Discrete Subspace Multiwindow Gabor Frames and Their Duals

Abstract

This paper addresses discrete subspace multiwindow Gabor analysis. Such a scenario can model many practical signals and has potential applications in signal processing. In this paper, using a suitable Zak transform matrix we characterize discrete subspace mixed multi-window Gabor frames (Riesz bases and orthonormal bases) and their duals with Gabor structure. From this characterization, we can easily obtain frames by designing Zak transform matrices. In particular, for usual multi-window Gabor frames (i.e., all windows have the same time-frequency shifts), we characterize the uniqueness of Gabor dual of type I (type II) and also give a class of examples of Gabor frames and an explicit expression of their Gabor duals of type I (type II).

1. Introduction

Let ℋ be a separable Hilbert space. An at most countable sequence {ℎ_𝑖}_𝑖∈ℐ in ℋ is called a frame for ℋ if there exist 0<𝐴≤𝐵<∞ such that 𝐴𝑓²≤_𝑖∈ℐ𝑓,ℎ_𝑖²≤𝐵𝑓²for𝑓∈ℋ, (1) where 𝐴 and 𝐵 are called frame bounds. The sequence {ℎ_𝑖}_𝑖∈ℐ is called a Bessel sequence in ℋ if the right-hand side inequality in (1) holds. In this case 𝐵 is called a Bessel bound. A frame for ℋ is said to be a Riesz basis if it ceases to be a frame for ℋ whenever an arbitrary element is removed. And in this case, the frame bounds are also called Riesz bounds. The fundamentals of frames can be found in [1–4]. We denote by ℤ the set of integers, by ℕ the set of positive integers, by {𝐞_𝑖} the vector in Euclidean spaces with the 𝑖th component being 1 and others being zero, and by ℕ_𝛾 the set {0,1,…,𝛾−1} for 𝛾∈ℕ. This paper addresses Gabor systems 𝐺(𝐠,𝐍,𝐌) of the form 𝐺(𝐠,𝐍,𝐌)=𝐸_𝑚/𝑀𝑇_𝑛𝑁𝑙𝑔_𝑙∶1≤𝑙≤𝐿,𝑚∈ℕ_𝑀,𝑛∈ℤ, (2) where 𝐿 is a fixed positive integer, 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), 𝐍=(𝑁₁,𝑁₂,…,𝑁_𝐿) with each 𝑁_𝑙∈ℕ, and 𝐌=(𝑀₁,𝑀₂,…,𝑀_𝐿) with each 𝑀_𝑙∈ℕ. Throughout this paper, we work under the following assumptions.

Assumption 1. 𝐿 is a positive integer.

Assumption 2. 𝑀₁=𝑀₂=⋯=𝑀_𝐿=𝑀, and 𝑁_𝑙/𝑀=𝑝_𝑙/𝑞_𝑙 with 𝑝_𝑙 and 𝑞_𝑙 being relatively prime positive integers for 1≤𝑙≤𝐿.

We always denote by 𝑁 the least common multiple of 𝑁_𝑙 with 1≤𝑙≤𝐿, by 𝜆_𝑙 the positive integer satisfying 𝑁=𝜆_𝑙𝑁_𝑙 for each 1≤𝑙≤𝐿, by 𝑝 and 𝑞 two relatively prime positive integers satisfying 𝑁/𝑀=𝑝/𝑞, and by 𝑄 the number 𝑄=𝑞∑^𝐿_𝑙=1𝜆_𝑙. Obviously, they are all uniquely determined by Assumptions 1 and 2.

Remark 3. We do not lose generality by making Assumptions 1 and 2. Let us check a general Gabor system of the form 𝐺(𝐠,𝐍,𝐌)=𝐸_{𝑚𝑙/𝑀𝑙}𝑇_𝑛𝑁𝑙𝑔_𝑙∶1≤𝑙≤𝐿,𝑚_𝑙∈ℕ_𝑀𝑙,𝑛∈ℤ (3) with 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) satisfying 𝑔_𝑙∈𝑙²(ℤ), 𝐍=(𝑁₁,𝑁₂,…,𝑁_𝐿), and 𝐌=(𝑀₁,𝑀₂,…,𝑀_𝐿)(𝑀₁,𝑀₂,…,𝑀_𝐿 are not necessarily all the same). Suppose 𝑀 is the greatest common divisor of 𝑀₁,𝑀₂,…,𝑀_𝐿, and 𝑀_𝑙=𝛼_𝑙𝑀 with 𝛼_𝑙∈ℕ for 1≤𝑙≤𝐿. Define 𝑔^(𝛽𝑙)_𝑙(⋅)=𝑒^{2𝜋𝑖(𝛽𝑙/𝑀𝑙)⋅}𝑔_𝑙(⋅) for 1≤𝑙≤𝐿 and 𝛽_𝑙∈ℕ_𝛼𝑙. Then, by a simple computation, 𝐺(𝐠,𝐍,𝐌) is a frame for a closed subspace ℳ of 𝑙²(ℤ) if and only if the Gabor system 𝐸_𝑚/𝑀𝑇_𝑛𝑁𝑙𝑔^(𝛽𝑙)_𝑙∶1≤𝑙≤𝐿,𝛽_𝑙∈ℕ_𝛼𝑙,𝑚∈ℕ_𝑀,𝑛∈ℤ (4) is a frame for ℳ. Therefore, the study of (3) is reduced to the study of (4).

We denote by ℳ(𝐠,𝐍,𝐌) the closed linear span of 𝐺(𝐠,𝐍,𝐌). A Gabor system 𝐺(𝐠,𝐍,𝐌) of the form (2) is called a mixed multi-window Gabor system if 𝐿>1 since 𝑁₁,𝑁₂,… and 𝑁_𝐿 are not necessarily all the same. (In particular, this system is the usual multi-window Gabor system when 𝑁₁=𝑁₂=⋯=𝑁_𝐿). It is called a subspace mixed multi-window Gabor frame if it is a frame for ℳ(𝐠,𝐍,𝐌) and 𝐿>1. Similarly, a Gabor system 𝐺(𝐠,𝐍,𝐌) is called a single-window Gabor system if 𝐿=1 and called a single-window subspace Gabor frame if it is a frame for ℳ(𝐠,𝐍,𝐌) and 𝐿=1.

For a Bessel sequence 𝐺(𝐠,𝐍,𝐌) in 𝑙²(ℤ) of the form (2), we define the associated synthesis operator 𝒯_𝐠∶𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿)→𝑙²(ℤ) by 𝒯_𝐠𝑐=^𝐿_{𝑙=1𝑚∈ℕ𝑀𝑛∈ℤ}𝑐_{𝑙,𝑚,𝑛}𝐸_𝑚/𝑀𝑇_𝑛𝑁𝑙𝑔_𝑙 (5) for 𝑐=(𝑐₁,𝑐₂,…,𝑐_𝐿)∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿). Then it is a bounded operator, and its adjoint operator 𝒯^∗_𝐠 (so-called analysis operator) is given by 𝒯^∗_𝐠𝑓=𝑐₁𝑓,𝑐₂𝑓,…,𝑐_𝐿𝑓for𝑓∈𝑙²(ℤ), (6) where 𝑐_𝑙(𝑓)={⟨𝑓,𝐸_𝑚/𝑀𝑇_𝑛𝑁𝑙𝑔_𝑙⟩}_{𝑚∈ℕ𝑀,𝑛∈ℤ} for each 1≤𝑙≤𝐿. For a Bessel sequence 𝐺(𝐡,𝐍,𝐌) in 𝑙²(ℤ) with 𝐡=(ℎ₁,ℎ₂,…,ℎ_𝐿), we associate it with 𝒯_𝐡 similarly. Write 𝒮_𝐡,𝐠=𝒯_𝐠𝒯^∗_𝐡; that is, 𝒮_𝐡,𝐠𝑓=^𝐿_{𝑙=1𝑚∈ℕ𝑀𝑛∈ℤ}𝑓,𝐸_𝑚/𝑀𝑇_𝑛𝑁𝑙ℎ_𝑙𝐸_𝑚/𝑀𝑇_𝑛𝑁𝑙𝑔_𝑙 (7) for 𝑓∈𝑙²(ℤ). Let 𝐺(𝐠,𝐍,𝐌) be a frame for ℳ(𝐠,𝐍,𝐌) and let 𝐺(𝐡,𝐍,𝐌) be a Bessel sequence in 𝑙²(ℤ). 𝐺(𝐡,𝐍,𝐌) is called an oblique Gabor dual for 𝐺(𝐠,𝐍,𝐌) if 𝒮_𝐡,𝐠𝑓=𝑓for𝑓∈ℳ(𝐠,𝑁,𝑀). (8) Here it is not required that each ℎ_𝑙 belongs to ℳ(𝐠,𝐍,𝐌). In particular, an oblique Gabor dual 𝐺(𝐡,𝐍,𝐌) for 𝐺(𝐠,𝐍,𝐌) is said to be a Gabor dual of type 𝐼 for 𝐺(𝐠,𝐍,𝐌) if ℎ_𝑙∈ℳ(𝐠,𝐍,𝐌) for each 1≤𝑙≤𝐿 and is said to be a Gabor dual of type II for 𝐺(𝐠,𝐍,𝐌) if range(𝒯^∗_𝐡)⊂range(𝒯^∗_𝐠). These notions of duals are a generalization of the ones in [5, 6]. They are borrowed from [7, 8] which dealt with Gabor frames in 𝐿²(ℝ). For a Gabor dual of type II, it is not required to be in ℳ(𝐠,𝐍,𝐌), but a containment relation between the ranges of analysis operators is required. Observe that the canonical daul 𝐺(𝒮⁻¹_𝐠,𝐠𝐠,𝐍,𝐌) for 𝐺(𝐠,𝐍,𝐌) belongs to any one of the three Gabor duals.

In the past more than twenty years, the theory of frames has been growing rapidly. Gabor frames are a class of important frames among all kinds of frames. For continuous Gabor frames, single-window Gabor frames for 𝐿²(ℝ) have been studied extensively [2, 3, 9, 10]; multi-window Gabor frames for 𝐿²(ℝ) were firstly studied by Zibulski and Zeevi in [11], and then by others in [12–17]; single-window subspace (of 𝐿²(ℝ)) Gabor frames were studied in [7, 8, 18–23]. In digital signal processing, one usually encounters discrete signals instead of continuous ones. One can obtain discrete Gabor frames for 𝑙²(ℤ) via Gabor frames for 𝐿²(ℝ) through sampling under certain additional assumptions [24–26], though these assumptions are artificial and too technical. So it should be more reasonable to consider Gabor frames in 𝑙²(ℤ) without referring to frames in 𝐿²(ℝ). Intuitively, the general theory of discrete Gabor analysis is similar to the continuous case. Sometimes major differences occur. In 1989, Heil in [27] showed that while Gabor frames in the continuous case are bases only if they are generated by functions that are not smooth or have poor decay, it is possible in the discrete case to construct Gabor frames that are bases and are generated by sequences with good decay. The sampled Gaussian gives an example of such a signal. Also due to its good potential for digital signal processing, since then, Gabor analysis on 𝑙²(ℤ) has attracted many researchers (see [2, 3, 24–34] and the references therein). In [6, 35], single-window Gabor frames on discrete periodic subsets of ℤ were investigated. It is well known that a single-window Gabor expansion is not enough to analyze the dynamic time-frequency contents of signals that contain a wide range of spatial and frequency components. A multi-window Gabor expansion is capable of extracting local frequencies in an adaptive manner, in which wide windows are responsible for slow-changing components and narrow windows are designed to extract transient and rapid-changing components of a signal. See [15] and the references therein for details. In addition, Example 38 and the arguments before it in the last section show that, for Gabor duals, multi-window Gabor frames behave differently from single-window ones. Motivated by the above works, we in this paper study subspace Gabor frame of the form (2) under Assumptions 1 and 2.

The rest of this paper is organized as follows. Section 2 is an auxiliary one to following sections. In this section, we introduce the notion of Zak transform matrix associated with a Gabor system of the form (2) and investigate its properties. In terms of Zak transform matrices, we in Section 3 characterize subspace multi-window Gabor frames (Riesz bases, orthonormal bases) and in Section 4 Gabor duals of type I (type II) for subspace multi-window Gabor frames. Section 5 focuses on subspace multi-window Gabor frames with all the windows having the same time-frequency shifts. We characterize the uniqueness of Gabor duals of type I (type II) and obtain a class of examples of subspace multi-window Gabor frames (Riesz bases, orthonormal bases) and their Gabor duals of type I (type II) (see Theorems 36 and 37).

2. Zak Transforms and Zak Transform Matrices

This section focuses on some properties of Zak transforms and Zak transform matrices, which is an auxiliary one to following sections. We start with some notations and notions. For 𝑠, 𝑡∈ℕ, we denote by ℳ_𝑠,𝑡 the set of 𝑠×𝑡 complex matrices. Given 𝛾∈ℕ, two subsets 𝑆₁ and 𝑆₂ of ℤ are said to be 𝛾ℤ congruent if there exists a sequence of subsets {𝑆_1,𝑘}_𝑘∈ℤ of 𝑆₁ such that ⋃_𝑘∈ℤ𝑆_1,𝑘=𝑆₁, 𝑆_1,𝑘1∩𝑆_1,𝑘2=∅ for 𝑘₁≠𝑘₂ in ℤ and that 𝑆₂=⋃_𝑘∈ℤ(𝑆_1,𝑘+𝛾𝑘), (𝑆_1,𝑘1+𝛾𝑘₁)∩(𝑆_1,𝑘2+𝛾𝑘₂)=∅ for 𝑘₁≠𝑘₂ in ℤ. If 𝑆₁ or 𝑆₂ is a finite set in addition, only finitely many 𝑆_1,𝑘 in the above are nonempty, while the others are empty. For two vectors 𝑥 and 𝑦 in a Hilbert space ℋ, we always write their inner product as ⟨𝑥,𝑦⟩ instead of ⟨𝑥,𝑦⟩_ℋ when it causes no confusions. Let 𝐸 be a finite set in ℤ and let Γ∈ℕ. We denote by 𝐿²(𝐸×[0,1),ℂ^Γ) the Hilbert space of vector-valued functions 𝑓 satisfying 𝑓(𝑗,⋅)∈𝐿²([0,1),ℂ^Γ) for each 𝑗∈𝐸 endowed with the inner product 𝑓,𝑓=_𝑗∈𝐸¹₀𝑓𝑗,𝑣,𝑓𝑗,𝑣𝑑𝑣 (9) for 𝑓, 𝑓∈𝐿²(𝐸×[0,1),ℂ^Γ), where ⟨𝑓(𝑗,𝑣),𝑓(𝑗,𝑣)⟩ denotes their inner product in ℂ^Γ; by 𝑙²(𝐸×ℤ,ℂ^Γ) the Hilbert space of sequences 𝑐=(𝑐₁,𝑐₂,…,𝑐_Γ) satisfying 𝑐_𝛾∈𝑙²(𝐸×ℤ) endowed with the inner product ⟨𝑐,𝑐⟩̃=^Γ_𝛾=1𝑐_𝛾,𝑐_𝛾̃ (10) for 𝑐, 𝑐̃∈𝑙²(𝐸×ℤ,ℂ^Γ), where ⟨𝑐_𝛾,𝑐_𝛾̃⟩ denotes their inner product in 𝑙²(𝐸×ℤ) and by 𝑙₀(𝐸×ℤ,ℂ^Γ) the set of sequences 𝑐=(𝑐₁,𝑐₂,…,𝑐_Γ) with each 𝑐_𝛾 being a finitely supported sequence defined on 𝐸×ℤ. For simplicity, we write 𝐿²(𝐸×[0,1)) for 𝐿²(𝐸×[0,1),ℂ^Γ), 𝑙²(𝐸×ℤ) for 𝑙²(𝐸×ℤ,ℂ^Γ), and 𝑙₀(𝐸×ℤ) for 𝑙₀(𝐸×ℤ,ℂ^Γ) when Γ=1.

For 𝑓∈𝑙²(ℤ), define the discrete Zak transform 𝒵_𝑞𝑁𝑓 of 𝑓 by 𝒵_𝑞𝑁𝑓𝑗,𝑣∶=_ℓ∈ℤ𝑓𝑗+ℓ𝑞𝑁𝑒^{2𝜋𝑖ℓ𝑣} (11) for 𝑗∈ℤ and a.e. 𝑣∈ℝ. It is easy to check that 𝒵_𝑞𝑁 has quasi-periodicity: 𝒵_𝑞𝑁𝑓𝑗+𝑘𝑞𝑁,𝑣+ℓ=𝑒^{−2𝜋𝑖𝑘𝑣}𝒵_𝑞𝑁𝑓𝑗,𝑣 (12) for 𝑘,ℓ,𝑗∈ℤ and a.e. 𝑣∈ℝ.

Definition 4. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), we associate it with a matrix-valued function Ψ_𝐠(𝑗,𝑣)∶ℤ×ℝ→ℳ_𝑄,𝑝 by Ψ_𝐠=𝐺𝑗,𝑣¹𝐺𝑗,𝑣₂⋮𝐺𝑗,𝑣_𝐿𝑗,𝑣, (13) where 𝑄∶=𝑞∑^𝐿_𝑙=1𝜆_𝑙 and 𝐺_𝑙(𝑗,𝑣) is a block matrix of the form 𝐺_𝑙=𝒢𝑗,𝑣^𝑙𝒢𝑗,𝑣_𝑙𝑗−𝑁_𝑙⋮𝒢,𝑣_𝑙𝑗−𝜆_𝑙−1𝑁_𝑙,𝑣 (14) with 𝒢_𝑙(𝑗,𝑣)∶ℤ×ℝ→ℳ_𝑞,𝑝 for 1≤𝑙≤𝐿 and 𝒢_𝑙𝑗,𝑣_𝑟,𝑘=𝒵_𝑞𝑁𝑔_𝑙𝑗−𝑟𝑁+𝑘𝑀,𝑣 (15) for 𝑟∈ℕ_𝑞, 𝑘∈ℕ_𝑝.

By the quasi-periodicity of 𝒵_𝑞𝑁, an arbitrary 𝑓∈𝑙²(ℤ) is uniquely determined by the values of (𝒵_𝑞𝑁𝑓)(⋅,⋅) on 𝑆×[0,1) with 𝑆 being a set 𝑞𝑁ℤ congruent to ℕ_𝑞𝑁. So an arbitrary function 𝐹∈𝐿²(𝑆×[0,1)) determines a unique 𝑓∈𝑙²(ℤ) by 𝒵_𝑞𝑁𝑓𝑗,𝑣=𝐹𝑗,𝑣for𝑗,𝑣∈𝑆×[0,1). (16) Let us make another assumption that 𝑁₁=𝑁₂=⋯=𝑁_𝐿 in Definition 4. Then Ψ_𝐠(𝑗,𝑣) is an 𝐿𝑞×𝑝-matrix-valued function of the form Ψ_𝐠=𝒢𝑗,𝑣¹𝒢𝑗,𝑣₂⋮𝒢𝑗,𝑣_𝐿,𝒢𝑗,𝑣_𝑙𝑗,𝑣_𝑟,𝑘=𝒵_𝑞𝑁𝑔_𝑙𝑗−𝑟𝑁+𝑘𝑀,𝑣 (17) for 1≤𝑙≤𝐿, 𝑟∈ℕ_𝑞, and 𝑘∈ℕ_𝑝. Also observe that ℕ_𝑀/𝑞−𝑁ℕ_𝑞+𝑀ℕ_𝑝 is 𝑞𝑁ℤ congruent to ℕ_𝑞𝑁. We have if 𝑁₁=𝑁₂=⋯=𝑁_𝐿, then an arbitrary function 𝑀(𝑗,𝑣)∶ℕ_𝑀/𝑞×[0,1)→ℳ_{𝐿𝑞,𝑝} with all entries in 𝐿²(ℕ_𝑀/𝑞×[0,1)) determines a unique 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ) by Ψ_𝐠𝑗,𝑣=𝑀𝑗,𝑣for𝑗∈ℕ_𝑀/𝑞,a.e.𝑣∈[0,1). (18) But if 𝑁_𝑙, 1≤𝑙≤𝐿, are not all the same, a function 𝑀(𝑗,𝑣)∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑄,𝑝 with all entries in 𝐿²(ℕ_𝑀/𝑞×[0,1)) does not necessarily determine a 𝐠. We show it by an example. In this case, there exists 1≤𝑙≤𝐿 such that 𝜆_𝑙>1. We may as well assume that 𝜆₁>1. Choose 𝑀=𝑀𝑗,𝑣¹𝑀𝑗,𝑣₂⋮𝑀𝑗,𝑣_𝐿,𝑀𝑗,𝑣_𝑙=𝑀𝑗,𝑣^𝑙,0𝑀𝑗,𝑣_𝑙,1⋮𝑀𝑗,𝑣_{𝑙,𝜆𝑙−1}𝑗,𝑣 (19) for 1≤𝑙≤𝐿 and (𝑗,𝑣)∈ℕ_𝑀/𝑞×[0,1) such that every entry of 𝑀(𝑗,𝑣) belongs to 𝐿²(ℕ_𝑀/𝑞×[0,1)) and that 𝑀_1,0𝑗,𝑣≠0,𝑀_1,1𝑗,𝑣=0for𝑗,𝑣∈ℕ_𝑀/𝑞×[0,1). (20) Suppose there exists 𝐠 such that Ψ_𝐠(𝑗,𝑣)=𝑀(𝑗,𝑣) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1). Then 𝒢₁𝑗,𝑣=𝑀_1,0,𝒢𝑗,𝑣₁𝑗−𝑁₁,𝑣=𝑀_1,1𝑗,𝑣 (21)(22) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1), where 𝒢₁𝑗,𝑣_𝑟,𝑘=𝒵_𝑞𝑁𝑔₁,𝒢𝑗−𝑟𝑁+𝑘𝑀,𝑣₁𝑗−𝑁₁,𝑣_𝑟,𝑘=𝒵_𝑞𝑁𝑔₁𝑗−𝑁₁−𝑟𝑁+𝑘𝑀,𝑣 (23) for (𝑟,𝑘)∈ℕ_𝑞×ℕ_𝑝. Since the sets ℕ_𝑀/𝑞−𝑁ℕ_𝑞+𝑀ℕ_𝑝 and ℕ_𝑀/𝑞−𝑁ℕ_𝑞+𝑀ℕ_𝑝−𝑁₁ are both 𝑞𝑁ℤ congruent to ℕ_𝑞𝑁, we have 𝒵_𝑞𝑁𝑔₁(𝑗₀,𝑣)≠0 for some 𝑗₀∈ℤ and 𝑣∈𝐸⊂[0,1) with |𝐸|>0 by (20), (21), and the quasi-periodicity of 𝒵_𝑞𝑁, while 𝒵_𝑞𝑁𝑔₁(𝑗,𝑣)=0 for 𝑗∈ℤ and a.e. 𝑣∈[0,1) by (20), (22), and the quasi-periodicity of 𝒵_𝑞𝑁. This is a contradiction. Therefore, we must be careful when we define 𝐠 by a function 𝑀(𝑗,𝑣)∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑄,𝑝 if 𝑁_𝑙, 1≤𝑙≤𝐿, are not all the same.

Definition 5. Define 𝐙_𝑞𝑁∶𝑙²(ℤ)→𝐿²(ℕ_𝑀×[0,1),ℂ^𝑝) by 𝐙_𝑞𝑁𝑓=𝒵𝑗,𝑣^𝑞𝑁𝑓𝒵𝑗,𝑣_𝑞𝑁𝑓⋮𝒵𝑗+𝑀,𝑣_𝑞𝑁𝑓𝑗+𝑝−1𝑀,𝑣for𝑗,𝑣∈ℕ_𝑀×[0,1). (24)

By the quasi-periodicity of 𝒵_𝑞𝑁 and [35, Theorem 2.1], we have the following lemmas.

Lemma 6. 𝐙_𝑞𝑁 is a unitary operator from 𝑙²(ℤ) onto 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑝), and 𝒵_𝑞𝑁 is a unitary operator from 𝑙²(ℤ) onto 𝐿²(𝐸×[0,1)) for an arbitrary subset 𝐸 of ℤ which is 𝑞𝑁ℤ congruent to ℕ_𝑞𝑁.

Definition 7. Define the Fourier transform ℱ∶𝑙²(ℕ_𝑀×ℤ)→𝐿²(ℕ_𝑀×[0,1)) by =1ℱ𝑐𝑗,𝑣√𝑀_{𝑚∈ℕ𝑀𝑛∈ℤ}𝑐_𝑚,𝑛𝑒^{2𝜋𝑖(𝑚/𝑀)𝑗}𝑒^{2𝜋𝑖𝑛𝑣} (25) for 𝑐∈𝑙²(ℕ_𝑀×ℤ), 𝑗∈ℕ_𝑀, and a.e. 𝑣∈[0,1).

Definition 8. Define 𝒥∶𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿)→𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄) by 𝒥𝑐𝑗,𝑣=𝒞𝑗,𝑣 (26) for 𝑐=(𝑐₁,𝑐₂,…,𝑐_𝐿)∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿), 𝑗∈ℕ_𝑀, and a.e. 𝑣∈ℝ, where 𝒞=𝒞𝑗,𝑣¹𝒞𝑗,𝑣₂⋮𝒞𝑗,𝑣_𝐿,𝒞𝑗,𝑣_𝑙=𝒞𝑗,𝑣^𝑙(0)𝒞𝑗,𝑣_𝑙(1)⋮𝒞𝑗,𝑣_{(𝜆𝑙𝑙−1)},𝒞𝑗,𝑣_{(𝜏𝑙)𝑙}=∑𝑗,𝑣^{𝑚∈ℕ𝑀}_𝑛∑_∈ℤ𝑐_{𝑙,𝑚,𝑛𝑞𝜆𝑙+𝜏𝑙}𝑒^{2𝜋𝑖(𝑚/𝑀)𝑗}𝑒^{2𝜋𝑖𝑛𝑣}_𝑚∈∑_ℕ𝑀∑_𝑛∈ℤ𝑐_{𝑙,𝑚,(𝑛𝑞+1)𝜆𝑙+𝜏𝑙}𝑒^{2𝜋𝑖(𝑚/𝑀)𝑗}𝑒^{2𝜋𝑖𝑛𝑣}⋮_𝑚∈∑_ℕ𝑀∑_𝑛∈ℤ𝑐_{𝑙,𝑚,(𝑛𝑞+𝑞−1)𝜆𝑙+𝜏𝑙}𝑒^{2𝜋𝑖(𝑚/𝑀)𝑗}𝑒^{2𝜋𝑖𝑛𝑣} (27) for 1≤𝑙≤𝐿 and 𝜏_𝑙∈ℕ_𝜆𝑙.

Similarly, for an arbitrary 𝑑∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿), we associate it with 𝒟(𝑗,𝑣).

Lemma 9. (i)  ℱ is a unitary operator from 𝑙²(ℕ_𝑀×ℤ) onto 𝐿²(ℕ_𝑀×[0,1));
(ii)  (1/√𝑀)𝒥 is a unitary operator from 𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿) onto 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄).

Proof. Since {(1/√𝑀)𝑒^{2𝜋𝑖(𝑚/𝑀)𝑗}𝑒^{2𝜋𝑖𝑛𝑣}∶𝑚∈ℕ_𝑀,𝑛∈ℤ} is an orthonormal basis for 𝐿²(ℕ_𝑀×[0,1)), we have (i). (ii) is an immediate consequence of (i).

Lemma 10. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with 𝑔_𝑙∈𝑙²(ℤ) for 1≤𝑙≤𝐿, one has(i)𝒵_𝑞𝑁(𝐸_𝑚/𝑀𝑇_{𝑛𝑞𝑁+𝜏𝑙𝑁𝑙+𝑟𝑁}𝑔_𝑙)(𝑗,𝑣)=𝑒^{2𝜋𝑖(𝑚/𝑀)𝑗}𝑒^{2𝜋𝑖𝑛𝑣}𝒵_𝑞𝑁𝑔_𝑙(𝑗−𝜏_𝑙𝑁_𝑙−𝑟𝑁,𝑣) for 𝑟∈ℕ_𝑞;(ii)𝐙_𝑞𝑁𝑓𝑗,𝑣=Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣 (28) for 𝑓=𝒯_𝐠𝑐 with 𝑐∈𝑙₀(ℕ_𝑀×ℤ,ℂ^𝐿), where 𝒯_𝐠𝑐 is as in (5).

Proof. (i) can be obtained by a direct computation. Applying (i), we have 𝒵_𝑞𝑁𝑓𝑗,𝑣=𝒵_𝑞𝑁^𝐿_{𝑙=1𝜏𝑙∈ℕ𝜆𝑙𝑟∈ℕ𝑞𝑚∈ℕ𝑀𝑛∈ℤ}𝑐_{𝑙,𝑚,(𝑛𝑞+𝑟)𝜆𝑙+𝜏𝑙}×𝐸_𝑚/𝑀𝑇_{𝑛𝑞𝑁+𝜏𝑙𝑁𝑙+𝑟𝑁}𝑔_𝑙=^𝐿_{𝑙=1𝜏𝑙∈ℕ𝜆𝑙𝑟∈ℕ𝑞}𝒞^(𝜏𝑙)_𝑙𝑗,𝑣_𝑟𝒵_𝑞𝑁𝑔_𝑙𝑗−𝜏_𝑙𝑁_𝑙−𝑟𝑁,𝑣 (29) for (𝑗,𝑣)∈ℤ×ℝ. This implies (ii).

Remark 11. When 𝐺(𝐠,𝐍,𝐌) in Lemma 10 is a Bessel sequence in 𝑙²(ℤ), by the same procedure as the above we can prove (28) holds for 𝑓=𝒯_𝐠𝑐 with 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿).

Lemma 12. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with 𝑔_𝑙∈𝑙²(ℤ) for 1≤𝑙≤𝐿, one has Ψ_𝐠𝑀𝑗+𝑞𝑛,𝑣=𝑒^{−2𝜋𝑖𝑚𝑛𝑣}𝒞_𝑛(𝑣)Ψ_𝐠𝑗,𝑣𝒟_𝑛(𝑣) (30) for 𝑗∈ℤ, 𝑛=𝑘_𝑛𝑞+(𝑚_𝑛𝑞−𝑟_𝑛)𝑝 with (𝑘_𝑛,𝑟_𝑛,𝑚_𝑛)∈ℕ_𝑝×ℕ_𝑞×ℤ, and a.e. 𝑣∈ℝ, where 𝒟_𝑛(𝑣)=^{0𝑒−2𝜋𝑖𝑣𝐼𝑘𝑛}_{𝐼𝑛𝑝−𝑘0}, 𝒞_𝑛(𝑣)=diag(𝒞_1,𝑛(𝑣),𝒞_2,𝑛(𝑣),…,𝒞_𝐿,𝑛(𝑣)), and 𝒞_𝑙,𝑛(𝑣) denotes the block matrix (with 𝜆_𝑙 blocks) of the form diag(𝐶_𝑛(𝑣),𝐶_𝑛(𝑣),…,𝐶_𝑛(𝑣)) with 𝐶_𝑛(𝑣)=^{0𝐼𝑛𝑞−𝑟}_{𝑒2𝜋𝑖𝑣𝐼𝑟𝑛0}.

Proof. By [6, Lemma 5], we have 𝒢_𝑙𝑀𝑗+𝑞𝑛,𝑣=𝑒^{−2𝜋𝑖𝑚𝑛𝑣}𝐶_𝑛(𝑣)𝒢_𝑙𝑗,𝑣𝒟_𝑛(𝑣) (31) for 1≤𝑙≤𝐿, 𝑗∈ℤ, 𝑛=𝑘_𝑛𝑞+(𝑚_𝑛𝑞−𝑟_𝑛)𝑝 with (𝑘_𝑛,𝑟_𝑛,𝑚_𝑛)∈ℕ_𝑝×ℕ_𝑞×ℤ, and a.e. 𝑣∈ℝ. This leads to the lemma.

3. Frame Characterization

This section is devoted to characterization of subspace Gabor frames 𝐺(𝐠,𝐍,𝐌) of the form (2).

Theorem 13. For any 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), the following are equivalent.(i)𝐺(𝐠,𝐍,𝐌) is a Bessel sequence in 𝑙²(ℤ) with Bessel bound 𝐵.(ii)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼 for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1).(iii)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼 for 𝑗∈ℤ and a.e. 𝑣∈ℝ.

Proof. By Lemma 12, conditions (ii) and (iii) are equivalent. So, to finish the proof, we only need to prove that (i) holds if and only if Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠≤𝐵𝑗,𝑣𝑀𝐼for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1), (32) since 𝐺(𝐠,𝐍,𝐌) is a Bessel sequence with Bessel bound 𝐵 if and only if 𝒯_𝐠𝑐²≤𝐵‖𝑐‖² (33) for 𝑐∈𝑙₀(ℕ_𝑀×ℤ,ℂ^𝐿), which can be rewritten as _{𝑗∈ℕ𝑀}¹₀Ψ^∗_𝐠(𝑗,𝑣)𝒥𝑐(𝑗,𝑣)²≤𝐵𝑑𝑣𝑀_{𝑗∈ℕ𝑀}¹₀𝒥𝑐(𝑗,𝑣)²𝑑𝑣 (34) by Lemmas 6–10.
Next we prove the equivalence between (32) and (34) to finish the proof. It is obvious that (32) implies (34). Now we turn to the converse implication. Suppose (34) holds. Since every entry in Ψ_𝐠(𝑗,⋅) belongs to 𝐿²([0,1)) and thus belongs to 𝐿¹([0,1)) for 𝑗∈ℕ_𝑀, almost every point in (0,1) is a Lebesgue point for every entry in Ψ^∗_𝐠(𝑗,𝑣) and 𝑗∈ℕ_𝑀. Let 𝑣₀ be such a point and 𝑗₀∈ℕ_𝑀. Fix 𝑥∈ℂ^𝑄 and 𝜖>0 with (𝑣₀−𝜖,𝑣₀+𝜖)⊂(0,1). Define 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿) by =1𝒥𝑐𝑗,𝑣√𝜒2𝜖^{{𝑗0}×(𝑣0−𝜖,𝑣0+𝜖)}𝑗,𝑣𝑥 (35) for 𝑗∈ℕ_𝑀, 𝑣∈[0,1). By Lemma 9 and the density of 𝑙₀(ℕ_𝑀×ℤ,ℂ^𝐿) in 𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿), (34) holds for 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿). Substituting (35) into (34), we obtain that 12𝜖_{(𝑣0−𝜖,𝑣0+𝜖)}Ψ^∗_𝐠𝑗₀,𝑣𝑥²𝐵𝑑𝑣≤𝑀‖𝑥‖². (36) Letting 𝜖→0 leads to Ψ^∗_𝐠𝑗₀,𝑣₀𝑥²≤𝐵𝑀‖𝑥‖²; (37) that is, Ψ_𝐠𝑗₀,𝑣₀Ψ^∗_𝐠𝑗₀,𝑣₀≤B𝑥,𝑥𝑀‖𝑥‖². (38) This gives (32) by the arbitrariness of 𝑗₀, 𝑣₀, and 𝑥. The proof is completed.

Lemma 14 ([1, Theorem A.6.5]). Let 𝑈₁, 𝑈₂, 𝑈₃ be self-adjoint operators on a Hilbert space ℋ. If 𝑈₁≤𝑈₂, and 𝑈₃≥0, and 𝑈₃ commutes with 𝑈₁ and 𝑈₂, then 𝑈₁𝑈₃≤𝑈₂𝑈₃.

Remark 15. The inequality Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼 in Theorem 13 can be replaced by any one of the following inequalities:(i)(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))²≤(𝐵/𝑀)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣);(ii)Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼;(iii)(Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣))²≤(𝐵/𝑀)Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣).
Indeed, we have (i) by Lemma 14 if Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼. From (i), we have Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝑥,Ψ^∗_𝐠𝑥≤𝐵𝑗,𝑣𝑀Ψ^∗_𝐠𝑗,𝑣𝑥,Ψ^∗_𝐠𝑗,𝑣𝑥 (39) for 𝑥∈ℂ^𝑄, and thus Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼 on range(Ψ^∗_𝐠(𝑗,𝑣)). This implies (ii) due to the fact that ℂ^𝑝=range(Ψ^∗_𝐠(𝑗,𝑣))⊕ker(Ψ_𝐠(𝑗,𝑣)). It is obvious that (ii) implies (iii) by Lemma 14. By the same procedure as in “(i)⇒(ii)”, we can prove that (iii) implies Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼.

Remark 16. Condition (iii) in Theorem 13 is equivalent to ‖Ψ_𝐠(𝑗,𝑣)‖≤𝐵/𝑀 for a.e. (𝑗,𝑣)∈ℤ×ℝ when Ψ_𝐠(𝑗,𝑣) is understood as an operator from ℂ^𝑝 into ℂ^𝑄. However, such norm is equivalent to the one obtained by taking the maximum of the absolute values of all entries of a matrix. So 𝐺(𝐠,𝐍,𝐌) is a Bessel sequence if and only if 𝒵_𝑞𝑁𝑔_𝑙∈𝐿^∞(ℤ×ℝ) for 1≤𝑙≤𝐿 by Theorem 13.

Lemma 17. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) be a Bessel sequence in 𝑙²(ℤ). For 𝑓∈𝑙²(ℤ), write 𝑑^(𝑓)=𝒯^∗_𝐠𝑓. Then 𝒥𝑑^(𝑓)𝑗,𝑣=𝑀Ψ_𝐠𝑗,𝑣𝐙_𝑞𝑁𝑓𝑗,𝑣 (40) for (𝑗,𝑣)∈ℕ_𝑀×ℝ when either 𝐺(𝐠,𝐍,𝐌) is a Bessel sequence or 𝑓∈𝑙₀(ℤ).

Proof. By Lemmas 6 and 10, we have 𝑑^(𝑓)_{𝑙,𝑚,(𝑛𝑞+𝑟)𝜆𝑙+𝜏𝑙}=_{𝑘∈ℕ𝑝𝑗∈ℕ𝑀}¹₀𝒵_𝑞𝑁𝑓×𝑗+𝑘𝑀,𝑣𝒵_𝑞𝑁𝑔_𝑙𝑗−𝜏_𝑙𝑁_𝑙−𝑟𝑁+𝑘𝑀,𝑣×𝑒^{−2𝜋𝑖𝑛𝑣}𝑑𝑣𝑒^{−2𝜋𝑖(𝑚/𝑀)𝑗}=_{𝑗∈ℕ𝑀}¹₀𝒢_𝑙𝑗−𝜏_𝑙𝑁_𝑙,𝑣𝐙_𝑞𝑁𝑓𝑗,𝑣_𝑟×𝑒^{−2𝜋𝑖𝑛𝑣}𝑑𝑣𝑒^{−2𝜋𝑖(𝑚/𝑀)𝑗} (41) for each 𝑟∈ℕ_𝑞, 𝜏_𝑙∈ℕ_𝜆𝑙, and 1≤𝑙≤𝐿. When either 𝐺(𝐠,𝐍,𝐌) is a Bessel sequence or 𝑓∈𝑙₀(ℤ), the integrand in (41) belongs to 𝐿²([0,1)) as a function about 𝑣 by Remark 16. It follows that 𝒟^(𝜏𝑙)𝑗,𝑣=𝑀𝒢_𝑙𝑗−𝜏_𝑙𝑁_𝑙,𝑣𝐙_𝑞𝑁𝑓𝑗,𝑣 (42) for (𝑗,𝑣)∈ℕ_𝑀×ℝ. The lemma therefore follows.

Theorem 18. Let 𝐺(𝐠,𝐍,𝐌) and 𝐺(𝐡,𝐍,𝐌) be both Bessel sequences in 𝑙²(ℤ). Then 𝐙_𝑞𝑁𝒮_𝐡,𝐠𝑓𝑗,𝑣=𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣𝐙_𝑞𝑁𝑓𝑗,𝑣 (43) for 𝑗∈ℕ_𝑀 and a.e. 𝑣∈[0,1).

Proof. Write 𝑑=𝒯^∗_𝐡𝑓. Then 𝐙_𝑞𝑁𝒮_𝐡,𝐠𝑓𝑗,𝑣=Ψ^∗_𝐠𝑗,𝑣𝒥𝑑𝑗,𝑣 (44) by Remark 11. Applying Lemma 17 to 𝐡, we have 𝒥𝑑𝑗,𝑣=𝑀Ψ^∗_𝐡(𝑗,𝑣)𝐙_𝑞𝑁𝑓𝑗,𝑣, (45) and thus (43) holds by (44).

Theorem 19. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), the following are equivalent:(i)𝐺(𝐠,𝐍,𝐌) is a frame for ℳ(𝐠,𝐍,𝐌) with frame bounds 𝐴 and 𝐵;(ii)(𝐴/𝑀)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))²≤(𝐵/𝑀)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1);(iii)(𝐴/𝑀)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))²≤(𝐵/𝑀)Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣) for 𝑗∈ℤ and a.e. 𝑣∈ℝ.

Proof. By Theorem 13 and Remark 15, we may as well assume that 𝐺(𝐠,𝐍,𝐌) is a Bessel sequence, and we need to treat the “lower frame bound” part under this assumption. By a similar argument to beginning proof of Theorem 13, we only need to prove that 𝒮_𝐠,𝐠𝑓,𝑓≥𝐴𝑓²for𝑓∈ℳ(𝐠,𝐍,𝐌) (46) if and only if Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)²≥𝐴𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣for𝑗∈ℕ_𝑀,𝑣∈[0,1). (47) Since the linear span of 𝐺(𝐠,𝐍,𝐌) is dense in ℳ(𝐠,𝐍,𝐌), (46) holds if and only if 𝒮_𝐠,𝐠𝑓,𝑓≥𝐴𝑓² (48) for 𝑓=𝒯_𝐠𝑐 with 𝑐∈𝑙₂(ℕ_𝑀×ℤ,ℂ^𝐿) by [1, Lemma 5.1.7]. By Lemma 10, Theorem 18, and Remark 11, (48) can be rewritten as 𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²𝒥𝑐𝑗,𝑣,𝒥𝑐𝑗,𝑣≥𝐴Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣,𝒥𝑐𝑗,𝑣, (49) equivalently, 𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²𝒥𝑐𝑗,𝑣,Ψ𝒥𝑐𝑗,𝑣≥𝐴_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣,𝒥𝑐𝑗,𝑣 (50) for 𝑗∈ℕ_𝑀, 𝑣∈[0,1), and 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿). By the same procedure as in Theorem 13, we can prove the equivalence between (47) and (50). The proof is completed.

Remark 20. By an argument similar to Remark 15, (Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))² and Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣) in Theorem 19 can be replaced by (Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣))² and Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣), respectively.

Definition 21. Given 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) be a Bessel sequence in 𝑙²(ℤ). We say that 𝐺(𝐠,𝐍,𝐌) has Riesz property if, for 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿), we must have 𝑐=0 whenever 𝒯_𝐠𝑐=0.

By an easy application of the spectral theorem of self-adjoint matrices, we have the following lemma (see also [36, page 978]).

Lemma 22. Given a measurable set 𝐸 in ℝ with |𝐸|>0, let 𝒜∶𝐸→ℳ_𝑠,𝑡 be a matrix-valued measurable function. Define by 𝒫(𝑣) the orthogonal projection of ℂ^𝑡 onto ker(𝒜(𝑣)). Then 𝒫(𝑣)=lim_𝑛→∞exp−𝑛𝒜^∗(𝑣)𝒜(𝑣) (51) for 𝑣∈𝐸, and thus 𝒫(𝑣) is measurable.

Theorem 23. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), the following are equivalent:(i)𝐺(𝐠,𝐍,𝐌) has Riesz property;(ii)rank(Ψ_𝐠(𝑗,𝑣))=𝑄 for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1);(iii)rank(Ψ_𝐠(𝑗,𝑣))=𝑄 for 𝑗∈ℤ and a.e. 𝑣∈ℝ.

Proof. By Lemma 12, (ii) and (iii) are equivalent. So we only need to prove that 𝑐=0 is a unique solution to 𝒯_𝐠𝑐=0 (52) in 𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿) if and only if rankΨ_𝐠𝑗,𝑣=𝑄for𝑗∈ℕ_𝑀,𝑣∈[0,1). (53) By Remark 11, (52) can be rewritten as Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣=0for𝑗∈ℕ_𝑀,𝑣∈[0,1). (54) So 𝑐=0 is a unique solution to (52) in 𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿) if and only if 𝑑(𝑗,𝑣)=0 is a unique solution to Ψ^∗_𝐠𝑗,𝑣𝑑𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1) (55) in 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄) by Lemma 9. It is obvious that (53) implies that 𝑑(𝑗,𝑣)=0 is a unique solution to (55) in 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄). Next we prove the converse implication. Suppose there exist 𝑗₀∈ℕ_𝑀 and 𝐸⊂[0,1) with |𝐸|>0 such that rank(Ψ_𝐠(𝑗₀,𝑣))<𝑄 on 𝐸. Let 𝒫(𝑗₀,𝑣) be the orthogonal projection of ℂ^𝑄 onto ker(Ψ^∗_𝐠(𝑗₀,𝑣)). Then there exist 𝑖₀∈ℕ_𝑄 and 𝐸^′⊂𝐸 with |𝐸^′|>0 such that 𝒫(𝑗₀,𝑣)𝐞_𝑖0≠0 for 𝑣∈𝐸^′. Define 𝑑(𝑗,𝑣)∈𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄) by 𝑑=𝒫𝐞𝑗,𝑣𝑗,𝑣_𝑖0,if𝑗=𝑗₀,𝑣∈𝐸^′,0,otherwise (56) for 𝑗∈ℕ_𝑀 and 𝑣∈[0,1). Then 𝑑(𝑗,𝑣) is well defined, and ‖𝑑(𝑗,𝑣)‖≤1 by Lemma 22. It follows that 𝑑(𝑗,𝑣) is a nonzero solution to (55) in 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄). This is a contradiction. The proof is completed.

Since a Riesz basis is exactly a frame having Riesz property, and an orthonormal basis is exactly a Riesz basis with Riesz bound 1, we have the following theorem by Theorems 19 and 23.

Theorem 24. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), the following are equivalent:(i)𝐺(𝐠,𝐍,𝐌) is a Riesz basis for 𝐺(𝐠,𝐍,𝐌) with Riesz bounds 𝐴 and 𝐵 (an orthonormal basis);(ii)(𝐴/𝑀)𝐼≤Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)=(1/𝑀)𝐼) for 𝑗∈ℕ_𝑀/𝑞 a.e. 𝑣∈[0,1);(iii)(𝐴/𝑀)𝐼≤Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)≤(𝐵/𝑀)𝐼(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)=(1/𝑀)𝐼) for 𝑗∈ℤ and a.e. 𝑣∈ℝ.

Next we turn to examples of Theorems 19 and 24. Suppose 𝐍=(6,4),𝐌=(2,2). (57) Then 𝑁/𝑀=6 with 𝑝=6, 𝑞=1, and 𝑄=5. For 𝐠=(𝑔₁,𝑔₂) with 𝑔₁,𝑔₂∈𝑙²(ℤ), we associate it with Ψ_𝐠 as in Definition 4. Then Ψ_𝐠=𝐺𝑗,𝑣¹𝐺𝑗,𝑣₂𝐺𝑗,𝑣with₁=𝒢𝑗,𝑣¹𝒢𝑗,𝑣₁𝑗−6,𝑣,𝐺₂=𝒢𝑗,𝑣𝒢²𝑗,𝑣₂𝒢𝑗−4,𝑣₂,𝑗−8,𝑣 (58) where 𝒢_𝑙𝑗,𝑣=𝒵₁₂𝑔_𝑙𝑗+2𝑘,𝑣_0,𝑘∈ℳ_1,6with𝑗∈ℕ₂,𝑘∈ℕ₆ (59) for 𝑙=1,2. By the quasi-periodicity of Zak transform, we have 𝒢₁=𝑒𝑗−6,𝑣_{2𝜋𝑖𝑣}𝒵₁₂𝑔₁𝑗+6,𝑣,𝑒^{2𝜋𝑖𝑣}𝒵₁₂𝑔₁,𝑒𝑗+8,𝑣_{2𝜋𝑖𝑣}𝒵₁₂𝑔₁𝑗+10,𝑣,𝒵₁₂𝑔₁,𝒵𝑗,𝑣₁₂𝑔₁𝑗+2,𝑣,𝒵₁₂𝑔₁,𝒢𝑗+4,𝑣₂=𝑒𝑗−4,𝑣_{2𝜋𝑖𝑣}𝒵₁₂𝑔₂𝑗+8,𝑣,𝑒^{2𝜋𝑖𝑣}𝒵₁₂𝑔₂,𝒵𝑗+10,𝑣₁₂𝑔₂𝑗,𝑣,𝒵₁₂𝑔₂,𝒵𝑗+2,𝑣₁₂𝑔₂𝑗+4,𝑣,𝒵₁₂𝑔₂,𝒢𝑗+6,𝑣₂=𝑒𝑗−8,𝑣_{2𝜋𝑖𝑣}𝒵₁₂𝑔₂𝑗+4,𝑣,𝑒^{2𝜋𝑖𝑣}𝒵₁₂𝑔₂,𝑒𝑗+6,𝑣_{2𝜋𝑖𝑣}𝒵₁₂𝑔₂𝑗+8,𝑣,𝒵₁₂𝑔₂,𝒵𝑗+10,𝑣₁₂𝑔₂𝑗,𝑣,𝒵₁₂𝑔₂𝑗+2,𝑣. (60) Thus for each 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1), 𝒢₁(𝑗−6,𝑣), and 𝒢₂(𝑗−4,𝑣), 𝒢₂(𝑗−8,𝑣) are uniquely determined by 𝒢₁(𝑗,𝑣) and 𝒢₂(𝑗,𝑣), respectively. Observe that ℕ₂+2ℕ₆ is 12ℤ congruent to ℕ₁₂. It follows that 𝐠 is uniquely determined by the values of Ψ_𝐠(𝑗,𝑣) for 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1). Therefore, an arbitrary matrix 5×6 matrix-valued function 𝒦(𝑗,𝑣) for 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1) with all entries being in 𝐿²(ℕ₂×[0,1)) determines a unique 𝐠 by Ψ_𝐠𝑗,𝑣=𝒦𝑗,𝑣for𝑗∈ℕ₂,a.e.𝑣∈[0,1). (61)

Let𝒦=𝑎𝜆𝑎𝜆𝑗,𝑣𝑗,𝑣𝑗,𝑣𝑗,𝑣²𝑎𝜆𝑎𝜆𝑗,𝑣𝑗,𝑣000000𝑎𝑗,𝑣𝑗,𝑣𝑗,𝑣²𝑎−𝑗,𝑣𝑗,𝑣𝜆𝑗,𝑣𝑎𝑗,𝑣𝑎𝑗,𝑣000000𝜆𝑗,𝑣𝑎𝑗,𝑣𝑎𝑗,𝑣000000−𝜆𝑗,𝑣𝑎𝑗,𝑣𝑎𝑗,𝑣 (62)for 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1). Define 𝐠=(𝑔₁,𝑔₂) by Ψ_𝐠𝑗,𝑣=𝒦𝑗,𝑣for𝑗∈ℕ₂,a.e.𝑣∈[0,1]. (63) Then we obtain the following example.

Example 25. Let 𝐍 and 𝐌 be defined as in (57) and define 𝐠=(𝑔₁,𝑔₂) by (63), where 𝜆(𝑗,⋅), 𝑎(𝑗,⋅) are continuous on [0,1] for 𝑗∈{0,1}. Assume that 𝜆𝑎𝑗,𝑣<1,𝑗,𝑣²1−2𝜆𝑗,𝑣⁴−2𝜆𝑗,𝑣⁶−𝜆𝑗,𝑣⁸<1−𝜆𝑗,𝑣³,𝑎𝑗,𝑣²1+2𝜆𝑗,𝑣³+2𝜆𝑗,𝑣⁵+𝜆𝑗,𝑣⁷<1+𝜆𝑗,𝑣³ (64) for (𝑗,𝑣)∈{0,1}×[0,1] satisfying 𝑎(𝑗,𝑣)≠0. Then supp𝑔₁⊂ℕ₆+12ℤ,supp𝑔₂⊂ℕ₄+12ℤ, (65) and 𝐺(𝐠,𝐍,𝐌) is a frame for ℳ(𝐠,𝐍,𝐌).

Proof. Equation (65) is an immediate consequence of (63). Next we prove that 𝐺(𝐠,𝐍,𝐌) is a frame for ℳ(𝐠,𝐍,𝐌). By a simple computation, we have Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²=𝑎𝑥,𝑥𝑗,𝑣²⁵_𝑙=1𝛼_𝑙𝑗,𝑣𝑥_𝑙²,Ψ+𝛽𝑗,𝑣_𝐠𝑗,𝑣Ψ^∗_𝐠=𝑎𝑗,𝑣𝑥,𝑥𝑗,𝑣²⁵_𝑙=1𝛼_𝑙𝑗,𝑣𝑥_𝑙²+𝛽𝑗,𝑣 (66) for (𝑗,𝑣)∈{0,1}×[0,1] and 𝑥∈ℂ⁵, where 𝛼₁𝑗,𝑣=𝜆𝑗,𝑣³+1+𝜆𝑗,𝑣²+𝜆𝑗,𝑣⁴²𝑎𝑗,𝑣²,𝛼₂𝑗,𝑣=1+1+𝜆𝑗,𝑣²+𝜆𝑗,𝑣⁴²×𝑎𝑗,𝑣²,𝛼₄𝑗,𝑣=1+𝜆𝑗,𝑣³+1+𝜆𝑗,𝑣²²𝑎𝑗,𝑣²,𝛼₄𝑗,𝑣=1+𝜆𝑗,𝑣³+1+𝜆𝑗,𝑣²²𝑎𝑗,𝑣²,𝛼₃𝑗,𝑣=𝛼₅𝑗,𝑣=1+𝜆𝑗,𝑣²²𝑎𝑗,𝑣²,𝛽𝑗,𝑣=2Re𝜆𝑗,𝑣³×2+2𝜆𝑗,𝑣²+𝜆𝑗,𝑣⁴×𝑎𝑗,𝑣²𝑎𝑗,𝑣²𝑥₁𝑥₄+2Re2+2𝜆𝑗,𝑣²+𝜆𝑗,𝑣⁴×𝑎𝑗,𝑣²𝑎𝑗,𝑣²𝑥₂𝑥₄−2Re𝜆𝑗,𝑣³𝑎𝑗,𝑣²𝑥₁𝑥₂,𝛼₁𝑗,𝑣=𝛼₂𝑗,𝑣=1+𝜆𝑗,𝑣²+𝜆𝑗,𝑣⁴,𝛼₃𝑗,𝑣=𝛼₄𝑗,𝑣=𝛼₅𝑗,𝑣=1+𝜆𝑗,𝑣²,𝛽𝑗,𝑣=−2Re𝜆𝑗,𝑣³𝑎𝑗,𝑣²𝑥₁𝑥₄+2Re𝑎𝑗,𝑣²𝑥₂𝑥₄. (67) It is easy to check that 𝛽𝑗,𝑣≤𝑎𝑗,𝑣²𝛽₁𝑗,𝑣𝑥₁²+𝛽₂𝑗,𝑣𝑥₂²+𝛽₄𝑗,𝑣𝑥₄², (68) where 𝛽₁𝑗,𝑣=𝜆𝑗,𝑣³+𝑎𝑗,𝑣²×2𝜆𝑗,𝑣³+2𝜆𝑗,𝑣⁵+𝜆𝑗,𝑣⁷,𝛽₂𝑗,𝑣=𝜆𝑗,𝑣³+𝑎𝑗,𝑣²×2+2𝜆𝑗,𝑣²+𝜆𝑗,𝑣⁴,𝛽₄𝑗,𝑣=𝑎𝑗,𝑣²×2+2𝜆𝑗,𝑣²+2𝜆𝑗,𝑣³+𝜆𝑗,𝑣⁴+2𝜆𝑗,𝑣⁵+𝜆𝑗,𝑣⁷.𝛽𝑗,𝑣≤𝑎𝑗,𝑣²×𝛽₁𝑗,𝑣𝑥₁²+𝛽₂𝑗,𝑣𝑥₂²+𝛽₄𝑗,𝑣𝑥₄², (69) where 𝛽₁𝑗,𝑣=𝜆𝑗,𝑣³,𝛽₂𝛽𝑗,𝑣=1,₄𝑗,𝑣=1+𝜆𝑗,𝑣³. (70) Write 𝐶_𝑙𝑗,𝑣=𝛼_𝑙𝑗,𝑣+𝛽_𝑙,𝐶𝑗,𝑣_𝑙𝑗,𝑣=𝛼_𝑙𝑗,𝑣−𝛽_𝑙𝐶=𝜆𝑗,𝑣for𝑙=1,2,4,𝑗,𝑣1+𝑗,𝑣²²𝑎𝑗,𝑣²,𝐷_𝑙𝑗,𝑣=𝛼_𝑙𝑗,𝑣+𝛽_𝑙,𝐷𝑗,𝑣_𝑙𝑗,𝑣=𝛼_𝑙𝑗,𝑣−𝛽_𝑙𝐷=𝜆𝑗,𝑣for𝑙=1,2,4,𝑗,𝑣𝑗,𝑣²+1. (71) Take 𝐶=min𝐶𝑗,𝑣,𝐶_𝑙𝑗,𝑣∶𝑙=1,2,4,𝑗∈ℕ₂[,𝐶,𝑣∈0,1]𝐶=max𝑗,𝑣,𝐶_𝑙𝑗,𝑣∶𝑙=1,2,4,𝑗∈ℕ₂[,𝐷,𝐷,𝑣∈0,1]𝐷=min𝑗,𝑣_𝑙𝑗,𝑣∶𝑙=1,2,4,𝑗∈ℕ₂[,𝐷,𝑣∈0,1]𝐷=max𝑗,𝑣,𝐷_𝑙𝑗,𝑣∶𝑙=1,2,4,𝑗∈ℕ₂,𝑣∈[0,1]. (72) Then Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²≤𝑎𝑥,𝑥𝑗,𝑣²𝐶₁𝑗,𝑣𝑥₁²+𝐶₂𝑗,𝑣𝑥₂²+𝐶₄𝑗,𝑣𝑥₄²×𝑥+𝐶𝑗,𝑣₃₂+𝑥₅²≤𝐶𝑎𝑗,𝑣²‖𝑥‖²,Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²≥𝑎𝑥,𝑥𝑗,𝑣²𝐶₁𝑗,𝑣𝑥₁²+𝐶₂𝑗,𝑣𝑥₂²+𝐶₄𝑗,𝑣𝑥₄²×𝑥+𝐶𝑗,𝑣₃₂+𝑥₅²≥𝐶𝑎𝑗,𝑣²‖𝑥‖²,Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠≤𝑎𝑗,𝑣𝑥,𝑥𝑗,𝑣²𝐷₁𝑗,𝑣𝑥₁²+𝐷₂𝑗,𝑣𝑥₂²+𝐷₄𝑗,𝑣𝑥₄²×𝑥+𝐷𝑗,𝑣₃₂+𝑥₅²≤𝐷𝑎𝑗,𝑣²‖𝑥‖²,Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠≥𝑎𝑗,𝑣𝑥,𝑥𝑗,𝑣²𝐷₁𝑗,𝑣𝑥₁²+𝐷₂𝑗,𝑣𝑥₂²+𝐷₄𝑗,𝑣𝑥₄²+𝐷𝑗,𝑣𝑥₃²+𝑥₅²≥𝐷𝑎𝑗,𝑣²‖𝑥‖². (73) It follows that 𝐴2Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝑥,𝑥≤Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²≤𝐵𝑥,𝑥2Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝑥,𝑥, (74) for 𝑥∈ℂ⁵, 𝑗∈ℕ₂, and a.e. 𝑣∈[0,1), and thus 𝐴2Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣≤Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣²≤𝐵2Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣 (75) for 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1), where 𝐴=2𝐶/𝐷, 𝐵=2𝐶/𝐷. By Theorem 19, 𝐺(𝐠,𝐍,𝐌) is a frame for ℳ(𝐠,𝐍,𝐌) with frame bounds 𝐴 and 𝐵.

Remark 26. By a simple computation, we have rank(Ψ_𝐠(𝑗,𝑣))=5 if and only if 𝑎(0,𝑣)𝑎(1,𝑣)𝜆(0,𝑣)𝜆(1,𝑣)≠0fora.e.𝑣∈[0,1). (76) Also observe that rank(Ψ_𝐠(𝑗,𝑣))=5 is equivalent to Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣) being invertible, and that (75) can be reduced to 𝐴𝐼2≤Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣≤𝐵𝐼2for𝑗∈ℕ₂,a.e.𝑣∈[0,1) (77) in this case. Therefore, 𝐺(𝐠,𝐍,𝐌) is a Riesz basis for ℳ(𝐠,𝐍,𝐌) with Riesz bounds 𝐴 and 𝐵 if and only if (76) holds.

4. Gabor Dual Characterization

Let 𝐺(𝐠,𝐍,𝐌) be a frame for ℳ(𝐠,𝐍,𝐌). In this section, we discuss three kinds of duals with Gabor structure. We establish characterizations of Gabor duals of type I and type II and obtain a sufficient condition for Gabor duals of type II and oblique Gabor duals.

By an argument similar to [5, Lemma 2.6], we have the following lemmas.

Lemma 27. For 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) and 𝐡=(ℎ₁,ℎ₂,…,ℎ_𝐿) with each 𝑔_𝑙, ℎ_𝑙∈𝑙²(ℤ), the following are equivalent:(i)ℎ_𝑙∈ℳ(𝐠,𝐍,𝐌) for each 1≤𝑙≤𝐿;(ii)there exists a function 𝐴∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑄,𝑄 such that Ψ_𝐡𝑗,𝑣=𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣 (78) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1);(iii) there exists a function 𝐴∶ℤ×ℝ→ℳ_𝑄,𝑄 such that Ψ_𝐡𝑗,𝑣=𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣 (79) for 𝑗∈ℤ and a.e. 𝑣∈ℝ.

Lemma 28. Given 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), 𝐡=(ℎ₁,ℎ₂,…,ℎ_𝐿) with each ℎ_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) and 𝐺(𝐡,𝐍,𝐌) be both Bessel sequences in 𝑙²(ℤ). Then the following are equivalent:(i)range(𝒯^∗_𝐡)⊂range(𝒯^∗_𝐠); (ii)there exists a function 𝐵∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑝,𝑝 such that Ψ_𝐡𝑗,𝑣=Ψ_𝐠𝑗,𝑣𝐵𝑗,𝑣𝑓𝑜𝑟𝑗,𝑣∈ℕ_𝑀/𝑞×[0,1); (80)(iii)there exists a function 𝐵∶ℤ×ℝ→ℳ_𝑝,𝑝 such that Ψ_𝐡𝑗,𝑣=Ψ_𝐠𝑗,𝑣𝐵𝑗,𝑣𝑓𝑜𝑟𝑗,𝑣∈ℤ×ℝ. (81)

Proof. If (ii) holds, we can obtain (iii) by choosing 𝐵∶ℤ×ℝ→ℳ_𝑝,𝑝 as 𝐵𝑀𝑗+𝑞𝑛,𝑣=𝒟^∗_𝑛(𝑣)𝐵𝑗,𝑣𝒟_𝑛(𝑣) (82) for (𝑗,𝑣)∈ℕ_𝑀/𝑞×ℝ, where 𝒟_𝑛(𝑣) is defined as in Lemma 12. So we only need to prove that (i) holds if and only if there exists a function 𝐵∶ℕ_𝑀×[0,1)→ℳ_𝑝,𝑝 such that Ψ_𝐡𝑗,𝑣=Ψ_𝐠𝑗,𝑣𝐵𝑗,𝑣for𝑗,𝑣∈ℕ_𝑀×[0,1). (83) For 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿), 𝑐 is orthogonal to range(𝒯^∗_𝐠) (range(𝒯^∗_𝐡)) if and only if 𝒥𝑐𝑗,𝑣⟂Ψ_𝐠𝑗,𝑣𝐙_𝑞𝑁𝑓⟂𝑗,𝑣𝒥𝑐𝑗,𝑣Ψ_𝐡𝑗,𝑣𝐙_𝑞𝑁𝑓𝑗,𝑣 (84) in 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄)-inner product by Lemmas 9 and 17, equivalently, Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣⟂𝐙_𝑞𝑁𝑓𝑗,𝑣Ψ^∗_𝐡𝑗,𝑣𝒥𝑐𝑗,𝑣⟂𝐙_𝑞𝑁𝑓𝑗,𝑣for𝑓∈𝑙²(ℤ) (85) in 𝐿²(ℕ_𝑀×[0,1),ℂ^𝑝)-inner. Again by Lemma 6, (85) is equivalent to Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣=0Ψ^∗_𝐡j,𝑣𝒥𝑐𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1). (86) So we only need to prove that (83) holds if and only if, for 𝒥𝑐(𝑗,𝑣)∈𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄), Ψ^∗_𝐡𝑗,𝑣𝒥𝑐𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1) (87) whenever Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1). (88) Obviously the necessity holds. Next we prove that a contradiction will occur if (83) is violated. Suppose (83) does not hold. Then there exist some 𝑗₀∈ℕ_𝑀 and 𝐸⊂[0,1) with |𝐸|>0, on which some 𝑟₀th column ℎ_𝑟0 of Ψ_𝐡(𝑗₀,𝑣) satisfies ℎ_𝑟0∉rangeΨ_𝐠𝑗₀,𝑣for𝑣∈𝐸. (89) Let 𝒫(𝑗₀,𝑣) be the orthogonal projection of ℂ^𝑄 onto ker(Ψ^∗_𝐠(𝑗₀,𝑣)). Define 𝒥𝑐(𝑗,𝑣)∈𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄) by =𝒫𝑗𝒥𝑐𝑗,𝑣⁰,𝑣ℎ_𝑟0𝑗₀,𝑣,if𝑗=𝑗₀,𝑣∈𝐸,0,otherwise (90) for (𝑗,𝑣)∈ℕ_𝑀×[0,1). Then 𝒥𝑐(𝑗,𝑣) solves (88) and 𝒥𝑐(𝑗₀,𝑣)≠0 for 𝑣∈𝐸. Also observe that the 𝑟₀th component of Ψ^∗_𝐡(𝑗₀,𝑣)𝒥𝑐(𝑗₀,𝑣) is exactly ‖𝒥𝑐(𝑗₀,𝑣)‖² for 𝑣∈𝐸. It follows that 𝒥𝑐(𝑗,𝑣) fails to solve (87). This finishes the proof.

Lemma 29. Given 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿), 𝐡=(ℎ₁,ℎ₂,…,ℎ_𝐿) with each 𝑔_𝑙, ℎ_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) and 𝐺(𝐡,𝐍,𝐌) be both Bessel sequences in 𝑙²(ℤ). Then 𝒮_𝐡,𝐠𝑓=𝑓 (91) for 𝑓∈ℳ(𝐠,𝐍,𝐌) if and only if Ψ^∗_𝐠𝑗,𝑣=𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣 (92) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1).

Proof. By Lemma 12, (92) holds for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1) if and only if it holds for 𝑗∈ℕ_𝑀 and a.e. 𝑣∈[0,1). Next we prove that (91) holds for 𝑓∈ℳ(𝐠,𝐍,𝐌) if and only if (92) holds for 𝑗∈ℕ_𝑀 and a.e. 𝑣∈[0,1).
Since range(𝒯_𝐠) is dense in ℳ(𝐠,𝐍,𝐌), (91) holds for 𝑓∈ℳ(𝐠,𝐍,𝐌) if and only if it holds for 𝑓∈range(𝒯_𝐠), equivalently, Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣=𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝒥𝑐𝑗,𝑣 (93) for 𝑐∈𝑙²(ℕ_𝑀×ℤ,ℂ^𝐿) by Remark 11 and Theorem 18. This is also equivalent to Ψ^∗_𝐠𝑗,𝑣𝑑𝑗,𝑣=𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝑑𝑗,𝑣 (94) for 𝑑(𝑗,𝑣)∈𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄), 𝑗∈ℕ_𝑀, and a.e. 𝑣∈[0,1) by Lemma 9. It is obvious that (94) holds if (92) holds for 𝑗∈ℕ_𝑀 and a.e. 𝑣∈[0,1). Now suppose (94) holds. For an arbitrarily fixed 𝑥∈ℂ^𝑄, choose 𝑑(𝑗,𝑣) as 𝑑𝑗,𝑣=𝑥for𝑗,𝑣∈ℕ_𝑀×[0,1). (95) Then 𝑑(𝑗,𝑣)∈𝐿²(ℕ_𝑀×[0,1),ℂ^𝑄), and thus Ψ^∗_𝐠(𝑗,𝑣)𝑥=𝑀Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐡(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)𝑥 for 𝑗∈ℕ_𝑀 and a.e. 𝑣∈[0,1) by (94). So (92) holds for 𝑗∈ℕ_𝑀 and a.e. 𝑣∈[0,1) by the arbitrariness of 𝑥. The proof is completed.

By Lemmas 27–29, we have the following theorem which characterizes the Gabor duals of type I (resp., type II).

Theorem 30. Given 𝐠=(g₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) be a frame for ℳ(𝐠,𝐍,𝐌). Then, for any Bessel sequence 𝐺(𝐡,𝐍,𝐌) in 𝑙²(ℤ), 𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type I (type II) for 𝐺(𝐠,𝐍,𝐌) if and only if the following hold:(i) there exists 𝐴∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑄,𝑄(𝐵∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑝,𝑝) such that Ψ_𝐡𝑗,𝑣=𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣=Ψ_𝐠𝑗,𝑣𝐵𝑗,𝑣; (96)(ii)Ψ^∗_𝐠(𝑗,𝑣)=𝑀Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐡(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1).

Theorem 31. Given 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) be a frame for ℳ(𝐠,𝐍,𝐌). Then, for any Bessel sequence 𝐺(𝐡,𝐍,𝐌) in 𝑙²(ℤ),(i) 𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type I for 𝐺(𝐠,𝐍,𝐌) if and only if there exists 𝒜∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑄,𝑄 such that Ψ_𝐡=1𝑗,𝑣𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†Ψ_𝐠×𝑗,𝑣𝐼−𝑀Ψ^∗_𝐠𝑗,𝑣𝒜𝑗,𝑣Ψ_𝐠Ψ𝑗,𝑣+𝒜𝑗,𝑣_𝐠𝑗,𝑣 (97) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1);(ii) 𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type II for 𝐺(𝐠,𝐍,𝐌) if Ψ_𝐡=1𝑗,𝑣𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣^†×𝐼−𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣𝒜𝑗,𝑣+Ψ_𝐠𝑗,𝑣𝒜𝑗,𝑣 (98) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1);(iii)𝐺(𝐡,𝐍,𝐌) is an oblique dual of 𝐺(𝐠,𝐍,𝐌) if one of the following conditions holds:(a)Ψ_𝐡(𝑗,𝑣)=(1/𝑀)Ψ_𝐠(𝑗,𝑣)(Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣))^†[𝐼−𝑀Ψ^∗_𝐠(𝑗,𝑣)𝒜(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣)]+𝒜(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1);(b)Ψ_𝐡(𝑗,𝑣)=(1/𝑀)(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))^†Ψ_𝐠(𝑗,𝑣)[𝐼−𝑀Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣)𝒜(𝑗,𝑣)]+Ψ_𝐠(𝑗,𝑣)𝒜(𝑗,𝑣) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1).

Proof. The items (ii) and (iii) can be proved similarly to item (i). Next we prove item (i). First we assume that (97) holds. Then 𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣=Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†×Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣−𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†×Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣𝒜𝑗,𝑣×Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣+𝑀Ψ^∗_𝐠𝑗,𝑣𝒜𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣 (99) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1). Also observe that (Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))^†Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣) is the projection from ℂ^𝑄 onto range(Ψ_𝐠(𝑗,𝑣)). It follows that 𝑀Ψ^∗_𝐠(𝑗,𝑣)Ψ_𝐡(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣)=Ψ^∗_𝐠(𝑗,𝑣), and thus 𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type I for 𝐺(𝐠,𝐍,𝐌) by Theorem 30.
Now we turn to the converse implication. Suppose 𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type I for 𝐺(𝐠,𝐍,𝐌). Then there exists 𝐵(𝑗,𝑣)∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑄,𝑄 such that Ψ_𝐡𝑗,𝑣=𝐵𝑗,𝑣Ψ_𝐠,Ψ𝑗,𝑣_∗𝐠𝑗,𝑣=𝑀Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣 (100) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1). It follows that 1𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠=Ψ𝑗,𝑣_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†Ψ_𝐠𝑗,𝑣×Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣 (101) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1), and thus 1𝑀Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†Ψ_𝐠=Ψ𝑗,𝑣_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†×Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣 (102) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1) due to the fact that ℂ^𝑝=range(Ψ^∗_𝐠(𝑗,𝑣))⊕ker(Ψ_𝐠(𝑗,𝑣)). Put 𝒜(𝑗,𝑣)=𝐵(𝑗,𝑣)−(Ψ_𝐠(𝑗,𝑣)Ψ^∗_𝐠(𝑗,𝑣))^†. Then the right-hand side of (97) equals Ψ_𝐡𝑗,𝑣+Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†𝒫_{range(Ψ𝐠(𝑗,𝑣))}Ψ_𝐠−Ψ𝑗,𝑣_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†Ψ_𝐠𝑗,𝑣 (103) by (102), and this is exactly Ψ_𝐡(𝑗,𝑣) for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1) again by the fact that Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣^†=𝒫_{range(Ψ𝐠(𝑗,𝑣))}. (104)

5. The Case of 𝑁₁=𝑁₂=⋯=𝑁_𝐿

Theorems 30 and 31 characterize duals with Gabor structure for general 𝐍=(𝑁₁,𝑁₂,…,𝑁_𝐿). This section deals with the case 𝑁₁=𝑁₂=⋯=𝑁_𝐿. We start with a remark on the case of 𝑁_𝑙, 1≤𝑙≤𝐿, being not all the same.

Remark 32. Not every subspace Gabor frame 𝐺(𝐠,𝐍,𝐌) admits an oblique Gabor dual.

We show it by revisiting Example 25. Let us make an additional assumption that 𝜆(0,𝑣)=𝜆(1,𝑣)=0 and 𝑎(0,𝑣)𝑎(1,𝑣)≠0 for 𝑣∈[0,1]. Then 𝐺(𝐠,𝐍,𝐌) is a frame but not a Riesz basis for ℳ(𝐠,𝐍,𝐌) by Remark 26. Suppose 𝐺(𝐡,𝐍,𝐌) with 𝐡=(ℎ₁,ℎ₂) is an oblique Gabor dual for 𝐺(𝐠,𝐍,𝐌). Then Ψ^∗_𝐠𝑗,𝑣=2Ψ^∗_𝐠𝑗,𝑣Ψ_𝐡𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣for𝑗∈ℕ₂,a.e.𝑣∈[0,1) (105) by Theorem 30. Writing out (0,0), (3,1), and (5,4) entries of both sides, we have 2𝑎𝑗,𝑣²𝒵₁₂ℎ₁𝑗,𝑣=𝑎,2𝑗,𝑣𝑎𝑗,𝑣²𝒵₁₂ℎ₁𝑎𝑗,𝑣+2𝑗,𝑣²𝒵₁₂ℎ₂𝑗+2,𝑣=2𝑎𝑎(𝑗,𝑣),𝑗,𝑣²𝒵₁₂ℎ₂𝑗+2,𝑣=𝑎𝑗,𝑣 (106)(107)(108) for 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1). By (107) and (108), we have 2𝑎(𝑗,𝑣)𝒵₁₂ℎ₁(𝑗,𝑣)=0 for 𝑗∈ℕ₂ and a.e. 𝑣∈[0,1). This contradicts (106).

Observe that 𝑁₁≠𝑁₂ in Remark 32 (𝑁₁=6 and 𝑁₂=4). It is natural to ask the following question.

Does every subspace Gabor frame 𝐺(𝐠,𝐍,𝐌) admit no oblique Gabor dual whenever 𝑁_𝑙, 1≤𝑙≤𝐿, are not all the same?

The following example gives a negative answer to this question.

Example 33. Let 𝐍=(1,2) and 𝐌=(3,3). Assume that 𝒥(𝑣)=^𝒥1_𝒥^(𝑣)_2(𝑣), ℰ(𝑣)=^ℰ1_ℰ^(𝑣)_2(𝑣) have the form 𝒥₁(𝑣)=𝑎_0,0(𝑣)𝑎_0,1(𝑣)𝑎_1,0(𝑣)𝑎_1,1(𝑣)𝑎_2,0(𝑣)𝑎_2,1(𝑣)𝑎_2,1(𝑣)𝑒^{−2𝜋𝑖𝑣}𝑎_2,0(𝑣)𝑒^{2𝜋𝑖𝑣}𝑎_0,1(𝑣)𝑎_0,0(𝑣)𝑒^{2𝜋𝑖𝑣}𝑎_1,1(𝑣)𝑎_1,0(𝑣),ℰ₁(𝑣)=𝑐_0,0(𝑣)𝑐_0,1(𝑣)𝑐_1,0(𝑣)𝑐_1,1(𝑣)𝑐_2,0(𝑣)c_2,1(𝑣)𝑐_2,1(𝑣)𝑒^{−2𝜋𝑖𝑣}𝑐_2,0(𝑣)𝑒^{2𝜋𝑖𝑣}𝑐_0,1(𝑣)𝑐_0,0(𝑣)𝑒^{2𝜋𝑖𝑣}𝑐_1,1(𝑣)𝑐_1,0(𝑣),𝒥₂(𝑣)=𝑏_0,0(𝑣)𝑏_0,1(𝑣)𝑏_1,0(𝑣)𝑏_1,1(𝑣)𝑏_2,0(𝑣)𝑏_2,1(𝑣),ℰ₂(𝑣)=𝑑_0,0(𝑣)𝑑_0,1(𝑣)𝑑_1,0(𝑣)𝑑_1,1(𝑣)𝑑_2,0(𝑣)𝑑_2,1(𝑣), (109) for 𝑣∈[0,1) with all entries of 𝒥(𝑣) and ℰ(𝑣) in 𝐿^∞([0,1)), that (𝒥₁(𝑣))^∗ℰ₁(𝑣) has the form 𝐴(𝑣)00𝐴(𝑣) and (𝒥(𝑣))^∗𝒥(𝑣) has the form 𝐵(𝑣)00𝐵(𝑣) for 𝑣∈[0,1), and that 𝒥₂(𝑣)^∗ℰ₂(𝑣)=13(𝑣)001−𝐴3−𝐴(𝑣) (110) for 𝑣∈[0,1) satisfying 𝐵(𝑣)≠0. Define 𝐠=(𝑔₁,𝑔₂) and 𝐡=(ℎ₁,ℎ₂) by Ψ_𝐠(0,𝑣)=𝒥(𝑣),Ψ_𝐡(0,𝑣)=ℰ(𝑣)for𝑣∈[0,1). (111) Then 𝐠 and 𝐡 are well defined by the quasi-periodicity of the Zak transform 𝒵₆ and 6ℤ congruence between {0}−2ℕ₃+3ℕ₂ and ℕ₆, and 𝐺(𝐠,𝐍,𝐌) and 𝐺(𝐡,𝐍,𝐌) are both Bessel sequences by Remark 16. A simple computation shows that Ψ^∗_𝐠(0,𝑣)Ψ_𝐠(0,𝑣)=3Ψ^∗_𝐠(0,𝑣)Ψ_𝐡(0,𝑣)Ψ^∗_𝐠(0,𝑣)Ψ_𝐠(0,𝑣) (112) for 𝑣∈[0,1). This implies that Ψ^∗_𝐠(0,𝑣)=3Ψ^∗_𝐠(0,𝑣)Ψ_𝐡(0,𝑣)Ψ^∗_𝐠(0,𝑣) for 𝑣∈[0,1) due to the fact that ℂ⁹=range(Ψ_𝐠(0,𝑣))⊕ker(Ψ^∗_𝐠(0,𝑣)). So 𝐺(𝐡,𝐍,𝐌) is an oblique Gabor dual for 𝐺(𝐡,𝐍,𝐌) by Lemma 29.

Remark 32 and Example 33 show that the Gabor dual theory for subspace Gabor frames is complicated when 𝑁_𝑙, 1≤𝑙≤𝐿, are not all the same. There are still many unresolved problems in this direction. Next we work under the following assumption.

Assumption 34. 𝐿 is a positive integer, 𝐌=(𝑀,𝑀,…,𝑀), and 𝐍=(𝑁₁,𝑁₂,…,𝑁_𝐿) with 𝑁₁=𝑁₂=⋯=𝑁_𝐿=𝑁.

The following theorem characterizes the uniqueness of such duals.

Theorem 35. Given 𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) with each 𝑔_𝑙∈𝑙²(ℤ), let 𝐺(𝐠,𝐍,𝐌) be a frame for ℳ(𝐠,𝐍,𝐌). Then the following are equivalent:(i)𝐺(𝐠,𝐍,𝐌) has a unique Gabor dual of type I (type II);(ii)rank(Ψ_𝐠(𝑗,𝑣))∈{0,𝐿𝑞}(rank(Ψ_𝐠(𝑗,𝑣))∈{0,𝑝}) for a.e. (𝑗,𝑣)∈ℕ_𝑀/𝑞×[0,1);(iii)rank(Ψ_𝐠(𝑗,𝑣))∈{0,𝐿𝑞}(rank(Ψ_𝐠(𝑗,𝑣))∈{0,𝑝}) for a.e. (𝑗,𝑣)∈ℤ×ℝ.

Proof. We first deal with the Gabor dual of type I. By Lemma 12, (ii) and (iii) are equivalent. By Theorem 30 and Lemma 27, we only need to prove that (ii) holds if and only if a function 𝐴∶ℕ_𝑀×[0,1)→ℳ_{𝐿𝑞,𝐿𝑞} solves 𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1) (113) whenever it solves Ψ^∗_𝐠𝑗,𝑣𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1). (114) Suppose (ii) holds and 𝐴∶ℕ_𝑀×[0,1)→ℳ_{𝐿𝑞,𝐿𝑞} solves (114). If (𝑗,𝑣) is such that rank(Ψ_𝐠(𝑗,𝑣))=0, then Ψ_𝐠(𝑗,𝑣)=0, and thus 𝐴(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣)=0. If (𝑗,𝑣) is such that rank(Ψ_𝐠(𝑗,𝑣))=𝐿𝑞, then (114) implies that 𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣=0, (115) which leads to 𝐴(𝑗,𝑣)=0 and thus 𝐴(𝑗,𝑣)Ψ_𝐠(𝑗,𝑣)=0. Conversely, suppose (ii) does not hold. Then there exist some 𝑗₀∈ℕ_𝑀 and 𝐞_𝑘, 𝐞_𝑟 such that 𝒫𝑗₀,𝑣𝐞_𝑘≠0,𝐞^∗_𝑟Ψ_𝐠𝑗₀,𝑣≠0 (116) on some 𝐸⊂[0,1) with |𝐸|>0, where 𝒫(𝑗₀,𝑣) is the orthogonal projection of ℂ^𝐿𝑞 onto ker(Ψ^∗_𝐠(𝑗₀,𝑣)). Define 𝐴𝑗₀=𝒫𝑗,𝑣⁰,𝑣𝐞_𝑘𝐞^∗_𝑟,if𝑗=𝑗₀,𝑣∈𝐸,0,otherwise (117) for (𝑗,𝑣)∈ℕ_𝑀×[0,1). Then (114) holds, but (113) fails to hold.
Next we turn to the Gabor duals of type II. Similarly to the above arguments, by Theorem 30 and Lemmas 27 and 28, we only need to prove that (ii) holds if and only if a function 𝐵∶ℕ_𝑀×[0,1)→ℳ_𝑝,𝑝 solves Ψ_𝐠𝑗,𝑣𝐵𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1) (118) whenever it solves Ψ^∗_𝐠𝑗,𝑣Ψ_𝐠𝑗,𝑣𝐵𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣=0for𝑗∈ℕ_𝑀,a.e.𝑣∈[0,1). (119) The necessity can be proved similarly to the case of type I. Now we suppose (ii) does not hold. Then there exist some 𝑗₀∈ℕ_𝑀 and 𝐞_𝑘, 𝐞_𝑟 such that 𝒫𝑗₀,𝑣𝐞_𝑘≠0,Ψ_𝐠𝑗₀,𝑣𝐞_𝑟≠0 (120) on some 𝐸⊂[0,1) with |𝐸|>0, where 𝒫(𝑗₀,𝑣) is the orthogonal projection of ℂ^𝑝 onto ker(Ψ_𝐠(𝑗₀,𝑣)). Define 𝐵(𝑗₀,𝑣) by 𝐵^∗𝑗₀=𝒫𝑗,𝑣⁰,𝑣𝐞_𝑘𝐞^∗_𝑟,if𝑗=𝑗₀,𝑣∈𝐸,0,otherwise (121) for (𝑗,𝑣)∈ℕ_𝑀×[0,1). Then (119) holds, but (118) fails to hold. The proof is completed.

It is well known that every scalar 𝑘×𝑛 matrix 𝑀 with rank 𝑟≥1 has a decomposition 𝑀=𝑈𝑉𝐷000^∗, (122) where 𝑈 and 𝑉 are, respectively, 𝑘×𝑘 and 𝑛×𝑛 unitary matrices, 𝐷000 is a 𝑘×𝑛 block matrix in which 𝐷 is an 𝑟×𝑟 diagonal matrix with positive entries in the diagonal (see [1, Theorem 1.5.4]). Observe that rank(𝑀) may change in variables if 𝑀 is a matrix-valued function. Next we restrict ourselves to Gabor systems 𝐺(𝐠,𝐍,𝐌) such that Ψ_𝐠𝐷0𝑗,𝑣=𝑈𝑗,𝑣𝑉𝑗,𝑣00^∗𝑗,𝑣for𝑗∈ℕ_𝑀/𝑞,a.e.𝑣∈[0,1), (123) where 𝑈(𝑗,𝑣) and 𝑉(𝑗,𝑣) are, respectively, 𝐿𝑞×𝐿𝑞 and 𝑝×𝑝 unitary matrices for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1), and D(𝑗,𝑣)000 is an 𝐿𝑞×𝑝 block matrix in which 𝐷(𝑗,𝑣) is of the form 𝐷𝑗,𝑣=diag𝜆₁𝑗,𝑣,𝜆₂𝑗,𝑣,…,𝜆_𝑚𝑗,𝑣 (124) with 𝜆_𝑖(𝑗,𝑣)≥0 for 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1) and ⋃_{𝑗∈ℕ𝑀/𝑞}{𝑣∈[0,1)∶𝜆_𝑖(𝑗,𝑣)>0} being of positive measure for each 1≤𝑖≤𝑚.

As an immediate consequence of Theorems 19, 24, and 35, we have the following theorem.

Theorem 36. Let 𝟎≠𝐠=(𝑔₁,𝑔₂,…,𝑔_𝐿) be defined as in (123). Then(i)𝐺(𝐠,𝐍,𝐌) is a frame for ℳ(𝐠,𝐍,𝐌) with frame bounds 𝐴, 𝐵 if and only if 𝐴𝑀≤𝜆_𝑖≤𝑗,𝑣𝐵𝑀𝑜𝑛𝑗,𝑣∈ℕ_𝑀/𝑞×[0,1)∶𝜆_𝑖𝑗,𝑣>0 (125) for 1≤𝑖≤𝑚;(ii)𝐺(𝐠,𝐍,𝐌) is a Riesz basis (an orthonormal basis) for ℳ(𝐠,𝐍,𝐌) with Riesz bounds 𝐴, 𝐵 if and only if 𝑚=𝐿𝑞,𝐴𝑀≤𝜆_𝑖𝑗,𝑣≤𝐵𝑀𝜆_𝑖𝑗,𝑣=1𝑀 (126) for 1≤𝑖≤𝑚, 𝑗∈ℕ_𝑀/𝑞, and a.e. 𝑣∈[0,1);(iii)𝐺(𝐠,𝐍,𝐌) is a frame for ℳ(𝐠,𝐍,𝐌) with frame bounds 𝐴 and 𝐵 and simultaneously admits a unique Gabor dual of type I (type II) if and only if 𝑚=𝐿𝑞 (𝑚=𝑝), and for each 𝑗∈ℕ_𝑀/𝑞 and a.e. 𝑣∈[0,1), either 𝜆_𝑖(𝑗,𝑣)=0 for each 1≤𝑖≤𝑚 or 𝐴/𝑀≤𝜆_𝑖(𝑗,𝑣)≤𝐵/𝑀 for each 1≤𝑖≤𝑚.

Theorem 37. Let 𝐠 be defined as in (123) and let 𝐺(𝐠,𝐍,𝐌) be a frame for ℳ(𝐠,𝐍,𝐌). Then, for 𝐡=(ℎ_𝑙,ℎ₂,…,ℎ_𝐿) with each ℎ_𝑙∈𝑙²(ℤ), we have(i)𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type I for 𝐺(𝐠,𝐍,𝐌) if and only if Ψ_𝐡 has the form Ψ_𝐡=1𝑗,𝑣𝑀𝑈𝐶D0𝑗,𝑣𝑗,𝑣𝑉𝑗,𝑣00^∗𝑗,𝑣𝑓𝑜𝑟𝑗∈ℕ_𝑀/𝑞,𝑎.𝑒.𝑣∈[0,1), (127) where 𝐶∶ℕ_𝑀/𝑞×[0,1)→ℳ_{𝐿𝑞,𝐿𝑞} satisfies 𝐶(𝑗,𝑣)=(𝐶₁(𝑗,𝑣),0) with 𝐶₁(𝑗,𝑣)∈ℳ_{𝐿𝑞,𝑚}, the entries of the 𝑖th column of 𝐶₁(𝑗,𝑣) are essentially bounded on {(𝑗,𝑣)∈ℕ_𝑀/𝑞×[0,1)∶𝜆_𝑖(𝑗,𝑣)>0} for 1≤𝑖≤𝑚, and 𝐶𝑗,𝑣_𝑖,𝑛=0,𝑖≠𝑛,𝜆_𝑖𝑗,𝑣𝜆_𝑛𝜆𝑗,𝑣≠0,_𝑖𝑗,𝑣⁻²,𝑖=𝑛,𝜆_𝑖𝑗,𝑣≠0 (128) for 1≤𝑖, 𝑛≤𝑚;(ii)𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type II for 𝐺(𝐠,𝐍,𝐌) if and only if Ψ_𝐡 has the form Ψ_𝐡=1𝑗,𝑣𝑀𝑈𝐷0𝑗,𝑣𝐶𝑉𝑗,𝑣00𝑗,𝑣^∗𝑗,𝑣𝑓𝑜𝑟𝑗∈ℕ_𝑀/𝑞,𝑎.𝑒.𝑣∈[0,1), (129)where 𝐶∶ℕ_𝑀/𝑞×[0,1)→ℳ_𝑝,𝑝 satisfies 𝐶(𝑗,𝑣)=^𝐶1₀^(𝑗,𝑣) with 𝐶₁(𝑗,𝑣)∈ℳ_𝑚,𝑝, the entries of the 𝑖th row of 𝐶₁(𝑗,𝑣) are essentially bounded on {(𝑗,𝑣)∈ℕ_𝑀/𝑞×[0,1)∶𝜆_𝑖(𝑗,𝑣)>0} for 1≤𝑖≤𝑚, and 𝐶𝑗,𝑣_𝑖,𝑛=0,𝑖≠𝑛,𝜆_𝑖𝑗,𝑣𝜆_𝑛𝜆𝑗,𝑣≠0,_𝑖(𝑗,𝑣)⁻²,𝑖=𝑛,𝜆_𝑖𝑗,𝑣≠0 (130) for 1≤𝑖, 𝑛≤𝑚.

Proof. We only prove item (i), and (ii) can be proved similarly. By Theorem 30, 𝐺(𝐡,𝐍,𝐌) is a Gabor dual of type I for 𝐺(𝐠,𝐍,𝐌) if and only if there exists a function 𝐴∶ℕ_𝑀/𝑞×[0,1)→ℳ_{𝐿𝑞,𝐿𝑞} such that Ψ_𝐡𝑗,𝑣=𝐴𝑗,𝑣Ψ_𝐠,Ψ𝑗,𝑣_∗𝐠𝑗,𝑣=𝑀Ψ^∗_𝐠𝑗,𝑣𝐴𝑗,𝑣Ψ_𝐠𝑗,𝑣Ψ^∗_𝐠𝑗,𝑣 (131)(132) for 𝑗∈ℕ_𝑀/𝑞 and a.e. [0,1), and for each 𝑗∈ℕ_𝑀/𝑞 the entries of Ψ_𝐡(𝑗,𝑣) belong to 𝐿^∞([0,1)). Write 𝐶𝑗,𝑣=𝑀𝑈^∗𝐴𝑈=𝐶𝑗,𝑣𝑗,𝑣𝑗,𝑣₁𝑗,𝑣,𝐶₂𝑗,𝑣, (133) where 𝐶₁(𝑗,𝑣)∈ℳ_{𝐿𝑞,𝑚}. A simple computation shows that (132) is equivalent to 𝐷𝑗,𝑣=𝐷𝑗,𝑣𝐶₁₁𝑗,𝑣𝐷²𝑗,𝑣for𝑗∈ℕ_𝑀/𝑞,a.e.𝑣∈[0,1), (134) where 𝐶₁(𝑗,𝑣)=^𝐶11_𝐶^(𝑗,𝑣)_{21(𝑗,𝑣)}, 𝐶₁₁(𝑗,𝑣)∈ℳ_𝑚,𝑚. It is obvious that (134) is equivalent to (128), which implies that (132) is equivalent to (128). By (133), (132) can be rewritten as Ψ_𝐡=1𝑗,𝑣𝑀𝑈𝑗,𝑣𝐶₁𝑗,𝑣𝐷𝑗,𝑣,0𝑉^∗𝑗,𝑣. (135) Since 𝐶₂(𝑗,𝑣) does not appear in (135), we assume that 𝐶₂(𝑗,𝑣)=0 without loss of generality. By the unitary properties of 𝑈(𝑗,𝑣) and 𝑉(𝑗,𝑣), for each 𝑗∈ℕ_𝑀/𝑞, the entries of Ψ_𝐡(𝑗,𝑣) belong to 𝐿^∞([0,1)) if and only if the entries of 𝐶₁(𝑗,𝑣)𝐷(𝑗,𝑣) belong to 𝐿^∞([0,1)); that is, the entries of the 𝑖th column of 𝐶₁(𝑗,𝑣) are essentially bounded on {(𝑗,𝑣)∈ℕ_𝑀/𝑞×[0,1)∶𝜆_𝑖(𝑗,𝑣)>0} for 1≤𝑖≤𝑚. The proof is completed.

Next we discuss a special case of Theorem 36. Let 𝐍=1, 𝐌=2, 𝟏=(1,1), and 𝟐=(2,2) and write 𝑉₁(0,𝑣)=𝑉(0,𝑣)=1 and 𝑈₁()=10,𝑣√1+cos²,𝐷2𝜋𝑣−1cos2𝜋𝑣cos2𝜋𝑣1₁()=√0,𝑣1+cos²0,𝑈()=12𝜋𝑣0,𝑣√3𝐷()=√−1𝑖−𝑖cos2𝜋𝑣−𝑖sin2𝜋𝑣cos2𝜋𝑣𝑖sin2𝜋𝑣−𝑖𝑖−1−𝑖−𝑖sin2𝜋𝑣𝑖cos2𝜋𝑣−𝑖sin2𝜋𝑣−cos2𝜋𝑣−1−10,𝑣3000 (136) for 𝑣∈[0,1). Define 𝐠 and 𝐠=(𝑔₁,𝑔₂) as Ψ_𝐠(0,𝑣)=𝑈₁(0,𝑣)𝐷₁(0,𝑣)𝑉^∗₁(),Ψ𝐠()=𝑈()𝐷()𝑉0,𝑣0,𝑣0,𝑣0,𝑣^∗(0,𝑣) (137) by letting 𝑈=𝑈₁, 𝐷=𝐷₁, and 𝑉=𝑉₁ and 𝑈=𝑈, 𝐷=𝐷, and 𝑉=𝑉 in (123), respectively. Then Ψ_𝐠()=,Ψ𝐠()=[).0,𝑣−1cos2𝜋𝑣0,𝑣−1cos2𝜋𝑣−1−𝑖sin2𝜋𝑣for𝑣∈0,1 (138) This implies that 𝐠=𝑔₁=−𝜒_{0}+(1/2)𝜒_{−3,1}, 𝑔₂=−𝜒_{0}−(1/2)𝜒_{1}+(1/2)𝜒_{−3} and that ℳ(𝐠,1,2)=ℳ𝐠,𝟏,𝟐=𝑙²(ℤ) (139) by [5, Theorem 3.1]. So 𝐺(𝐠,1,2) and 𝐺(𝐠,𝟏,𝟐) are both frames for 𝑙²(ℤ), but neither is a Riesz basis by Theorem 36. Define 𝐡∈𝑙²(ℤ) and 𝐡=(ℎ₁,ℎ₂) with ℎ₁,ℎ₂∈𝑙²(ℤ) by Ψ_𝐡(0,𝑣)=(1/2)Ψ_𝐠(0,𝑣)(Ψ^∗_𝐠(0,𝑣)Ψ_𝐠(0,𝑣))^† and Ψ𝐡𝐠(0,𝑣)=(1/2)Ψ(0,𝑣)(Ψ𝐠^∗(0,𝑣)Ψ𝐠(0,𝑣))^† for 𝑣∈[0,1). Then 𝐺(𝐡,1,2) and 𝐺(𝐡,𝟏,𝟐) are, respectively, the canonical duals for 𝐺(𝐠,1,2) and 𝐺(𝐠,𝟏,𝟐) by [5, Remark 5.5], and Ψ_𝐡()=10,𝑣21+cos²Ψ𝐡()=12𝜋𝑣−1cos2𝜋𝑣,0,𝑣6−1cos2𝜋𝑣−1−𝑖sin2𝜋𝑣 (140)(141) for 𝑣∈[0,1). It follows that −1𝐡=6𝜒_{0}+1𝜒12^{−3,1}1,−6𝜒_{0}−1𝜒12^{1}+1𝜒12^{−3}, (142) and 𝐡 is infinitely supported. It is interesting that 𝐠 and 𝐡 are both finitely supported, but 𝐡 is not although 𝐠 is the first component of 𝐠. Therefore, there exist significant differences in Gabor duals between multi-window Gabor frames and single-window ones. Now we conclude this paper by summarizing the above arguments as the following example.

Example 38. Let 𝟏=(1,1) and 𝟐=(2,2), 𝐠=𝑔₁=−𝜒_{0}+12𝜒_{−3,1},𝑔₂=−𝜒_{0}−12𝜒_{1}+12𝜒_{−3}. (143)𝐡 and 𝐡, as in (140) and (142) and write 𝐠=(𝑔₁,𝑔₂). Then(i)𝐺(𝐠,1,2) and 𝐺(𝐠,𝟏,𝟐) are both frames for 𝑙²(ℤ), but neither is a Riesz basis for 𝑙²(ℤ);(ii)𝐺(𝐡,1,2) and 𝐺(𝐡,𝟏,𝟐) are, respectively, the canonical duals for 𝐺(𝐠,1,2) and 𝐺(𝐠,𝟏,𝟐);(iii)𝐡 is finitely supported, and 𝐡 is not.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11271037), Beijing Natural Science Foundation (Grant no. 1122008), and the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant no. KM201110005030).