# Characterizations of Irregular Multigenerator Gabor Frame on Periodic Subsets of ℝ

Authors: D. H. Yuan; Y. Feng; Y. F. Shen; S. Z. Yang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the multigenerator system{EmblTnalφl,m,n∈ℤ,l=0,…,r-1} for φ0,…,φr-1∈L2(𝕊) and a0,b0,…,ar-1, br-1>0, where the parameters b0,…,br-1>0 are not necessary the same. With the help of frame theory, we provide some sufficient or necessary conditions for the system to be a frame for L2(𝕊). Moreover, we present some characterizations for this system to be a Parseval frame.


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## Body

## 1. Introduction

Fora,b∈ℝ, consider the translation operator (Tag)(x)=g(x-a) and the modulation operator (Ebg)(x)=e2πibxg(x), both acting on g∈L2(ℝ). We say that the system {EmbTnag,m,n∈ℤ} is a Gabor frame for L2(ℝ) if there exist two constants A,B>0 such that
(1.1)A‖f‖2⩽∑m,n∈Z‍|〈f,EmbTnag〉|2⩽B‖f‖2
holds for every f∈L2(ℝ).Gabor analysis is a pervasive signal processing method for decomposing and reconstructing signals from their time-frequency (TF) projections, and Gabor representation is used in many applications ranging from speech processing and texture segmentation to pattern and object recognition, among others. However, as it is widely recognized, a single-windowed Gabor expansion is not enough to analyze the dynamic TF contents of signals that contain a wide range of spatial and frequency components, the resolution of which is normally very poor. Therefore, if one could incorporate a set of multiple windows of various TF localizations in a frame system, the representation of signals of multiple and/or time-varying frequencies would have their corresponding windowing templates and resolutions to relate to. To this purpose, one of the best choices may be the multigenerator Gabor system.Multigenerator Gabor system is firstly presented by Zibulski and Zeevi in [1]. Utilizing the piecewise Zak transform (PZT), they [2] discussed the frame operator associated with the multigenerator Gabor frame. They pointed out that the so-called Balian-Low theorem for multigenerator Gabor frame is generalized to consideration of a scheme of multigenerator which makes it possible to overcome in a way the constraint imposed by the original theorem in the case of a single window. Since then, researchers are interested in the study of both theory and application aspects of multigenerator Gabor frame; for detail, see [2–7]. Multigenerator Gabor systems may be both interesting and useful since they can increase the degree of freedom by incorporating windows of various types and widths.Note thataℤ-periodic set in ℝ can be used to model a signal that appears periodically but intermittently. Recently, some authors concerned Gabor analysis in L2(𝕊), where 𝕊 is an aℤ-periodic set in ℝ. Although classical Gabor analysis tools in L2(𝕊) can be adjusted to treat such a scenario by padding with zeros outside the set 𝕊, Gabor systems that fit exactly such a scenario might have been more efficient. Gabardo and Li [8] obtained density results for Gabor systems associated with periodic subsets of the real line. Lian and Li [9] studied the Gabor frame sets for subspaces. They pointed out that only periodic 𝕊 in ℝ  is  suitable for Gabor analysis.Motivated by [7–9], we address the issue about the multigenerator Gabor frame in this paper. With the help of frame theory, we provide some sufficient or necessary conditions for the multigenerator Gabor frame system to be a frame for L2(𝕊), and we obtain the characterization for the multigenerator Gabor system to be a Parseval frame.


## 2. Notations

In this section, we present some notations and lemmas, which will be needed in the rest of the paper. Let𝕊 be aℤ-periodic subset of ℝ. Then, 𝕊 is aqℤ-periodic subset of ℝ for any given q∈ℤ. Denote 𝕊0=[0,a)⋂‍𝕊 and
(2.1)Li(S):={f∈Li(R)∣supp⁡(f)⊂S},
where supp(f)={t∈ℝ:f(t)≠0} and i=1,2. Given a measurable set 𝔽 in ℝ and a constant c>0. Define.
(2.2)Fc,x={y∈F:y=x+cjforsomej∈Z},x∈R.
Consider the relation R between x, y in 𝔽: xRy if and only if
(2.3)card(Fc,x)=card(Fc,y),
where card(𝔼) denotes the cardinality of 𝔼 for a set 𝔼. Then, it is easy to check that 𝔽 has an equivalence relation R. Moreover, define
(2.4)F(c,k):={x∈F:card(Fc,x)=k},k∈N∪{+∞}.
Note that {𝔽(c,k)}k∈ℕ∪{+∞} is an equivalence class under the relation R or a partition of 𝔽. Thus
(2.5)F(c,k)⋂‍F(c,k′)=∅
for k,k′∈ℕ∪{+∞} and k≠k′. Obviously, 𝔽⋂(𝔽(c,k′)+cj)⊂𝔽(c,k′) for given j∈ℤ∖{0}. It follows that
(2.6)F(c,k)⋂(F(c,k′)+cj)=∅
for j∈ℤ∖{0}, k,k′∈ℕ∪{+∞} and k≠k′.Example 2.1.
Define𝔽n=(n-1/2|n|+2,n+1/2|n|+2)⊂ℝ for n∈ℤ. Consider 𝔽=⋃n∈ℤ‍𝔽n. Then
(2.7)μ(F)=∑n∈Z‍μ(Fn)=2∑n∈N‍μ(Fn)-12=∑n∈N‍12n-12=32<∞.
However, it is easy to check that 𝔽(c,∞)=ℤ, which means 𝔽(c,∞)≠∅.Remark 2.2.
Note that𝔽(c,k) is the subset as defined in (2.2) of [9]. We point out that Proposition  2.1 (v) in [9] is incorrect.Definition 2.3.
Letgn∈L2(𝕊) for n∈ℤ. We say that the system {gn,n∈ℤ} is a frame for L2(𝕊) if there exist two constants A,B>0 such that
(2.8)A‖f‖2⩽∑n∈Z‍|〈f,gn〉|2⩽B‖f‖2
holds for every f∈L2(𝕊); moreover, we say the frame {gn,n∈ℤ} for L2(𝕊) is tight if A=B. In particular, the frame {gn,n∈ℤ} for L2(𝕊) is Parseval if A=B=1.Given a frame{gn∈L2(𝕊),n∈ℤ} for L2(𝕊), a dual frame of {gn∈L2(𝕊),n∈ℤ} for L2(𝕊) is a frame {hn∈L2(𝕊),n∈ℤ} for L2(𝕊) which satisfies the reconstruction property
(2.9)f=∑n∈Z‍〈f,gn〉hn,∀f∈L2(S).For fixed positive integerr, let φ0,ψ0,…,φr-1,ψr-1∈L2(𝕊). For given a0,b0,…,ar-1,br-1∈ℝ, we say that the system {EmblTnalφl,m,n∈ℤ,l=0,…,r-1} is a multigenerator Gabor frame for L2(𝕊) if it is a frame for L2(𝕊). Given a multigenerator Gabor frame {EmblTnalφl,m,n∈ℤ,l=0,…,r-1} for L2(𝕊), a dual multigenerator Gabor frame {EmblTnalψl,m,n∈ℤ,l=0,…,r-1} for L2(𝕊) is a multigenerator Gabor frame
(2.10)f=∑l=0r-1‍∑m,n∈Z‍〈f,EmblTnalφl〉EmblTnalψl
for any f∈L2(𝕊).The following lemma follows from general characterizations of shift-invariant frames, see [10, Corollary  1.6.2]. Alternatively, it can be proved similarly to [11, Theorem  8.4.4].Lemma 2.4.
Letgn∈L2(ℝ), n∈ℤ, b>0 and suppose that
(2.11)B:=1bsup⁡x∈R∑k∈Z‍|∑n∈Z‍gn(x)gn(x-kb)¯|<∞.
Then {Embgn,m,n∈ℤ} is a Bessel sequences with upper frame bound B for L2(ℝ). If also
(2.12)A:=1binf⁡x∈R(∑n∈Z‍|gn(x)|2-∑k≠0‍|∑n∈Z‍gn(x)gn(x-kb)¯|)>0,
then {Embgn,m,n∈ℤ} is a frame for L2(ℝ) with bounds A and B.


## 3. Sufficient and Necessary Conditions

In this section, we provide some sufficient and necessary conditions for a class of the multigenerator Gabor frame system to be a frame forL2(𝕊).Firstly, we obtain the following theorem for the multigenerator Gabor system with the parametersa and b, which discloses the relationship between the Gabor system L2(ℝ) and its subspace L2(𝕊).Theorem 3.1.
Letφ0,…,φr-1∈L2(𝕊) and a,b>0. Then the following results hold.(I)
If the Gabor system{EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a frame for L2(ℝ), then it is a frame for L2(𝕊).(II)
If the Gabor system{EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a Bessel sequence for L2(𝕊) with upper bound B, then it is a Bessel sequence for L2(ℝ) with the same upper bound.Proof.
The part (I) follows from the fact thatL2(𝕊)⊂L2(ℝ).
Next, we prove the second part. Suppose that the Gabor system{EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a Bessel sequence for L2(𝕊). Then there exists a constant B>0 such that
(3.1)∑l=0r-1‍∑n∈Z‍|〈f,EmbTnaφl〉|2⩽B‖f‖2,∀f∈L2(S).
Observe that
(3.2)〈f,EmbTnaφl〉=∫R‍f(x)φl(x-na)¯e-2πmbxdx=∫S‍f(x)φl(x-na)¯e-2πmbxdx,
since fφl¯∈L2(𝕊), l=0,…,r-1, for all f∈L2(ℝ). It follows that
(3.3)〈f,EmbTnaφl〉=〈fχS,EmbTnaφl〉.
Therefore,
(3.4)∑l=0r-1‍∑n∈Z‍|〈f,EmbTnaφl〉|2=∑l=0r-1‍∑n∈Z‍|〈fχS,EmbTnaφl〉|2⩽B‖fχS‖2⩽B‖f‖2
for all f∈L2(ℝ). This implies that {EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a Bessel sequence for L2(ℝ) with the same upper bound B.Moreover, we have the following sufficient condition for the multigenerator Gabor system with the parametersa and b.Theorem 3.2.
Letφ0,…,φr-1∈L2(𝕊) and a,b>0. Moreover, suppose that
(3.5)B:=1bsup⁡x∈[0,1/b)∑k∈Z‍|∑l=0r-1‍∑n∈Z‍Tnaφl(x)Tnaφl(x+kb)¯|<∞.
Then {EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a Bessel sequence for L2(𝕊) with upper frame bound B. If also
(3.6)A:=1binf⁡x∈[0,1/b)(∑n∈Z‍|∑l=0r-1‍Tnaφl(x)|2-∑k≠0‍|∑n∈Z‍∑l=0r-1‍Tnaφl(x)Tnaφl(x+kb)¯|)>0,
then {EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a frame for L2(𝕊) with bounds A and B. That means {EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a multigenerator Gabor frame for L2(𝕊).Proof .
Define(3.7)H1(x):=∑k∈ℤ‍|∑l=0r-1‍∑n∈ℤ‍Tnaφl(x)Tnaφl(x+kb)¯|,H2(x):=∑n∈Z‍|∑l=0r-1‍Tnaφl(x)|2-∑k≠0‍|∑n∈Z‍∑l=0r-1‍Tnaφl(x)Tnaφl(x+kb)¯|,
then H1 and H2 are 1/b-periodic functions. Thus
(3.8)B=1bsupx∈ℝ∑k∈ℤ‍|∑l=0r-1‍∑n∈ℤ‍Tnaφl(x)Tnaφl(x+kb)¯|<∞,A=1binf⁡x∈R(∑n∈Z‍|∑l=0r-1‍Tnaφl(x)|2-∑k≠0‍|∑n∈Z‍∑l=0r-1‍Tnaφl(x)Tnaφl(x+kb)¯|)>0.
Define
(3.9)gn(x):=(Tkaφl)(x),
where n=l+rk and l=0,…,r-1. Then, one obtains from (3.8) that
(3.10)B=1bsupx∈ℝ∑k∈ℤ‍|∑n∈ℤ‍gn(x)gn(x-kb)¯|<∞,A=1binf⁡x∈R(∑n∈Z‍|gn(x)|2-∑k≠0‍|∑n∈Z‍gn(x)gn(x-kb)¯|)>0,
respectively. Note that L2(𝕊)⊂L2(ℝ). By Lemma 2.4, one obtains the results.Remark 3.3.
Theorem3.2 is similar to [11, Theorem  8.4.4]. Note that our result extends [11, Theorem  8.4.4] to the multigenerator and the periodic subset cases.The following theorem gives necessary condition for the system{EmblTnaφl,m,n∈ℤ,l=0,…,r-1} to be a multigenerator Gabor frame for L2(𝕊). It depends on the interplay among the function φ0,…,φr-1, the corresponding translation parameters a, b0, …, br-1, and the subset 𝕊.Theorem 3.4.
Letφ0,…,φr-1∈L2(𝕊) and a,b0,…,br-1>0. Assume that {EmblTnaφl,m,n∈ℤ,l=0,…,r-1} is a multigenerator Gabor frame for L2(𝕊) with bounds A and B. Then,
(3.11)AχS(x)⩽∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)⩽BχS(x),a.e.R.Proof.
Firstly, note that𝕊 is a aℤ-periodic subset of ℝ. Therefore, φl(·-na)∈L2(𝕊) for all n∈ℤ and l=0,…,r-1. Thus
(3.12)∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)=0,a.e.R∖S.
The rest part of the proof is by contradiction. Assume that the upper condition in (3.11) is violated on 𝕊. Then there exists a measurable set Δ⊂𝕊 with measure μ(Δ)>0 such that
(3.13)∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)>B,a.e.Δ.
Similar to the discussion in [11, Proposition  8.3.2], we can assume that
(3.14)∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)>B+ϵ,a.e.Δ
for small ϵ>0. Note that 𝕊 is a aℤ-periodic subset of ℝ. We can assume further that Δ⊂𝕊0. Define
(3.15)1b0:=min{1bl,l=0,⋯,r-1},Δk=Δ⋂‍[k-1b0,kb0],k∈Z.
Then, there exists k0∈ℤ such that μ(Δk0)>0 and
(3.16)∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)>B+ϵ,a.e.Δk0.
If not, that is μ(Δk)=0 for all k∈ℤ, then
(3.17)μ(Δ)=μ(⋃k∈Z‍Δk)=∑k∈Z‍μ(Δk)=0.
This contradicts to μ(Δ)>0. Therefore, we can also assume that Δ is contained in an interval of length 1/b0 and that Δ is a subset of 𝕊0.
Now consider the functionf=χΔ and note that ∥f∥2=μ(Δ). Then for any n∈ℤ, the function fTnaφl¯ has support in Δ. Since the functions {blEmbl,m∈ℤ} constitute an orthonormal basis for L2(I) for every interval I of length 1/bl for fixed l=0,…,r-1, we have
(3.18)∑m∈Z‍|〈f,EmblTnaφl〉|2=∑m∈Z‍|〈fTnaφl¯,Embl〉|2=1bl∫-∞∞‍|f(x)|2|φl(x-na)|2dx.
Thus,
(3.19)∑m,n∈Z‍|〈f,EmblTnaφl〉|2=∫-∞∞‍|f(x)|2(1bl∑n∈Z‍|φl(x-na)|2)dx.
Therefore,
(3.20)∑l=0r-1‍∑m,n∈Z‍|〈f,EmblTnaφl〉|2=∫-∞∞‍|f(x)|2∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)dx>∫Δ‍(B+ϵ)dx=(B+ϵ)μ(Δ)=(B+ϵ)‖f‖2>B‖f‖2.
This contradicts to the assumption that B is an upper frame bound for{EmblTnaφl,m,n∈ℤ,l=0,…,r-1}. A similar proof shows that if the lower condition in (3.11) is violated, then A cannot be a lower frame bound for {EmblTnaφl,m,n∈ℤ,l=0,…,r-1}.


## 4. Parseval Multigenerater Gabor Frame

In applications of frames, it is inconvenient that the frame decomposition, stated in [12, Theorem  5.1.7], requires inversion of the frame operator. As we have seen in the discussion of general frame theory, one way of avoiding the problem is to consider tight frames. We will characterize Parseval multigenerater Gabor frames in this section. Noting that L2(𝕊)⊂L2(ℝ), we obtain from [11, Lemma  8.4.3] or [12, Lemma  9.1.4] the following lemma, which will be used in the rest of the section.Lemma 4.1.
Letf be a bounded measure function with compact support and g∈L2(𝕊). Then
(4.1)∑m,n∈Z‍|〈f,EmbTnag〉|2=1b∫R‍|f(x)|2∑n∈Z‍|g(x-na)|2dx+1b∑k∈Z‍∫R‍f(x)¯f(x-kb)∑n≠0‍g(x-na)g(x-na-kb)¯dx
for given a,b>0.Theorem 4.2.
Letφ0,…,φr-1∈L2(𝕊) and a,b0,…,br-1>0. Moreover, assume that {EmblTnaφl,m,n∈ℤ,l=0,…,r-1} is a tight frame for L2(𝕊) with A=1. Then,
(4.2)∑l=0r-1‍1bl(∑n∈Z‍|φl(x-na)|2)=χS(x),a.e.R.
Moreover, if b0=⋯=br-1 and denote b0 by b, then
(4.3)∑l=0r-1‍(∑n∈Z‍|φl(x-na)|2)=bχS(x),(4.4)∑l=0r-1‍(∑n∈Z‍φl(x-na)φl(x-na-kb)¯)=0,fork≠0
hold a.e. in ℝ.Proof.
Define(4.5)1b0:=min⁡{1bl,l=0,…,r-1}.
Consider
(4.6)CL2(S):={f:f∈L2(S)andsupp⁡f⊂[0,1b0)∩S}.
Note that {EmblTnaφl,m,n∈ℤ,l=0,…,r-1} is a tight frame for L2(𝕊) with A=1. Then
(4.7)∑l=0r-1‍∑m,n∈Z‍|〈f,EmblTnaφl〉|2=‖f‖2,∀f∈CL2(S).
Again, we obtain from Lemma 4.1 that
(4.8)∑m∈Z‍|〈f,EmblTnaφl〉|2=∑m∈Z‍|∫R‍f(x)φl(x-na)¯e-2πmblxdx|2=1bl∫R‍|f(x)φl(x-na)¯|2dx,=1bl∫01/b0‍|f(x)φl(x-na)|2dx,∀f∈CL2(S)
for fixed l and n. Thus,
(4.9)∫01/b0‍|f(x)|2dx=∫01/b0‍|f(x)|2(∑l=0r-1‍1bl∑n∈Z‍|φl(x-na)|2)dx
for any f∈CL2(𝕊). This implies that
(4.10)∑l=0r-1‍(1bl∑n∈Z‍|φl(x-na)|2)=1,a.e.[0,1/b0)∩S.
Note that
(4.11)S=⋃k∈Z‍([kb0,k+1b0)∩S),
and we obtain the desired result (4.2) and its special case  (4.3).
Next, we prove (4.4). For fixed l=0,…,r-1, we obtain from Lemma 4.1 that
(4.12)∑m,n∈Z‍|〈f,EmbTnaφl〉|2=1b∫R‍|f(x)|2∑n∈Z‍|φl(x-na)|2dx+1b∑k≠0‍∫R‍f(x)¯f(x-kb)∑n∈Z‍φl(x-na)φl(x-na-kb)¯dx.
Then,
(4.13)∑l=0r-1‍∑m,n∈Z‍|〈f,EmbTnaφl〉|2=∫R‍|f(x)|2∑l=0r-1‍1b∑n∈Z‍|φl(x-na)|2dx+1b∑k≠0‍∫R‍(f(x)¯f(x-kb)∑l=0r-1‍∑n∈Z‍(φl(x-na)φl(x-na-kb)¯))dx.
This, together with (4.3), follows that
(4.14)∑k≠0‍∫R‍(f(x)¯f(x-kb)∑n∈Z‍(φl(x-na)φl(x-na-kb)¯))dx=0.
A change of variable shows that the contribution in the above sum arising from any value of k is the complex conjugate of the contribution form the value -k. Therefore,
(4.15)∑k=1∞‍Re(∫R‍f(x)¯f(x-kb)Gk(x)dx)=0,
where Gk(x):=∑l=0r-1‍∑n∈ℤ‍φl(x-na)φl(x-na-k/b)¯ for k∈ℕ. Now we divide three cases to draw the result.
Case  1.
x
∉
𝕊. Note that 𝕊 is a aℤ-periodic set. Then x-na∉𝕊 for all n∈ℤ. Therefore,
(4.16)φl(x-na)=0,∀n∈Z.
Thus,
(4.17)∑l=0r-1‍∑n∈Z‍φl(x-na)φl(x-na-kb)¯=0,∀k≠0.
Case  2.
x
-
k
/
b
∉
𝕊 for fixed k∈ℤ∖{0}. Then x-k/b-na∉𝕊 for all n∈ℤ. Therefore,
(4.18)∑l=0r-1‍∑n∈Z‍φl(x-na)φl(x-na-kb)¯=0.
Case  3.
x
∈
𝕊 and x-k/b∈𝕊 for fixed k∈ℤ∖{0}. Consider k⩾1 and let I be any interval in ℝ of length at most 1/b. Denote I∩𝕊 by Δ0 and (I-k/b)∩𝕊+k/b by Δ′. If μ(Δ0∩Δ′)=0, then x∉Δ0 a.e. or x∉Δ′-k/b a.e., thus
(4.19)∑l=0r-1‍∑n∈Z‍φl(x-na)φl(x-na-kb)¯=0.
Now consider μ(Δ0∩Δ′)>0. Define a function f∈L2(𝕊) by
(4.20)f(x):={e-argGk0(x),x∈Δ0∩Δ′1,x∈Δ0∩Δ′-kb,0,otherwise.
Then, by (4.15),
(4.21)0=∑k=1∞‍Re(∫R‍f(x)¯f(x-kb)Gk(x)dx)=Re(∫R‍f(x)¯f(x-k0b)Gk0(x)dx)=∫Δ0∩Δ′‍|Gk0(x)|dx.
It follows that Gk0(x)=0, a.e. on Δ0∩Δ′. Since I is an arbitrary interval of length at most 1/b, we conclude that Gk0(x)=0, a.e. in 𝕊. A direct computation shows that
(4.22)G-k0(x)=Gk0(x+k0b).
Thus, we obtain the desired results.To proceed further, we need use the following symbols. Forb0,…,br-1>0, define
(4.23)1b0:=min{1bl,l=0,…,r-1},1b1:=min⁡{1bl:bl<b0,l=1,…,r-1},⋮1bq:=min{1bl:bl<bq-1,l=1,…,r-1},I0:={l:bl=b0,l=0,…,r-1},I1:={l:bl=b1,l=0,…,r-1},⋮Iq:={l:bl=bq,l=0,…,r-1}.
Then there exists a unique nonnegative integer q0 such that
(4.24)Iq≠∅,forq=0,…,q0,Iq1⋂‍Iq2=∅,forq1≠q2,⋃q=0q0‍Iq={0,…,r-1}.Theorem 4.3.
Letq0 be the unique nonnegative integer satisfying (4.24). Assume that φ0,…,φr-1∈L2(𝕊) and a,b0,…,br-1>0 satisfy
(4.25)∑l=0r-1‍1bl(∑n∈Z‍|φl(x-na)|2)=χS(x),(4.26)∑l∈Iq‍(∑n∈ℤ‍φl(x-na)φl(x-na-kbl)¯)=0,fork≠0,q=0,…,q0
a.e. in ℝ. Then {EmblTnaφl,m,n∈ℤ,l=0,…,r-1} is a tight frame for L2(𝕊) with A=1.Proof .
For fixedl=0,…,r-1, we obtain from Lemma 4.1 that
(4.27)∑m,n∈Z‍|〈f,EmblTnaφl〉|2=1bl∫R‍|f(x)|2∑n∈Z‍|φl(x-na)|2dx+1bl∑k≠0‍∫R‍f(x)¯f(x-kbl)∑n∈Z‍φl(x-na)φl(x-na-kbl)¯dx.
Then,
(4.28)∑l=0r-1‍∑m,n∈Z‍|〈f,EmblTnaφl〉|2=∫R‍|f(x)|2∑l=0r-1‍1bl∑n∈Z‍|φl(x-na)|2dx+(*),
where
(4.29)(*):=∫R‍∑l=0r-1‍{1bl∑k≠0‍[f(x)¯f(x-kbl)∑n∈Z‍(φl(x-na)φl(x-na-kbl)¯)]}dx.
This, together with (4.25), follows that
(4.30)∑l=0r-1‍∑m,n∈Z‍|〈f,EmblTnaφl〉|2=∫R‍|f(x)|2dx+(*).
Define
(4.31)Gkq(x):=∑l∈Iq‍(∑n∈Z‍φl(x-na)φl(x-na-kbl)¯),forq=0,…,q0.
Then, we obtain from (4.26) that
(4.32)(*)=∑k≠0‍(∫R‍∑q=0q0‍1bqf(x)¯f(x-kbq)Gkq(x)dx)=0.
This, together with (4.30), follows that
(4.33)∑l=0r-1‍∑m,n∈Z‍|〈f,EmblTnaφl〉|2=∫R‍|f(x)|2dx=‖f‖2,∀f∈L2(S).
Therefore, {EmblTnaφl,m,n∈ℤ,l=0,…,r-1} is a tight frame for L2(𝕊) with A=1.Note thatq0=0 when b0=b1=⋯=br-1=b. We obtain from Theorems 4.2 and 4.3 that following Theorem 4.4, which is a necessary and sufficient condition for multigenerater Gabor Parseval frame.Theorem 4.4.
Letφ0,…,φr-1∈L2(𝕊) and a,b>0. Then {EmbTnaφl,m,n∈ℤ,l=0,…,r-1} is a tight frame for L2(𝕊) with A=1 if and only if
(4.34)∑l=0r-1‍(∑n∈Z‍|φl(x-na)|2)=bχS(x),∑l=0r-1‍(∑n∈Z‍φl(x-na)φl(x-na-kb)¯)=0,fork≠0
hold a.e. in ℝ.Remark 4.5.
If𝕊=ℝ, then 𝕊 is aℤ-periodic set for any given a>0. In this case, {Em/aTnaφl,m,n∈ℤ,l=0,…,r-1} is a tight frame for L2(𝕊) with A=1 if and only if
(4.35)∑l=0r-1‍(∑n∈ℤ‍|φl(x-na)|2)=1aχ𝕊(x),∑l=0r-1‍(∑n∈Z‍φl(x-na)φl(x-na-ka)¯)=0,fork≠0
hold a.e. in ℝ.



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*Source: 746823-2012-07-02.xml*