1. Introduction
Note that the classical Fourier atoms e2πint cannot expose the time-varying property of nonstationary signals [1]. Recently, a kind of specific nonlinear phase function θa(2πt) is introduced in [2–6]. 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. [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 {eimθ(2πt)g(t-n),m,n∈ℤ} 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 [7]. 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∈ℤ} proposed by [8] 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 [2–6]. Using the nonlinear Fourier atoms in [2–6], 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(ℝ,dθ) and L2(ℝ) 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 L2(ℝ,dθ) and L2(ℝ). Basing on this relationship, 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 these results for general Gabor system {eimθ(2πt)g(t-n),m,n∈ℤ}, 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 L2(ℝ,dθ) and L2(ℝ); we also depict some properties of general frame {eimθ(2πt)gn (t),m,n∈ℤ} for L2(ℝ,dθ). In Section 4, we establish the relationship between Gabor 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; 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 θ in ℝ, consider the space L2(ℝ,dθ) of square integrable functions in ℝ with respect to θ and the finite norm:
(1)∥f∥θ=(1θ(2π)-θ(0)∫-∞∞|f(x)|2dθ(2πx))1/2
induced by the inner product
(2)〈f,g〉θ:=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∈ℤ}, Fu et al. introduced some assumptions including the Assumptions 2.1 and 2.2 in [8] for a nonlinear phase function θ. Combining these two assumptions together, we obtain the following Assumption 1.

Assumption 1.
Let function θ:ℝ→ℝ be a measure on ℝ and satisfy
(3)θ(x+2kπ)=θ(x)+2kπ,
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 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∈ℝ and k∈ℤ. 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 ℝ, denote by
(7)fθ(t):=f(12πθ-1(2πt))
through the rest of paper.

For a,b∈ℝ, consider the translation operator (Tag)(x)=g(x-a) and the modulation operator (Ebθg)(x)=eibθ(2πx)g(x), both acting on L2(ℝ,dθ). In [8], Fu et al. proposed a general Gabor frame for L2(ℝ,dθ). We say that the system {EmθTng, m,n∈ℤ} is a general Gabor frame for L2(ℝ,dθ) if there exist two constants A,B>0 such that
(8)A∥f∥θ2⩽∑m, n∈ℤ|〈f,EmbθTnag〉θ|2⩽B∥f∥θ2
holds for all f∈L2(ℝ,dθ). To further study the general Gabor frame as defined in [8], we introduce a general frame concept as follows.

Definition 2.
Let gn∈L2(ℝ,dθ), n∈ℤ. One says that the system {Emθgn,m,n∈ℤ} is a general frame for L2(ℝ,dθ) if there exist two constants A,B>0 such that
(9)A∥f∥θ2⩽∑m, n∈ℤ|〈f,Emθgn〉θ|2⩽B∥f∥θ2
holds for all f∈L2(ℝ,dθ); moreover, one says that the frame {Emθgn,m,n∈ℤ} is tight if A=B; in particular, the frame {Emθgn,m,n∈ℤ} is Parseval if A=B=1.

Given a frame {Emθgn,m,n∈ℤ} for L2(ℝ,dθ), a dual frame is a frame {Emθhn,m,n∈ℤ} of L2(ℝ,dθ) which satisfies the reconstruction property
(10)f=∑n, m∈ℤ〈f,Emθgn〉θEmθhn, ∀f∈L2(ℝ,dθ),
and we say that the systems {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} constitute a pair of dual frames for L2(ℝ,dθ), where the convergence is in the L2 sense. Note that if θ(2πx)=2πx for all x∈ℝ, then the frame {Emθgn,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:
(11)[f,g]bθ(x):=∑k∈ℤfθ(x+kb)gθ(x+kb)¯, a.e. x∈ℝ.
If θ(2πx)=2πx for x∈ℝ, then [f,g]bθ is bracket product (denote by [f,g]b) introduced by Ron and Shen in [11]. 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 [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 {Embhn,m,n∈ℤ} be Bessel sequences in L2(ℝ). Define
(13)Sf=∑m, n∈ℤ〈f,Embgn〉Embhn, ∀f∈L2(ℝ).
Then, the following holds:
(14)(Sf)(x)=1b∑n∈ℤ[f,gn]b(x)hn(x)=1b∑n∈ℤ∑k∈ℤf(x+kb)gn(x+kb)¯hn(x),ccccccccccccccccccccccciccc∀f∈L2(ℝ),
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
(15)∑k∈ℤgn(x+kb)hn(x+kb)¯=bδn,0, a.e. x∈ℝ.

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 gn∈L2(ℝ), n∈ℤ, b>0, and
(16)B:=1bsupx∈ℝ∑k∈ℤ|∑n∈ℤgn(x)gn(x-kb)¯|<∞.
Then, {Embgn,m,n∈ℤ} is a Bessel sequence with upper frame bound B. If also
(17)A:=1binfx∈ℝ(∑n∈ℤ|gn(x)|2-∑k≠0|∑n∈ℤgn(x)gn(x-kb)¯|)>0,
then {Embgn,m,n∈ℤ} constitutes a frame L2(ℝ) with bounds A and B.

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In this section, we discuss frames for L2(ℝ,dθ). Here, we will establish the relationship between frames for L2(ℝ,dθ) and L2(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) as follows.

Theorem 5.
Let f,g be functions defined on ℝ. Then, 〈f,g〉θ=〈fθ,gθ〉; in particular, ∥f∥θ= ∥fθ∥, which means that f∈L2(ℝ,dθ) if and only if fθ∈L2(ℝ).

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,g〉θ=〈fθ,gθ〉.
Thus, we can obtain the desired result.

Theorem 6.
Let gn,n∈ℤ be functions defined on ℝ. Then, the system {Emθgn,m,n∈ℤ} constitutes a frame for L2(ℝ,dθ) if and only if the system {Emgnθ,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 gnθ∈L2(ℝ) for n∈ℤ. Note that
(20)〈f,Emθgn〉θ=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)A∥f∥θ2⩽∑m, n∈ℤ|〈f,Emθgn〉θ|2⩽B∥f∥θ2, ∀f∈L2(ℝ,dθ)
is equivalent to
(22)A∥fθ∥2⩽∑m, n∈ℤ|〈fθ,Emgnθ〉|2⩽B∥fθ∥2, ∀fθ∈L2(ℝ).
Now, we can obtain the desired results.

Theorem 7.
Let gn,n∈ℤ be functions defined on ℝ. Then, the systems {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} constitute a pair of dual frames for L2(ℝ,dθ) if and only if the systems {Emgnθ,m,n∈ℤ} and {Emhnθ,m,n∈ℤ} constitute a pair of dual frames for L2(ℝ).

Proof.
“if” part. If the systems {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} constitute a pair of dual frames for L2(ℝ,dθ). Then, by Theorem 6, these two systems {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} are frames for L2(ℝ,dθ). Moreover, we obtain from (20) that
(23)f(x)=∑n, m∈ℤ〈fθ,Emgnθ〉Emθhn(x), x∈ℝ,
for any f∈L2(ℝ,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, m∈ℤ〈fθ,Emgnθ〉Emhnθ(x), x∈ℝ,
for any fθ∈L2(ℝ), where the convergence is in the L2 sense. Therefore, the systems {Emgnθ,m,n∈ℤ} and {Emhnθ,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 section, we can prove the following theorem.

Theorem 8.
Let gn,hn∈L2(ℝ,dθ) for n∈ℤ. Let {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} be Bessel sequences in L2(ℝ,dθ). Define
(25)Sf=∑m, n∈ℤ〈f,Emθgn〉θEmθhn,
for f∈L2(ℝ,dθ). Then,
(26)(Sfθ)(x)=∑n∈ℤ[f,gn]θ(x)hnθ(x) ×∑n∈ℤ∑k∈ℤfθ(x+k)gnθ(x+k)¯hnθ(x)
holds for f∈L2(ℝ,dθ), where the convergence is in the L2 sense. Moreover, the systems {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} constitute a pair of dual frames for L2(ℝ,dθ) if and only if
(27)∑k∈ℤgnθ(x+k)hnθ(x+k)¯=δn,0, a.e. x∈ℝ.

Proof.
For fixed f∈L2(ℝ,dθ), one obtains from (20) that
(28)Sf(x)=∑m, n∈ℤ〈f,Emθgn〉θEmθhn(x)=∑m, n∈ℤ〈fθ,Emgnθ〉Emθhn(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, n∈ℤ〈fθ,Emgnθ〉Emhnθ(x), ∀fθ∈L2(ℝ).
Note that the systems {Emθgn,m,n∈ℤ} and {Emθhn,m,n∈ℤ} are Bessel sequences in L2(ℝ,dθ). We can deduce that the systems {Emgnθ,m,n∈ℤ} and {Emhnθ,m,n∈ℤ} are Bessel sequences in L2(ℝ). Therefore, one obtains (26) from (14).

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

Theorem 9.
Let gn∈L2(ℝ,dθ), n∈ℤ, and suppose that
(30)B:=supx∈ℝ∑k∈ℤ|∑n∈ℤgnθ(x)gnθ(x+k)¯|<∞.
Then, the system {Emθgn,m,n∈ℤ} is a Bessel sequence with upper frame bound B for L2(ℝ,dθ). If also
(31)A:=infx∈ℝ(∑n∈ℤ|gnθ(x)|2-∑k≠0|∑n∈ℤgnθ(x)gnθ(x+k)¯|)>0,
then the system {Emθgn,m,n∈ℤ} constitutes a frame for L2(ℝ,dθ) with bounds A and B.

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

4. Gabor Frame for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M255"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo mathvariant="bold">(</mml:mo><mml:mi>ℝ</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mi>d</mml:mi><mml:mi>θ</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>
In this section, Gabor frames for L2(ℝ,dθ) are discussed. We establish the relationship between the generalized Gabor frame {eimθ(2πt)g(t-n),m,n∈ℤ} for L2(ℝ,dθ) and the classical 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 different viewpoint.

Theorem 10.
Let g∈L2(ℝ,dθ). Then,
(32)(Tng)θ(x)=Tngθ(x).

Proof.
Since
(33)θ-1(x+2kπ)=θ-1(x)+2kπ, ∀x∈ℝ, k∈ℤ,
then
(34)θ-1(2π(x+k))=θ-1(2πx+2kπ)=θ-1(2πx)+2kπ, ∀x∈ℝ, k∈ℤ.
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 ℝ. Then, the general Gabor system {EmθTng,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):=Tng(x), n∈ℤ. From Theorem 6, one obtains that the general Gabor system {EmθTng,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 from Theorem 10.

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

Theorem 12.
Consider g, h∈L2(ℝ,dθ). Let the systems {EmθTng,m,n∈ℤ} and {EmθTnh,m,n∈ℤ} be Bessel sequences in L2(ℝ,dθ). Define
(36)Sf=∑m, n∈ℤ〈f,EmθTng〉EmθTnh, ∀f∈L2(ℝ,dθ).
Then, for any f∈L2(ℝ,dθ),
(37)(Sfθ)(x)=∑n∈ℤ[fθ,Tngθ](x)Tnhθ(x)=∑n∈ℤ∑k∈ℤfθ(x+k)Tngθ(x+k)¯ Tnhθ(x),
where the convergence is in the L2 sense. Moreover, the systems {EmθTng,m,n∈ℤ} and {EmθTnh,m,n∈ℤ} constitute a pair of dual frames for L2(ℝ,dθ) if and only if
(38)∑k∈ℤTngθ(x+k)Tnhθ(x+k)¯=δn,0, a.e. x∈ℝ.

Proof.
Replacing gn and hn by Tng and Tnh in (26) and (27), respectively, we have
(39)(Sfθ)(x)=∑n∈ℤ[f,Tng]θ(x)(Tnh)θ(x)=∑n∈ℤ∑k∈ℤfθ(x+k)(Tng)θ(x+k)¯(Tnh)θ(x),(40)∑k∈ℤ(Tng)θ(x+k)(Tnh)θ(x+k)¯=δn, 0, a.e. x∈ℝ.
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 g∈L2(ℝ,dθ), n∈ℤ, and suppose that
(41)B:=supx∈[0,1]∑k∈ℤ|∑n∈ℤTngθ(x)Tngθ(x+k)¯|<∞.
Then, the system {EmθTng,m,n∈ℤ} is a Bessel sequence for L2(ℝ,dθ) with upper frame bound B. If also
(42)A:=infx∈[0,1](∑n∈ℤ|Tngθ(x)|2infx∈[0,1]ccccccc-∑k≠0|∑n∈ℤTngθ(x)Tngθ(x+k)¯|)>0,
then the system {EmθTng,m,n∈ℤ} constitutes a frame for L2(ℝ,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∈ℤ|∑n∈ℤ(Tng)θ(x)(Tng)θ(x+k)¯|,A=infx∈[0,1](∑n∈ℤ|(Tng)θ(x)|2infx∈[0,1]ccccc-∑k≠0|∑n∈ℤ(Tng)θ(x)(Tng)θ(x+k)¯|).
Define
(44)H1(x):=∑k∈ℤ|∑n∈ℤ(Tng)θ(x)(Tng)θ(x+k)¯|,H2(x):=∑n∈ℤ|(Tng)θ(x)|2 -∑k≠0|∑n∈ℤ(Tng)θ(x)(Tng)θ(x+k)¯|,
then H1 and H2 are 1-periodic functions. Thus,
(45)B=supx∈ℝ∑k∈ℤ|∑n∈ℤ(Tng)θ(x)(Tng)θ(x+k)¯|,A=infx∈ℝ(∑n∈ℤ|(Tng)θ(x)|2infx∈ℝccccc-∑k≠0|∑n∈ℤ(Tng)θ(x)(Tng)θ(x+k)¯|).
By Theorem 9, one obtains the results.

Theorem 14.
Let g∈L2(ℝ,dθ). Assume that the system {EmθTng,m,n∈ℤ} constitutes a generalized Gabor frame for L2(ℝ,dθ) with bounds A and B. Then,
(46)A⩽∑n∈ℤ|gθ(x-n)|2⩽B, a.e. x∈ℝ.

Proof.
If the system {EmθTng,m,n∈ℤ} constitutes a generalized Gabor frame for L2(ℝ,dθ) with bounds A and B. Then, by Theorem 7, {Em(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 {EmTngθ,m,n∈ℤ} constitutes a Gabor frame for L2(ℝ) with the same bounds A and B. Thus, one obtains from [14, Proposition 8.3.2] the desired result.

Theorem 15.
Let g∈L2(ℝ,dθ). Suppose that the system {EmθTng,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
(47)∫-∞∞x2|g(x)|2dθ(2πx)=∞
or
(48)∫-∞∞ξ2|g^(ξ)|2dθ(2πξ)=∞.

Proof.
Since {EmθTng,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
(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)∫0∞θ2(2πx)|g(x)|2dθ(2πx)=∞.
Since the derivative θ′ of function θ is continuous on ℝ and θ satisfies Assumption 1, then
(54)0<minx∈[0,2π]θ′(x)≤θ′(t)≤maxx∈[0,2π]θ′(x), t∈ℝ.
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, ∀x≥x0.
Note that
(57)∫0x0θ2(2πx)|g(x)|2dθ(2πx)<∞.
Therefore,
(58)∫x0∞C2x2|g(x)|2dθ(2πx) ≥∫x0∞θ2(2πx)|g(x)|2dθ(2πx)=∞.
That is,
(59)∫x0∞x2|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, ∀x≥x0,
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 g∈L2(ℝ,dθ). Suppose that the system {EmθTng,m,n∈ℤ} constitutes a general Gabor frame for L2(ℝ,dθ). Let β be a 2π-periodic function such that
(64)θ(x)=x+β(x), ∀x∈ℝ,|β(x)|≤Cx, ∀x≥x0,
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(ℝ,dθ).

Theorem 17.
Let g∈L2(ℝ,dθ). Then, the system {EmθTng,m,n∈ℤ} constitutes a tight frame for L2(ℝ,dθ) with A=1 if and only if
(65)∑n∈ℤ|gθ(x-n)|2=1,∑n∈ℤgθ(x-n)gθ(x-n-k)¯=0, for k≠0
holds a.e. in ℝ.

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