Characterizations of Irregular Multigenerator Gabor Frame on Periodic Subsets of ℝ

Abstract

We consider the multigenerator system {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝑙𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} for 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎₀,𝑏₀,…,𝑎_𝑟−1, 𝑏_𝑟−1>0, where the parameters 𝑏₀,…,𝑏_𝑟−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.

1. Introduction

For 𝑎,𝑏∈ℝ, consider the translation operator (𝑇_𝑎𝑔)(𝑥)=𝑔(𝑥−𝑎) and the modulation operator (𝐸_𝑏𝑔)(𝑥)=𝑒^{2𝜋𝑖𝑏𝑥}𝑔(𝑥), both acting on 𝑔∈𝐿²(ℝ). We say that the system {𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔,𝑚,𝑛∈ℤ} is a Gabor frame for 𝐿²(ℝ) if there exist two constants 𝐴,𝐵>0 such that 𝐴𝑓²⩽_{𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔²⩽𝐵𝑓² (1.1) 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 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 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. Let 𝕊 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 𝐿^𝑖(𝕊)∶=𝑓∈𝐿^𝑖(ℝ)∣supp𝑓⊂𝕊, (2.1) 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. 𝔽_𝑐,𝑥=𝑦∈𝔽∶𝑦=𝑥+𝑐𝑗forsome𝑗∈ℤ,𝑥∈ℝ. (2.2) Consider the relation 𝑅 between 𝑥, 𝑦 in 𝔽: 𝑥𝑅𝑦 if and only if card𝔽_𝑐,𝑥=card𝔽_𝑐,𝑦, (2.3) 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∶card𝔽_𝑐,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)𝔽𝑐,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)𝔽𝑐,𝑘^′+𝑐𝑗=∅ (2.6) for 𝑗∈ℤ⧵{0}, 𝑘,𝑘^′∈ℕ∪{+∞} and 𝑘≠𝑘^′.

Example 2.1. Define 𝔽_𝑛=(𝑛−1/2^|𝑛|+2,𝑛+1/2^|𝑛|+2)⊂ℝ for 𝑛∈ℤ. Consider 𝔽=⋃_𝑛∈ℤ𝔽_𝑛. Then 𝜇(𝔽)=_𝑛∈ℤ𝜇𝔽_𝑛=2_𝑛∈ℕ𝜇𝔽_𝑛−12=_𝑛∈ℕ12^𝑛−12=32<∞. (2.7) 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∈𝐿²(𝕊) for 𝑛∈ℤ. We say that the system {𝑔_𝑛,𝑛∈ℤ} is a frame for 𝐿²(𝕊) if there exist two constants 𝐴,𝐵>0 such that 𝐴𝑓²⩽_𝑛∈ℤ𝑓,𝑔_𝑛²⩽𝐵𝑓² (2.8) holds for every 𝑓∈𝐿²(𝕊); moreover, we say the frame {𝑔_𝑛,𝑛∈ℤ} for 𝐿²(𝕊) is tight if 𝐴=𝐵. In particular, the frame {𝑔_𝑛,𝑛∈ℤ} for 𝐿²(𝕊) is Parseval if 𝐴=𝐵=1.

Given a frame {𝑔_𝑛∈𝐿²(𝕊),𝑛∈ℤ} for 𝐿²(𝕊), a dual frame of {𝑔_𝑛∈𝐿²(𝕊),𝑛∈ℤ} for 𝐿²(𝕊) is a frame {ℎ_𝑛∈𝐿²(𝕊),𝑛∈ℤ} for 𝐿²(𝕊) which satisfies the reconstruction property 𝑓=_𝑛∈ℤ𝑓,𝑔_𝑛ℎ_𝑛,∀𝑓∈𝐿²(𝕊). (2.9)

For fixed positive integer 𝑟, let 𝜑₀,𝜓₀,…,𝜑_𝑟−1,𝜓_𝑟−1∈𝐿²(𝕊). For given 𝑎₀,𝑏₀,…,𝑎_𝑟−1,𝑏_𝑟−1∈ℝ, we say that the system {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝑙𝜑_𝑙,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 {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝑙𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} for 𝐿²(𝕊), a dual multigenerator Gabor frame {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝑙𝜓_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} for 𝐿²(𝕊) is a multigenerator Gabor frame 𝑓=^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝑙𝜑_𝑙𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝑙𝜓_𝑙 (2.10) 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. Let 𝑔_𝑛∈𝐿²(ℝ), 𝑛∈ℤ, 𝑏>0 and suppose that 1𝐵∶=𝑏sup_{𝑥∈ℝ𝑘∈ℤ𝑛∈ℤ}𝑔_𝑛(𝑥)𝑔_𝑛𝑘𝑥−𝑏<∞. (2.11) Then {𝐸_𝑚𝑏𝑔_𝑛,𝑚,𝑛∈ℤ} is a Bessel sequences with upper frame bound 𝐵 for 𝐿²(ℝ). If also 1𝐴∶=𝑏inf_{𝑥∈ℝ𝑛∈ℤ}𝑔_𝑛(𝑥)²−_{𝑘≠0𝑛∈ℤ}𝑔_𝑛(𝑥)𝑔_𝑛𝑘𝑥−𝑏>0, (2.12) then {𝐸_𝑚𝑏𝑔_𝑛,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 for 𝐿²(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 𝐿²(ℝ) and its subspace 𝐿²(𝕊).

Theorem 3.1. Let 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎,𝑏>0. Then the following results hold.(I)If the Gabor system {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,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 {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,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 that 𝐿²(𝕊)⊂𝐿²(ℝ).
Next, we prove the second part. Suppose that the Gabor system {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a Bessel sequence for 𝐿²(𝕊). Then there exists a constant 𝐵>0 such that ^𝑟−1_{𝑙=0𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙²⩽𝐵𝑓²,∀𝑓∈𝐿²(𝕊). (3.1) Observe that 𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙=_ℝ𝑓(𝑥)𝜑_𝑙(𝑥−𝑛𝑎)𝑒^{−2𝜋𝑚𝑏𝑥}𝑑𝑥=_𝕊𝑓(𝑥)𝜑_𝑙(𝑥−𝑛𝑎)𝑒^{−2𝜋𝑚𝑏𝑥}𝑑𝑥, (3.2) since 𝑓𝜑_𝑙∈𝐿²(𝕊), 𝑙=0,…,𝑟−1, for all 𝑓∈𝐿²(ℝ). It follows that 𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙=𝑓𝜒_𝕊,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙. (3.3) Therefore, ^𝑟−1_{𝑙=0𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙²=^𝑟−1_{𝑙=0𝑛∈ℤ}𝑓𝜒_𝕊,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙²⩽𝐵𝑓𝜒_𝕊²⩽𝐵𝑓² (3.4) for all 𝑓∈𝐿²(ℝ). This implies that {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,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 parameters 𝑎 and 𝑏.

Theorem 3.2. Let 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎,𝑏>0. Moreover, suppose that 1𝐵∶=𝑏sup_{𝑥∈[0,1/𝑏)𝑘∈ℤ}^𝑟−1_{𝑙=0𝑛∈ℤ}𝑇_𝑛𝑎𝜑_𝑙(𝑥)𝑇_𝑛𝑎𝜑_𝑙𝑘𝑥+𝑏<∞. (3.5) Then {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a Bessel sequence for 𝐿²(𝕊) with upper frame bound 𝐵. If also 1𝐴∶=𝑏inf_{𝑥∈[0,1/𝑏)𝑛∈ℤ}^𝑟−1_𝑙=0𝑇_𝑛𝑎𝜑_𝑙(𝑥)²−_{𝑘≠0𝑛∈ℤ}^𝑟−1_𝑙=0𝑇_𝑛𝑎𝜑_𝑙(𝑥)𝑇_𝑛𝑎𝜑_𝑙𝑘𝑥+𝑏>0, (3.6) then {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a frame for 𝐿²(𝕊) with bounds 𝐴 and 𝐵. That means {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a multigenerator Gabor frame for 𝐿²(𝕊).

Proof . Define 𝐻₁(𝑥)∶=_𝑘∈ℤ^𝑟−1_{𝑙=0𝑛∈ℤ}𝑇_𝑛𝑎𝜑_𝑙(𝑥)𝑇_𝑛𝑎𝜑_𝑙𝑘𝑥+𝑏,𝐻₂(𝑥)∶=_𝑛∈ℤ^𝑟−1_𝑙=0𝑇_𝑛𝑎𝜑_𝑙(𝑥)²−_{𝑘≠0𝑛∈ℤ}^𝑟−1_𝑙=0𝑇_𝑛𝑎𝜑_𝑙(𝑥)𝑇_𝑛𝑎𝜑_𝑙𝑘𝑥+𝑏, (3.7) then 𝐻₁ and 𝐻₂ are 1/𝑏-periodic functions. Thus 1𝐵=𝑏sup_{𝑥∈ℝ𝑘∈ℤ}^𝑟−1_{𝑙=0𝑛∈ℤ}𝑇_𝑛𝑎𝜑_𝑙(𝑥)𝑇_𝑛𝑎𝜑_𝑙𝑘𝑥+𝑏1<∞,𝐴=𝑏inf_{𝑥∈ℝ𝑛∈ℤ}^𝑟−1_𝑙=0𝑇_𝑛𝑎𝜑_𝑙(𝑥)²−_{𝑘≠0𝑛∈ℤ}^𝑟−1_𝑙=0𝑇_𝑛𝑎𝜑_𝑙(𝑥)𝑇_𝑛𝑎𝜑_𝑙𝑘𝑥+𝑏>0. (3.8) Define 𝑔_𝑛(𝑥)∶=𝑇_𝑘𝑎𝜑_𝑙(𝑥), (3.9) where 𝑛=𝑙+𝑟𝑘 and 𝑙=0,…,𝑟−1. Then, one obtains from (3.8) that 1𝐵=𝑏sup_{𝑥∈ℝ𝑘∈ℤ𝑛∈ℤ}𝑔_𝑛(𝑥)𝑔_𝑛𝑘𝑥−𝑏1<∞,𝐴=𝑏inf_{𝑥∈ℝ𝑛∈ℤ}𝑔_𝑛(𝑥)²−_{𝑘≠0𝑛∈ℤ}𝑔_𝑛(𝑥)𝑔_𝑛𝑘𝑥−𝑏>0, (3.10) respectively. Note that 𝐿²(𝕊)⊂𝐿²(ℝ). By Lemma 2.4, one obtains the results.

Remark 3.3. Theorem 3.2 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 {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,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,…,𝜑_𝑟−1, the corresponding translation parameters 𝑎, 𝑏₀, …, 𝑏_𝑟−1, and the subset 𝕊.

Theorem 3.4. Let 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎,𝑏₀,…,𝑏_𝑟−1>0. Assume that {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a multigenerator Gabor frame for 𝐿²(𝕊) with bounds 𝐴 and 𝐵. Then, 𝐴𝜒_𝕊(𝑥)⩽^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²⩽𝐵𝜒_𝕊(𝑥),a.e.ℝ. (3.11)

Proof. Firstly, note that 𝕊 is a 𝑎ℤ-periodic subset of ℝ. Therefore, 𝜑_𝑙(⋅−𝑛𝑎)∈𝐿²(𝕊) for all 𝑛∈ℤ and 𝑙=0,…,𝑟−1. Thus ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=0,a.e.ℝ⧵𝕊. (3.12)
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 ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²>𝐵,a.e.Δ. (3.13) Similar to the discussion in [11, Proposition  8.3.2], we can assume that ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²>𝐵+𝜖,a.e.Δ (3.14) for small 𝜖>0. Note that 𝕊 is a 𝑎ℤ-periodic subset of ℝ. We can assume further that Δ⊂𝕊₀. Define 1𝑏⁰1∶=min𝑏_𝑙,Δ,𝑙=0,⋯,𝑟−1_𝑘=Δ𝑘−1𝑏⁰,𝑘𝑏⁰,𝑘∈ℤ. (3.15) Then, there exists 𝑘₀∈ℤ such that 𝜇(Δ_𝑘0)>0 and ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²>𝐵+𝜖,a.e.Δ_𝑘0. (3.16) If not, that is 𝜇(Δ_𝑘)=0 for all 𝑘∈ℤ, then 𝜇(Δ)=𝜇_𝑘∈ℤΔ_𝑘=_𝑘∈ℤ𝜇Δ_𝑘=0. (3.17) 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 function 𝑓=𝜒_Δ and note that ‖𝑓‖²=𝜇(Δ). Then for any 𝑛∈ℤ, the function 𝑓𝑇_𝑛𝑎𝜑_𝑙 has support in Δ. Since the functions {𝑏_𝑙𝐸_𝑚𝑏𝑙,𝑚∈ℤ} constitute an orthonormal basis for 𝐿²(𝐼) for every interval 𝐼 of length 1/𝑏_𝑙 for fixed 𝑙=0,…,𝑟−1, we have _𝑚∈ℤ𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=_𝑚∈ℤ𝑓𝑇_𝑛𝑎𝜑_𝑙,𝐸_𝑚𝑏𝑙²=1𝑏_𝑙^∞_−∞𝑓(𝑥)²𝜑_𝑙(𝑥−𝑛𝑎)²𝑑𝑥. (3.18) Thus, _{𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=^∞_−∞𝑓(𝑥)²1𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²𝑑𝑥. (3.19) Therefore, ^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=^∞_−∞𝑓(𝑥)^2𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²>𝑑𝑥_Δ()()𝜇(Δ)=(𝑓𝐵+𝜖𝑑𝑥=𝐵+𝜖𝐵+𝜖)²>𝐵𝑓². (3.20) This contradicts to the assumption that 𝐵 is an upper frame bound for{𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_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_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,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. Let 𝑓 be a bounded measure function with compact support and 𝑔∈𝐿²(𝕊). Then _{𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝑔²=1𝑏_ℝ𝑓(𝑥)²_𝑛∈ℤ𝑔(x−𝑛𝑎)²+1𝑑𝑥𝑏_{𝑘∈ℤℝ}𝑓(𝑥)𝑓𝑘𝑥−𝑏_𝑛≠0𝑔(𝑥−𝑛𝑎)𝑔𝑘𝑥−𝑛𝑎−𝑏𝑑𝑥 (4.1) for given 𝑎,𝑏>0.

Theorem 4.2. Let 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎,𝑏₀,…,𝑏_𝑟−1>0. Moreover, assume that {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a tight frame for 𝐿²(𝕊) with 𝐴=1. Then, ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=𝜒_𝕊(𝑥),a.e.ℝ. (4.2) Moreover, if 𝑏₀=⋯=𝑏_𝑟−1 and denote 𝑏₀ by 𝑏, then ^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=𝑏𝜒_𝕊(𝑥),_{𝑟−1𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏=0,for𝑘≠0 (4.3)(4.4) hold a.e. in ℝ.

Proof. Define 1𝑏⁰1∶=min𝑏_𝑙,𝑙=0,…,𝑟−1. (4.5) Consider 𝐶𝐿²(𝕊)∶=𝑓∶𝑓∈𝐿²(𝕊)1andsupp𝑓⊂0,𝑏⁰∩𝕊. (4.6) Note that {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a tight frame for 𝐿²(𝕊) with 𝐴=1. Then ^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=𝑓²,∀𝑓∈𝐶𝐿²(𝕊). (4.7) Again, we obtain from Lemma 4.1 that _𝑚∈ℤ𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=_{𝑚∈ℤℝ}𝑓(𝑥)𝜑_𝑙(𝑥−𝑛𝑎)𝑒^{−2𝜋𝑚𝑏𝑙𝑥}𝑑𝑥²=1𝑏_𝑙ℝ𝑓(𝑥)𝜑_𝑙(𝑥−𝑛𝑎)²=1𝑑𝑥,𝑏_𝑙^1/𝑏0₀𝑓(𝑥)𝜑_𝑙(𝑥−𝑛𝑎)²𝑑𝑥,∀𝑓∈𝐶𝐿²(𝕊) (4.8) for fixed 𝑙 and 𝑛. Thus, ^1/𝑏0₀𝑓(𝑥)²𝑑𝑥=^1/𝑏0₀𝑓(𝑥)^2𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²𝑑𝑥 (4.9) for any 𝑓∈𝐶𝐿²(𝕊). This implies that ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=1,a.e.0,1/𝑏⁰∩𝕊. (4.10) Note that 𝕊=_𝑘∈ℤ𝑘𝑏⁰,𝑘+1𝑏⁰∩𝕊, (4.11) 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 _{𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙²=1𝑏_ℝ𝑓(𝑥)²_𝑛∈ℤ𝜑_𝑙(𝑥−𝑛𝑎)²+1𝑑𝑥𝑏_𝑘≠0ℝ𝑓(𝑥)𝑓𝑘𝑥−𝑏_𝑛∈ℤ𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏𝑑𝑥. (4.12) Then, ^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙²=_ℝ𝑓(𝑥)^2𝑟−1_𝑙=01𝑏_𝑛∈ℤ𝜑_𝑙(𝑥−𝑛𝑎)²+1𝑑𝑥𝑏_𝑘≠0ℝ𝑓(𝑥)𝑓𝑘𝑥−𝑏^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏𝑑𝑥. (4.13) This, together with (4.3), follows that _𝑘≠0ℝ𝑓(𝑥)𝑓𝑘𝑥−𝑏_𝑛∈ℤ𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏𝑑𝑥=0. (4.14) 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, ^∞_𝑘=1Re_ℝ𝑓(𝑥)𝑓𝑘𝑥−𝑏𝐺_𝑘(𝑥)𝑑𝑥=0, (4.15) where 𝐺_𝑘(𝑥)∶=∑^𝑟−1_𝑙=0∑_𝑛∈ℤ𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙(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, 𝜑_𝑙(𝑥−𝑛𝑎)=0,∀𝑛∈ℤ. (4.16) Thus, ^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏=0,∀𝑘≠0. (4.17)
Case  2. 𝑥−𝑘/𝑏∉𝕊 for fixed 𝑘∈ℤ⧵{0}. Then 𝑥−𝑘/𝑏−𝑛𝑎∉𝕊 for all 𝑛∈ℤ. Therefore, ^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏=0. (4.18)
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 ^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏=0. (4.19) Now consider 𝜇(Δ⁰∩Δ^′)>0. Define a function 𝑓∈𝐿²(𝕊) by 𝑓(𝑥)𝑒∶=^{−arg𝐺𝑘0(𝑥)},𝑥∈Δ⁰∩Δ^′1,𝑥∈Δ⁰∩Δ^′−𝑘𝑏,0,otherwise. (4.20) Then, by (4.15), 0=^∞_𝑘=1Re_ℝ𝑓(𝑥)𝑓𝑘𝑥−𝑏𝐺_𝑘(𝑥)𝑑𝑥=Re_ℝ𝑓(𝑥)𝑓𝑘𝑥−⁰𝑏𝐺_𝑘0(𝑥)𝑑𝑥=_Δ0∩Δ′𝐺_𝑘0(𝑥)𝑑𝑥. (4.21) It follows that 𝐺_𝑘0(𝑥)=0, a.e. on Δ⁰∩Δ^′. Since 𝐼 is an arbitrary interval of length at most 1/𝑏, we conclude that 𝐺_𝑘0(𝑥)=0, a.e. in 𝕊. A direct computation shows that 𝐺_−𝑘0(𝑥)=𝐺_𝑘0𝑘𝑥+⁰𝑏. (4.22) Thus, we obtain the desired results.

To proceed further, we need use the following symbols. For 𝑏₀,…,𝑏_𝑟−1>0, define 1𝑏⁰1∶=min𝑏_𝑙,1,𝑙=0,…,𝑟−1𝑏¹1∶=min𝑏_𝑙∶𝑏_𝑙<𝑏⁰,⋮1,𝑙=1,…,𝑟−1𝑏^𝑞1∶=min𝑏_𝑙∶𝑏_𝑙<𝑏^𝑞−1,𝐼,𝑙=1,…,𝑟−1₀∶=𝑙∶𝑏_𝑙=𝑏⁰,𝐼,𝑙=0,…,𝑟−1₁∶=𝑙∶𝑏_𝑙=𝑏¹,⋮𝐼,𝑙=0,…,𝑟−1_𝑞∶=𝑙∶𝑏_𝑙=𝑏^𝑞,𝑙=0,…,𝑟−1. (4.23) Then there exists a unique nonnegative integer 𝑞₀ such that 𝐼_𝑞≠∅,for𝑞=0,…,𝑞₀,𝐼_𝑞1𝐼_𝑞2=∅,for𝑞₁≠𝑞₂,_𝑞0𝑞=0𝐼_𝑞={0,…,𝑟−1}. (4.24)

Theorem 4.3. Let 𝑞₀ be the unique nonnegative integer satisfying (4.24). Assume that 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎,𝑏₀,…,𝑏_𝑟−1>0 satisfy ^𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=𝜒_𝕊(𝑥),_{𝑙∈𝐼𝑞𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏_𝑙=0,for𝑘≠0,𝑞=0,…,𝑞₀ (4.25)(4.26) a.e. in ℝ. Then {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a tight frame for 𝐿²(𝕊) with 𝐴=1.

Proof . For fixed 𝑙=0,…,𝑟−1, we obtain from Lemma 4.1 that _{𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=1𝑏_𝑙ℝ𝑓(𝑥)²_𝑛∈ℤ𝜑_𝑙(𝑥−𝑛𝑎)²+1𝑑𝑥𝑏_{𝑙𝑘≠0ℝ}𝑓(𝑥)𝑓𝑘𝑥−𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏_𝑙𝑑𝑥. (4.27) Then, ^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=_ℝ𝑓(𝑥)^2𝑟−1_𝑙=01𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²𝑑𝑥+(∗), (4.28) where (∗)∶=_ℝ^𝑟−1_𝑙=01𝑏_𝑙𝑘≠0𝑓(𝑥)𝑓𝑘𝑥−𝑏_{𝑙𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏_𝑙𝑑𝑥. (4.29) This, together with (4.25), follows that ^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=_ℝ𝑓(𝑥)²𝑑𝑥+(∗). (4.30) Define 𝐺^𝑞_𝑘(𝑥)∶=_{𝑙∈𝐼𝑞𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏_𝑙,for𝑞=0,…,𝑞₀. (4.31) Then, we obtain from (4.26) that (∗)=_𝑘≠0ℝ^𝑞0_𝑞=01𝑏^𝑞𝑓(𝑥)𝑓𝑘𝑥−𝑏_𝑞𝐺^𝑞_𝑘(𝑥)𝑑𝑥=0. (4.32) This, together with (4.30), follows that ^𝑟−1_{𝑙=0𝑚,𝑛∈ℤ}𝑓,𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙²=_ℝ𝑓(𝑥)²𝑑𝑥=𝑓²,∀𝑓∈𝐿²(𝕊). (4.33) Therefore, {𝐸_𝑚𝑏𝑙𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a tight frame for 𝐿²(𝕊) with 𝐴=1.

Note that 𝑞₀=0 when 𝑏₀=𝑏₁=⋯=𝑏_𝑟−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. Let 𝜑₀,…,𝜑_𝑟−1∈𝐿²(𝕊) and 𝑎,𝑏>0. Then {𝐸_𝑚𝑏𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a tight frame for 𝐿²(𝕊) with 𝐴=1 if and only if ^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=𝑏𝜒_𝕊(𝑥),_{𝑟−1𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙𝑘𝑥−𝑛𝑎−𝑏=0,for𝑘≠0 (4.34) hold a.e. in ℝ.

Remark 4.5. If 𝕊=ℝ, then 𝕊 is 𝑎ℤ-periodic set for any given 𝑎>0. In this case, {𝐸_𝑚/𝑎𝑇_𝑛𝑎𝜑_𝑙,𝑚,𝑛∈ℤ,𝑙=0,…,𝑟−1} is a tight frame for 𝐿²(𝕊) with 𝐴=1 if and only if ^𝑟−1_{𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)²=1𝑎𝜒_𝕊(𝑥),_{𝑟−1𝑙=0𝑛∈ℤ}𝜑_𝑙(𝑥−𝑛𝑎)𝜑_𝑙(𝑥−𝑛𝑎−𝑘𝑎)=0,for𝑘≠0 (4.35) hold a.e. in ℝ.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11071152) and the Natural Science Foundation of Guangdong Province (Grant nos. 10151503101000025 and S2011010004511); this research 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).