). Let β be a 2π-periodic function such that (64)θ(x)=x+β(x),x,|β(x)|Cx,xx0, where C 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 5.1.7], 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 L2(,dθ).

Theorem 17.

Let gL2(,dθ). Then, the system {EmθTng,m,n} constitutes a tight frame for L2(,dθ) with A=1 if and only if (65)n|gθ(x-n)|2=1,ngθ(x-n)gθ(x-n-k)¯=0,fork0 holds a.e. in .

Proof.

By Theorem 11, one obtains that the system {EmθTng,m,n} constitutes a tight frame for L2(,dθ) with A=1 if and only if {EmTngθ,m,n} constitutes a tight frame for L2() with A=1. From [14, Theorem 9.5.2], 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).

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