Subspace mixed rational time-frequency multiwindow Gabor frames and their Gabor duals

For a usual multiwindow Gabor system, all windows share common time-frequency shifts. A mixed multiwindow Gabor system is one of its generalizations, for which time-frequency shifts vary with the windows. This paper addresses subspace mixed multiwindow Gabor systems with rational time-frequency product lattices. It is a continuation of (Li and Zhang in Abstr. Appl. Anal. 2013:357242, 2013; Zhang and Li in J. Korean Math. Soc. 51:897–918, 2014). In (Li and Zhang in Abstr. Appl. Anal. 2013:357242, 2013) we dealt with discrete subspace mixed Gabor systems and in (Zhang and Li in J. Korean Math. Soc. 51:897–918, 2014) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}(\mathbb{R})$\end{document}L2(R) ones. In this paper, using a suitable Zak transform matrix method, we characterize subspace mixed multiwindow Gabor frames and their Gabor duals, obtain explicit expressions of Gabor duals, and characterize the uniqueness of Gabor duals. We also provide some examples, which show that there exist significant differences between mixed multiwindow Gabor frames and usual multiwindow Gabor frames.


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 constants 0 < A ≤ B < ∞ such that where A and B are called frame bounds; it 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. In this case, the frame bounds are also called Riesz bounds. The fundamentals of frames can be found in [3][4][5][6]. For λ ∈ R, define the modulation operator E λ and translation operator T λ on L 2 (R) respectively by E λ f (·) = e 2π iλ· f (·) and T λ f (·) = f (·λ) for f ∈ L 2 (R). This paper addresses Gabor systems of the form G(g, a, b) = {E mb l T na l g l : m, n ∈ Z, 1 ≤ l ≤ L}, ( 2 ) where L is a fixed positive integer, g = {g l : 1 ≤ l ≤ L} ⊂ L 2 (R), a = (a 1 , a 2 , . . . , a L ), and b = (b 1 , b 2 , . . . , b L ) with a l , b l > 0, 1 ≤ l ≤ L. We denote by M(g, a, b) the closed linear span of G(g, a, b) in L 2 (R). A Gabor system G (g, a, b) is called a single-window Gabor system if L = 1; it is called a mixed multiwindow Gabor system if L > 1 and a l (or b l ) with 1 ≤ l ≤ L are not all the same; it is called a multiwindow Gabor system if L > 1, a 1 = a 2 = · · · = a L , and b 1 = b 2 = · · · = b L . Similarly, G(g, a, b) is called a subspace single-window Gabor frame if it is a frame for M(g, a, b) and L = 1; it is called a subspace mixed multiwindow Gabor frame if it is a frame for M(g, a, b), L > 1, and a l (or b l ) with 1 ≤ l ≤ L are not all the same; it is called a subspace multiwindow Gabor frame if it is a frame for M(g, a, b), L > 1, a 1 = a 2 = · · · = a L , and b 1 = b 2 = · · · = b L . In particular, when M(g, a, b) = L 2 (R), these frames are usual frames, which have been extensively studied [7][8][9][10][11][12]. To distinguish from subspace frames, we call them whole space frames. For a Bessel sequence G(g, a, b) in L 2 (R), define the associated synthesis operator T g : l 2 (Z 2 , C L ) → L 2 (R) by T g c = L l=1 m∈Z n∈Z c l,m,n E mb l T na l g l (3) for c = (c 1 , c 2 , . . . , c L ) ∈ l 2 (Z 2 , C L ). Then it is a bounded operator, and its adjoint operator T * g (so-called analysis operator) is given by where c l (f ) = { f , E mb l T na l g l } m,n∈Z for 1 ≤ l ≤ L. Similarly, for a Bessel sequence G(h, a, b) Here we do not require that h ⊂ M(g, a, b). In particular, an oblique Gabor dual G(h, a, b) for G(g, a, b) is said to be a Gabor dual of type I for G(g, a, b) if h ⊂ M(g, a, b), and it is said to be a Gabor dual of type II for G(g, a, b) if range(T * h ) ⊂ range(T * g ). These notions are borrowed from [13] and [14]. Observe that a Gabor dual of type II is not required to be in M(g, a, b), but a moment containment relation is required.
For the whole space Gabor frames, single-window ones have been extensively studied in the past twenty years and more [4,5,7,9,15,16]. Multiwindow frames were firstly studied by Zibulski and Zeevi [10] and Zeevi, Zibulski, and Porat [11]. By introduction of a Zak transform they developed a matrix (so-called Zibulski-Zeevi matrix) algebraic tool for multiwindow Gabor frames and applied it to image processing and computer vision. Since then, many researchers have studied multiwindow Gabor frames and related applications [2,[17][18][19][20]. It was also pointed out in [12] that the Zibulski-Zeevi matrix method is not very efficient for mixed Gabor frames. In [2], with the help of a new Zak transform matrix, different from the Zibulski-Zeevi matrix, Zhang and Li investigated mixed rational time-frequency multiwindow Gabor frames (Riesz bases and orthonormal bases) and their Gabor duals in L 2 (R). For subspace Gabor frames, single-window ones have been considered by several papers [1,13,[21][22][23][24][25][26]. In [24,26], and [27], a Zak transform matrix different from the Zibulski-Zeevi matrix was introduced and used effectively to study Gabor systems on periodic subsets of the real line, whereas the Zibulski-Zeevi matrix method does not work well for such Gabor systems. A variation of this method was applied to Gabor systems on discrete periodic sets [28,29]. In [30], a density result for Gabor frames on periodic subsets of R d is obtained via the Haar measure of the group generated by lattices. In [31], subspace multiwindow Gabor frames and their Gabor duals were characterized. All works mentioned, except [1] and [2], have not concerned real mixed multiwindow Gabor systems. Motivated by these observations, this paper is devoted to studying mixed multiwindow Gabor systems of the form (2). We work under the following assumptions: Assumption 1 L is a positive integer; is a frame (a Riesz basis, an orthonormal basis) for M(g, a, b), where g (τ l ) l (·) = e 2π iτ l b l · g l (·). It is well known that b 1 , b 2 , . . . , b L are commensurable if they are all rational numbers or rational multiples of some fixed irrational number. We also remark that the restriction of "rational time-frequency" here is for using "finite-order" Zak transform matrix-valued functions. So Assumptions 1 and 2 are relatively general and reasonable to some extent.
Throughout this paper, p and q denote the least common multiple of p l and the greatest common divisor of q l with 1 ≤ l ≤ L, respectively. It is easy to check that p and q are relatively prime and that p q is the least common multiple of p l q l with 1 ≤ l ≤ L. So, for each 1 ≤ l ≤ L, there exists a unique λ l ∈ N such that This implies that λ l a l = p bq for 1 ≤ l ≤ L by Assumption 2. We write a = p bq and Q = q l = 1 L λ l , and we denote by N t the set and by I t the t × t identity matrix for t ∈ N. Hereinafter we use I to denote the identity matrix when we need not specify its size. Given a measurable set S in R, a collection {S k : up to a set of measure zero. For λ > 0 and measurable sets S, S ⊂ R, we say that S is λZcongruent to S if there exists a partition {S k : k ∈ Z} of S such that {S k + λk : k ∈ Z} is a partition of S . In particular, only finitely many S k among S k , k ∈ Z, are nonempty if, in addition, both S and S are bounded. Obviously, S is also λZ-congruent to S if S is λZcongruent to S . So, in this case, we usually say that S and S are λZ-congruent. For s, t ∈ N, we denote by M s,t the set of all s × t complex matrices. Let M ∈ M s,t , which we consider as a linear mapping from C t into C s , and define the mappingM: ThenM is a bijection, and thus it has an inverse (M) -1 . We The mapping M † is called the pseudo-inverse of M. The rest of this paper is organized as follows. In Sect. 2, using a suitable Zak transform matrix method, we characterize subspace mixed multiwindow Gabor frames, their Gabor duals of types I and II, and the uniqueness of Gabor duals and obtain explicit expressions of the Gabor duals. In Sect. 3, we give some examples and remarks. They show that there exist significant differences between mixed multiwindow Gabor frames and usual multiwindow Gabor frames. In particular, not every subspace mixed multiwindow Gabor frame G(g, a, b) admits an oblique Gabor dual. So there should be many challenging problems in this direction.

Frame and dual characterization
Let L, a, and b satisfy Assumptions 1 and 2. In this section, using a Zak transform matrix method, we characterize the Gabor systems G(g, a, b) that are frames for M(g, a, b) and Gabor systems G(h, a, b) that are duals of a frame G(g, a, b) of types I and II. We also characterize the uniqueness of Gabor duals. For for a.e. (t, v) ∈ R 2 and define It is easy to check that the Zak transform has quasi-periodicity: By Lemma 2.1 in [24], and by Lemma 2.1 in [2] we have the following: . . .
where G l (t, v) is a block matrix of the form It is easy to prove that if a 1 = a 2 = · · · = a L , then Q = Lq, and an arbitrary function However, it is not the case if a l , 1 ≤ l ≤ L, are not all the same. By an argument similar to that in [1], we have Example 2.1, which provides us with a counterexample. Therefore, we must be careful when we define g by a function M(t, v) are not all the same.
are not all the same. Then there exists 1 ≤ l ≤ L such that λ l > 1. We may as well assume that λ 1 > 1. Choose Suppose there exists g such that (9) and (11). This is a contradiction.
Define the Fourier transform F : . . .  (g, a, b) is a Bessel sequence in L 2 (R) with Bessel bound B.

Lemma 2.5 For
im n v C n G(t, v)D n for n = k n q + (m n qr n )p with (k n , r n , m n ) ∈ N p × N q × Z and a.e. (t, v) ∈ R 2 , where D n = 0 e -2π iv I k n I p-k n 0 , C n = diag(C 1,n (v), C 2,n (v), . . . , C L,n (v)), C l,n (v) denotes the block matrix (with λ l blocks) of the form diag(C n , C n , . . . , C n ) with C n = 0 I q-r n e 2π iv I r n 0 . g, a, b) is a Bessel sequence in L 2 (R), where T g c is as in (3). (iv) If G(g, a, b) is a Bessel sequence in L 2 (R), then Proof , v e -2π imbt e -2π inv dt dv for (r, β l ) ∈ N q × N λ l , 1 ≤ l ≤ L. When G(g, a, b) is a Bessel sequence, the integrand in (14) belongs to L 2 ([0, 1 b ) × [0, 1)) by Lemma 2.3(vi). It follows that 1). This leads to the lemma.

By an argument similar to Lemmas 27, 28 in [1] and Lemmas 3.3, 3.4 in [31]
, we have the following two lemmas.
Proof Since S h,g = T g T * h , applying Lemma 2.5(iii), (iv) leads to the lemma. let G(g, a, b) and G(h,  a, b) be Bessel sequences in L 2 (R). Then the following are equivalent: (h, a, b) is an oblique Gabor dual for G (g, a, b).

Lemma 2.9 Given
Proof By Lemma 2.5(ii), (ii) and (iii) are equivalent. So, to prove the lemma, it suffices to prove the equivalence between (i) and the following equation: Since range(T g ) is dense in M(g, a, b), (i) holds if and only if S h,g T g c = T g c for c ∈ l 2 Z 2 , C L or, equivalently, by Lemmas 2.1, 2.5, and 2.8, which is in turn equivalent to for Obviously, (16) implies (17). Now suppose (17) holds. For arbitrary fixed x ∈ C Q , choose d(t, v) as (17). So (16) holds by the arbitrariness of x. The proof is completed.
By the definition of pseudo-inverse, we have following two lemmas.

Lemma 2.10 For a d × d matrix A satisfying A * = A, we have
where P range(A) denotes the orthogonal projection from C d onto range(A).

Lemma 2.11 For an arbitrary s × t matrix A, we have
where P range(A) denotes the orthogonal projection from C s onto range(A).
So the matrix-valued function G(t, v) in Definition 2.1 is exactly g (t, v) by Definition 2.2 in [31]. It follows that G (g, a, b) and G(g, a, b) have the same frame properties. Therefore, using Theorems 2.9 and 2.14 and Remark 2.10 in [31], we have the following two theorems.
Theorem 2.1 For g = {g 1 , g 2 , . . . , g L } ⊂ L 2 (R), the following are equivalent: (g, a, b) is a frame for M(g, a, b) with frame bounds A and B. G(g, a,

b) is a Riesz basis for M(g, a, b) with Riesz bounds A and B (an orthonormal basis) if and only if
By Lemmas 2.6-2.9, we have following theorem, which characterizes the Gabor duals of type I (resp., type II): let G(g, a, b) be a frame for M(g, a, b). G(h, a, b) being a Bessel sequence in L 2 (R), G(h, a, b) is a Gabor dual of type I (type II) for G (g, a, b) if and only if the following hold: G(g, a, b) be a frame for M(g, a, b). Then the following are equivalent: (g, a, b) has a unique Gabor dual of type I (type II).

Theorem 2.4 Let
Proof We only prove "type I" part. The other part can be proved similarly. By Lemma 2.5(ii) we only need to prove the equivalence between (i) and (ii). By Lemmas 2.6 and 2.9, (i) holds if and only if for a function A : implies Next, we prove the equivalence between (ii) and the above implication. Obviously, (ii) leads to this implication. Next, we prove that the implication fails if (ii) is violated and thus finish the proof. Suppose (ii) does not hold. Then rank (G(t, v) where e i is the vector with the ith component being 1 and others zeros. Since rank (G(t, v) By the same procedure as in Lemma 4.1 in [31], there exist 1 ≤ k ≤ Q and )). By the argument of [8], p. 978, 1). Then A(·, ·) satisfies (18) but does not satisfy (19). The proof is completed.  (h, a, b) is a Gabor dual of type II for G(g, a, b) (h, a, b) is an oblique dual of G(g, a, b) if one of the following conditions holds: matrix method if the time shift parameters a 1 , a 2 , . . . , a L are not all the same. Example 3.2 shows that not every subspace mixed Gabor frame G (g, a, b) admits an oblique Gabor dual. Therefore, there exist significant differences between mixed multiwindow Gabor frames and usual multiwindow Gabor frames, and there should be many challenging problems in this direction. Definition 3.1 Given g = {g 1 , g 2 , . . . , g L } ⊂ L 2 (R), let G(g, a, b) be a Bessel sequence in L 2 (R). We say that G (g, a, b) has Riesz property if for c ∈ l 2 (Z 2 , C L ), we must have c = 0 whenever T g c = 0. By Lemma 2.5(ii) and Theorem 2.1 in [2] we have the following: let G(g, a, b) be a Bessel sequence in L 2 (R).

Conclusions
A mixed multiwindow Gabor system is one of generalizations of multiwindow Gabor systems, whose time-frequency shifts vary with the windows. This paper addresses subspace mixed multiwindow Gabor systems with rational time-frequency product lattices. Using a suitable Zak-transform matrix method, in this paper, we characterize subspace mixed multiwindow Gabor frames and their Gabor duals, obtain explicit expressions of Gabor duals, and characterize the uniqueness of Gabor duals. Some provided examples show that there exist significant differences between mixed multiwindow Gabor frames and usual multiwindow Gabor frames.