Characterizations of Irregular Multigenerator Gabor Frame on Periodic Subsets of R

and Applied Analysis 3 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 F c, k : {x ∈ F : card Fc,x k}, k ∈ N ∪ { ∞}. 2.4 Note that {F c, k }k∈N∪{ ∞} is an equivalence class under the relation R or a partition of F. Thus F c, k ⋂ F ( c, k′ ) ∅ 2.5 for k, k′ ∈ N ∪ { ∞} and k / k′. Obviously, F⋂ F c, k′ cj ⊂ F c, k′ for given j ∈ Z \ {0}. It follows that F c, k ⋂( F ( c, k′ ) cj ) ∅ 2.6 for j ∈ Z \ {0}, k, k′ ∈ N ∪ { ∞} and k / k′. Example 2.1. Define Fn n − 1/2|n| 2, n 1/2|n| 2 ⊂ R for n ∈ Z. Consider F ⋃ n∈Z Fn. Then


Introduction
For a, b ∈ R, consider the translation operator T a g x g x − a and the modulation operator E b g x e 2πibx g x , both acting on g ∈ L 2 R . We say that the system {E mb T na g, m, n ∈ Z} is a Gabor frame for L 2 R if there exist two constants A, B > 0 such that holds for every f ∈ L 2 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 singlewindowed 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 timevarying 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 that aZ-periodic set in R can be used to model a signal that appears periodically but intermittently. Recently, some authors concerned Gabor analysis in L 2 S , where S is an aZ-periodic set in R. Although classical Gabor analysis tools in L 2 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 L 2 S , and we obtain the characterization for the multigenerator Gabor system to be a Parseval frame.

Notations
In this section, we present some notations and lemmas, which will be needed in the rest of the paper. Let S be aZ-periodic subset of R. Then, S is aqZ-periodic subset of R for any given q ∈ Z. Denote S 0 0, a S and Note that {F c, k } k∈N∪{ ∞} is an equivalence class under the relation R or a partition of F. Thus However, it is easy to check that F c, ∞ Z, which means F c, ∞ / ∅.

Remark 2.2.
Note that F 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. Let g n ∈ L 2 S for n ∈ Z. We say that the system {g n , n ∈ Z} is a frame for L 2 S if there exist two constants A, B > 0 such that holds for every f ∈ L 2 S ; moreover, we say the frame {g n , n ∈ Z} for Given a frame {g n ∈ L 2 S , n ∈ Z} for L 2 S , a dual frame of {g n ∈ L 2 S , n ∈ Z} for L 2 S is a frame {h n ∈ L 2 S , n ∈ Z} for L 2 S which satisfies the reconstruction property f n∈Z f, g n h n , ∀f ∈ L 2 S .

2.9
For fixed positive integer r, let ϕ 0 , ψ 0 , . . . , ϕ r−1 , ψ r−1 ∈ L 2 S . For given a 0 , b 0 , . . ., a r−1 , b r−1 ∈ R, we say that the system {E mb l T na l ϕ l , m,n ∈ Z, l 0, . . . , r − 1} is a multigenerator Gabor frame for L 2 S if it is a frame for L 2 S . Given a multigenerator Gabor frame {E mb l T na l ϕ l , m, n ∈ Z, l 0, . . . , r−1} for L 2 S , a dual multigenerator Gabor frame {E mb l T na l ψ l , m, n ∈ Z, l 0, . . . , r − 1} for L 2 S is a multigenerator Gabor frame for any f ∈ L 2 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. Let g n ∈ L 2 R , n ∈ Z, b > 0 and suppose that

2.11
Then {E mb g n , m, n ∈ Z} is a Bessel sequences with upper frame bound B for L 2 R . If also then {E mb g n , m, n ∈ Z} is a frame for L 2 R with bounds A and B.

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 for L 2 S . Firstly, we obtain the following theorem for the multigenerator Gabor system with the parameters a and b, which discloses the relationship between the Gabor system L 2 R and its subspace L 2 S . Moreover, we have the following sufficient condition for the multigenerator Gabor system with the parameters a and b.

3.8
Define g n x : T ka ϕ l x , 3.9 where n l rk and l 0, . . . , r − 1. Then, one obtains from 3.8 that respectively. Note that L 2 S ⊂ L 2 R . By Lemma 2.4, one obtains the results. The following theorem gives necessary condition for the system {E mb l T na ϕ l , m, n ∈ Z, l 0, . . . , r − 1} to be a multigenerator Gabor frame for L 2 S . It depends on the interplay among the function ϕ 0 , . . . , ϕ r−1 , the corresponding translation parameters a, b 0 , . . ., b r−1 , and the subset S.

3.11
Proof. Firstly, note that S is a aZ-periodic subset of R. Therefore, ϕ l · − na ∈ L 2 S for all n ∈ Z and l 0, . . . , r − 1. 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
Similar to the discussion in 11, Proposition 8.3.2 , we can assume that , a.e. Δ 3.14 for small > 0. Note that S is a aZ-periodic subset of R. We can assume further that Δ ⊂ S 0 . Define

3.15
Then, there exists k 0 ∈ Z such that μ Δ k 0 > 0 and If not, that is μ Δ k 0 for all k ∈ Z, then This contradicts to μ Δ > 0. Therefore, we can also assume that Δ is contained in an interval of length 1/b 0 and that Δ is a subset of S 0 . Now consider the function f χ Δ and note that f 2 μ Δ . Then for any n ∈ Z, the function fT na ϕ l has support in Δ. Since the functions { b l E mb l , m ∈ Z} constitute an orthonormal basis for L 2 I for every interval I of length 1/b l for fixed l 0, . . . , r − 1, we have Abstract and Applied Analysis Therefore,

3.20
This contradicts to the assumption that B is an upper frame bound for{E mb l T na ϕ l , 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 {E mb l T na ϕ l , m, n ∈ Z, l 0, . . . , r − 1}.

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 L 2 S ⊂ L 2 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.