Admissibility Conditions for Multi-window Gabor Frames on Discrete Periodic Sets
Najib Khachiaa Mohamed Rossafi

	
	Admissibility for Multi-window  Gabor Frames]Admissibility Conditions for Multi-window  Gabor Frames on Discrete Periodic Sets
	
	
	
	
	
	
	
	
	
	
		
	
	
	[1]Najib Khachiaakhachiaa.najib@uit.ac.ma
	These authors contributed equally to this work.
	
	2]Mohamed Rossafimohamed.rossafi1@uit.ac.ma
	These authors contributed equally to this work.
	
	
	[1]Laboratory Partial Differential Equations, Spectral Algebra and Geometry, Department of Mathematics, Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco
	
	[2]Laboratory Partial Differential Equations, Spectral Algebra and Geometry, Higher School of Education and Training, University Ibn Tofail, Kenitra, Morocco
	
	
	
	
		In this paper, 𝒢(g,L,M,N) denotes a L-window Gabor system on a periodic set 𝕊, where L,M,M∈ℕ and g={g_l}_l∈ℕ_L⊂ℓ^2(𝕊).  We characterize which g generates  a complete multi-window  Gabor system and a multi-window Gabor frame 𝒢(g,L,M,N) on  𝕊 using the Zak transform.   Admissibility conditions for a periodic set to admit a  complete multi–window  Gabor system, multi-window  Gabor (Parseval) frame, and multi–window  Gabor (orthonormal) basis 𝒢(g,L,M,N) are given with respect to the parameters L, M and N.
	
	
	
	
	
	
	
	
	[MSC Classification]42C15; 42C40.
	
	[
    [
    September 9, 2024
=====================



§ INTRODUCTION AND PRELIMINARIES


When a signal appears periodically but intermittently, it can be considered within the entire space ℓ^2(ℤ) and analyzed in the standard manner. However, if the signal is only emitted for short periods, this method might not be the best approach. To perform Gabor analysis of the signal most efficiently while preserving all its features, Li and Lian studied single window Gabor systems on discrete periodic sets. They derived density results and frame characterizations. Compared to single window Gabor systems, multiwindow Gabor systems can be both interesting and beneficial, as they allow for more flexibility by using windows of different types and widths. For certain parameters N and M, there does not exist an associated Gabor frame with a single window. However, allowing the use of multiple windows guarantees the existence of Gabor frames. For example, for 𝕊=ℤ, a Gabor frame with one window exists only if N≤ M. N. Khachiaa, M. Rossafi, and S. Kabbaj showed in <cit.> that when it is not the case,  by allowing the use of multiple windows, the existence of Gabor frames associated with L-windows is ensured, where L is an integer satisfying N≤ LM (which, of course, exists).

A sequence {f_i}_i∈ℐ, where ℐ is a countable set, in a separable Hilbert space H is said to be frame if there exist 0< A≤ B<∞ ( called frame bounds) such that for all f∈ H, 
    Af^2≤∑_i∈ℐ|⟨ f,f_i ⟩|^2≤ Bf^2.
 If only the upper inequality holds,  {f_i}_i∈ℐ is called  a Bessel sequence with Bessel bound B. If A=B, the sequence is called a tight frame and if A=B=1, it is called a Parseval frame for H. For more details on frame theory, the reader can refer to <cit.>.

Denote by ℕ the set of positive integers, i.e. ℕ:={1,2,3,...} and for a given K∈ℕ, write ℕ_K:={0,1,...,K-1}. Let N,M,L∈ℕ and p,q∈ℕ such that pgcd(p,q)=1 and N/M=p/q. A nonempty subset 𝕊 of ℤ is said to be Nℤ-periodic set if for all j∈𝕊 and for all n∈ℤ, j+nN ∈𝕊. For K∈ℕ, write 𝕊_K:=𝕊∩ℕ_K. We denote by ℓ^2(𝕊) the closed subspace of ℓ^2(ℤ) defined by, 
    ℓ^2(𝕊):={f∈ℓ^2(ℤ):  f(j)=0  if  j∉𝕊}.
 
Define the modulation operator E_m/M with m∈ℤ and the translation operator T_nN with n∈ℤ for f∈ℓ^2(𝕊) by: 
    E_m/Mf(.):=e^2π i m/M. f(.),    T_nNf(.):=f(.-nN).
 
The modulation and translation  operators are unitary operators of ℓ^2(𝕊).
 For g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊), the associated multiwindow discrete Gabor system (M-D-G) is given by,

    𝒢(g,L,M,N):={E_m/MT_nNg_l}_m∈ℕ_M,n∈ℤ, l∈ℕ_L.
 
For j∈ℤ, we denote 𝒦_j={k∈ℕ_p:  j+kM∈𝕊} and  𝒦(j):=diag(χ_𝒦_j(0),χ_𝒦_j(1),...,χ_𝒦_j(p-1)).

Let K∈ℕ.  The discrete Zak tansform z_K of f∈ℓ^2(ℤ) for j∈ℤ and a.e θ∈ℝ is defined by, 	

    z_Kf(j,θ):=∑_k∈ℤf(j+kK)e^2π i k θ.

z_Kf is quasi-periodic. i.e. ∀ j, k,l∈ℤ, θ∈ℝ we have: 
    z_Kf(j+kK,θ+l)=e^-2π ikθz_Kf(j,θ).

Then z_K is, completely, defined by its values for j∈ℕ_M and θ∈ [0,1[. The reader can refer to <cit.> for more details on discrete Zak transform.

This paper is organized as follows. In section 2,  we will present some auxiliary lemmas to be used in the following sections. In section 3, we characterize which g∈ℓ^2(𝕊) generates a complete multi-window discrete Gabor system and a multi-window discrete Gabor frame 𝒢(g,L,M,N) for ℓ^2(𝕊) using the Zak transform. In section 4, we provide an admissibility characterization for complete multi-window discrete Gabor systems and multi-window discrete Gabor frames 𝒢(g,L,M,N) 
on a discrete periodic set 𝕊, and we finish with an example.



§ AUXILIARY LEMMAS

In this section, we present several lemmas and introduce the notations that will be utilized in the following sections. In addition to the notations introduced in the introduction, let ℳ_s,t denote the set of all s× t matrices with entries in ℂ. We use p∧ q to indicate that p and q are coprime. For a given matrix A, A^* is the conjugate transpose of A, N(A) represents its kernel, and A_s,t refers to its (s,t)-component. When A is a column matrix, we denote its r-component simply by A_r. Following this, we provide several definitions and results that will be useful throughout the rest of the paper.
<cit.>
	Let K∈ℕ, and 𝕊 be a Kℤ-periodic set in ℤ. Write 𝕊_K=𝕊∩ℕ_K. Then the restriction of z_K to ℓ^2(𝕊) is a unitary linear operator from ℓ^2(𝕊) to the Hilbert space  ℓ^2(Q) where Q=𝕊_K× [0,1[ and 
	
    ℓ^2(Q):={ψ:Q →ℂ: ∑_j∈𝕊_K∫_0^1 |ψ(j,θ)|^2   dθ< ∞}.
 
 

Let A,B⊂ℤ and K∈ℕ.  We say that A is Kℤ-congruent to B if there exists a partition {A_k}_k∈ℤ of A such that {A_k+kK}_k∈ℤ is a partition of B.
<cit.>
	Let N,M∈ℕ and p,q∈ℕ such that p∧ q=1 and N/M=p/q. Then 
	the set 
Δ:={j+kM-rN:  j∈ℕ_M/q,  k∈ℕ_p,  r∈ℕ_q} is qN-congruent to ℕ_pM.

For each f∈ℓ^2(ℤ), we associate a matrix-valued function Z_f:ℤ×ℝ→ℳ_q,p whose entry at the r-th row and the k-th column is defined by

    Z_f(j,θ)_r,k=z_pMf(j+kM-rN,θ).

<cit.>
	Let N,M∈ℕ and p,q∈ℕ such that p∧ q=1 and N/M=p/q. Then 
	z_pMf is completeley determined by the matrices Z_f(j,θ) for j∈ℕ_M/q and θ∈ [0,1[.

	Conversely, a matrix-valued function Z:ℕ_M/q× [0,1[→ℳ_q,p such that for all j∈ℕ_M/q, Z(j,.)_r,k∈ L^2([0,1[) also determines a unique f∈ℓ^2(ℤ) such that for all j∈ℕ_M/q, θ∈ [0,1[, Z_f(j,θ)=Z(j,θ).
	
	For g:={g_l}_l∈ℕ_L∈ℓ^2(ℤ), we associate the matrix-valued function Z_g:ℕ_N/M×ℝ→ℳ_qL,p defined for all j∈ℕ_M/q, θ∈ℝ by the block matrix: 
    Z_g(j,θ)=[   Z_g_0(j,θ);   Z_g_1(j,θ);            ⋮; Z_g_L-1(j,θ) ].


<cit.>
	Let p,q∈ℕ such that p∧ q=1. Then for all j∈ℤ, there exists a unique (k_0,l_0)∈ℕ_p×ℤ and a unique (k_0,m_0,r_0)∈ℕ_p×ℤ×ℕ_q such that j= k_0q+l_0p =k_0q+(m_0q+r_0)p.

<cit.>
	Let M, N ∈ℕ and p, q ∈ℕ such that  N/M=p/q and p∧ q = 1. Then, for
	all m ∈ℤ, there exists a unique (j, r, k, ℓ) ∈ℕ_M/q×ℕ_q ×ℕ_p ×ℤ such that m =
	j+kM -rN +ℓ qN.


The following proposition characterizes which Multi-window Gabor frames are Multi-window Gabor Riesz beses using the parameters L,M and N.
<cit.> 
	Let g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊).
	
		
  * 𝒢(g,L,M,N) is a frame for ℓ^2(𝕊) only when card(𝕊_N)/M≤ L.
		
  * Assume that 𝒢(g,L,M,N) is a frame for ℓ^2(𝕊). Then following statements are equivalent: 
		
			
  * 𝒢(g,L,M,N) is a  Riesz basis  (exact frame) for ℓ^2(𝕊).
			
  * card(𝕊_N)/M=L.
		
	


<cit.>
	Let M ∈ℕ and E ⊂ℤ. Then the following conditions are equivalent:
	
		
  * { e^2π im/M·χ_E(·) : m ∈ℕ_M } is a tight frame for ℓ^2(E) with frame bound M.
		
  * { e^2π im/M·χ_E(·) : m ∈ℕ_M } is complete in ℓ^2(E).
		
  * E is Mℤ-congruent to a subset of ℕ_M.
		
  * ∑_k ∈ℤχ_E(· + kM) ≤ 1 on ℤ.
	
 
<cit.>
	Let {f_i}_i∈ℐ, where ℐ is a countable sequence,  be a Parseval frame for a separable Hilbert space H. Then the following statements are equivalent:
	
		
  * {f_i}_i∈ℐ is a Riesz basis.
		
  * {f_i}_i∈ℐ is an orthonormal basis.
		
  * For all i∈ℐ, f_i=1.
	


	Let f∈ℓ^2(ℤ). If f∈ℓ^2(𝕊), then for all j∈ℤ, a.e θ∈ℝ, 
    Z_f(j,θ)𝒦(j)=Z_f(j,θ).

	


	Let s∈ℕ_q and t∈ℕ_p. We have: 
	
    [                                                                   (Z_f(j,θ)𝒦(j))_s,t                                                                                    =                                                        ∑_k=0^p-1Z_f(j,θ)_s,k𝒦(j)_k,t;                                                                                                                                                                         =                                                   ∑_k=0^p-1Z_f(j,θ)_s,kδ_k,tχ_𝒦_j(t);                                                                                                                                                                         =                                                                 Z_f(j,θ)_s,tχ_𝒦_j(t);                                                                                                                                                                         = {[              Z_f(j,θ)_s,t   if  t∈𝒦_j,;                         0   otherwise. ]. ]

	On the other hand, we have Z_f(j,θ)_s,t=z_pMf(j+tM-sN,θ)=∑_k∈ℤf(j+tM-sN+kpM)e^2π i kθ)=∑_k∈ℤf(j+tM-sN+kqN)e^2π i kθ since pM=qN. Then, if t∉𝒦_j, j+tM∉𝕊, then, for all k∈ℤ, j+tM-sN+kqN∉𝕊 by the Nℤ-periodicity of 𝕊, thus f(j+tM-sN+kqN)=0 for all k∈ℤ. Hence Z_f(j,θ)_s,t=0 if t∉𝒦_j. The proof is completed.



	For all j∈ℤ, 𝒦(j) is an orthogonal projection on ℂ^p. i.e.
	
		
  * 𝒦(j)^2=𝒦(j).
		
  * 𝒦(j)^*=𝒦(j).
	


	Let j∈ℤ. We have:  
    [                                      𝒦(j)^2                                           =  diag(χ_𝒦_j(0)^2,χ_𝒦_j(1)^2,…,χ_𝒦_j(p-1)^2);                                                                                       = diag(χ_𝒦_j(0),χ_𝒦_j(1),…, χ_𝒦_j(p-1))=𝒦(j). ]

	And 
    [                                      𝒦(j)^*                                           =        diag(χ_𝒦_j(0),χ_𝒦_j(1),…,χ_𝒦_j(p-1));                                                                                       = diag(χ_𝒦_j(0),χ_𝒦_j(1),…, χ_𝒦_j(p-1))=𝒦(j). ]




§ CHARACTERIZATIONS OF COMPLETE  MULTIWINDOW  DISCRETE GABOR SYSTEMS AND MULTIWINDOW DISCRETE GABOR FRAMES

In this section we use all the notations already introduced Without introducing them again. Let L,M,N∈ℕ and p,q∈ℕ such that p∧ q=1 and  N/M=p/q and denote 𝕊_N=𝕊∩ℕ_N. We characterise what g=:{g_l}_l∈ℕ_L⊂ℓ^2(𝕊) generates  a complete  Gabor system and  a Gabor frame 𝒢(g,L,M,N).
We first present the following proposition:

	Let g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊). Let M,N∈ℕ and p,q∈ℕ such that p∧ q=1. Then the integer-valued function (j,θ)→ rank(Z_g(j,θ)) is M/q-periodic with respect to j and 1-periodic with respect to θ. Moreover,  for all j∈ℤ and a.e θ∈ℝ, we have: 
	
    rank(Z_g(j,θ))≤  card(𝒦_j).


For the proof, we need the following lemma:

	For all j∈ℤ,  a.e θ∈ℝ, k'∈ℕ_p and r'∈ℕ_q, we have: 
	
    rank(Z_g(j,θ))=rank(Z_g(j+k'M+r'N,θ)).



	Let j∈ℤ, θ∈ℝ. Denote by C_k(j,θ) the k-th column of Z_g(j,θ).

	We have for all l∈ℕ_L and for all r∈ℕ_q, z_pMg_l((j+k'M)+kM-rN,θ)=z_pMg_l(j+(k+k_0)M-rN,θ).
	If 0≤ k≤ p-k'-1, then C_k(j+k_0M,θ)=C_k+k_0(j,θ). Otherwise  (p-k'≤ k≤ p-1), we have by quasi-periodicity of the Zak transform z_pMg_l((j+k'M)+kM-rN,θ)=e^-2π i θ.z_pMg_l(j+(k+k'-p)M-rN,θ). Then C_k(j+k'M,θ)=e^-2π iθ.C_k+k'-p(j,θ).

	Consider the map: 
    ϕ:[                                                            ℕ_p                                                              →                                                            ℕ_p;                                                              k                                                              ↦ {[  k+k'    if   0≤ k≤ p-k'-1,; k+k'-p    if  p-k'≤ k≤ p-1. ]. ]

	We show, easily, that ϕ is injective. It is, then, bijective. Hence: 
	
    [             rank(Z_g(j,θ))                          =     rank{C_k(j,θ):  k∈ℕ_p};                                                     = rank{C_k(j+k'M,θ):  k∈ℕ_p};                                                     =        rank(Z_g(j+k'M,θ)). ]

	Denote by R_r(j,θ) the r-th row of Z_g(j,θ). Then there exists a unique (l,r_0)∈ℕ_L×ℕ_q such that r=lq+r_0. Then (∀ j_0∈ℤ) R_r(j_0,θ) is the r_0-th row of Z_g_l(j_0,θ).
	We have z_pMg_l((j+r'N)+kM-r_0N,θ)=z_pMg_l(j+kM-(r_0-r')N,θ).

	If r'≤ r_0≤ q-1, then R_r(j+r'N,θ) is the (r_0-r')-th row of Z_g_l(j,θ), thus R_r(j+r'N,θ)=R_r-r'(j,θ). Otherwise (0≤ r_0≤ r'-1), since pM=qN and by quasi-periodicity of the Zak transform, we have: z_pMg_l((j+r'N)+kM-r_0N,θ)=z_pMg_l(j+kM-(r_0-r'+q)N,θ). Then R_r(j+r'N,θ) is the (r_0-r'+q)-th row of Z_g_l(j,θ), thus R_r(j+r'N,θ)=R_r-r'+q(j,θ).

	Consider the map: 
    ψ:[                                                                     ℕ_qL                                                                        →                                                                     ℕ_qL;                                                                        r                                                                        ↦ {[ r-r'    if   lq+r'≤ r≤ (l+1)q-1,;    r-r'+q    if  lq≤ r≤ lq+r'-1. ]. ]

	It is easy to show that ψ is injective. Then it is bijective. Hence: 
	
    [              rank(Z_g(j,θ))                           =     rank{R_r(j,θ):  r∈ℕ_qL};                                                       = rank{R_r(j+r'N,θ):  r∈ℕ_qL};                                                       =         rank(Z_g(j+r'N,θ)). ]

	Hence For all j∈ℤ,  θ∈ℝ, k'∈ℕ_p and r'∈ℕ_q, we have: 
	
    rank(Z_g(j,θ))=rank(Z_g(j+k'M+r'N,θ)).



	Let j∈ℤ,  θ∈ℝ. Given an arbitrary s∈ℤ. By lemma <ref>, there exists a unique (k_0,m_0,r_0)∈ℕ_p×ℤ×ℕ_q such that s=k_0q+(m_0q+r_0)p. Then: 
	
    [                         Z_g(j+M/qs,θ)                                     =              Z_g(j+k_0M+m_0pM+r_0N,θ);                                                                           =        e^2π i m_0θ.Z_g(j+k_0M+r_0N,θ);                                                                           = e^2π i m_0θ.Z_g(j,θ)     lemma <ref>. ]

	Hence: 
    rank(Z_g(j+M/qs,θ))=rank(Z_g(j,θ)).

	The 1-periodicity with respect to θ is simply due to the 1-periodicity of the Zak transform with respect to θ. 
	On the other hand, if k∉𝒦_j, i.e. k is such that j+kM∉𝕊, then, by Nℤ-periodicity of 𝕊, for all r∈ℤ, j+kM-rN∉𝕊, and thus for all l∈ℕ_L and for all r∈ℕ_q, z_pMg_l(j+kM-rN,θ)=0 since pM=qN. Then the k-th column of Z_g(j,θ) is identically zero. Hence rank(Z_g(j,θ))≤ card(𝒦_j).


	Let j∈ℤ. Since {𝕊_N+nN}_n∈ℤ is a partition of 𝕊, we have: 
	
    [                                       card(𝒦_j)                                               =                              ∑_k=0^p-1χ_𝕊(j+kM);                                                                                               =                    ∑_k=0^p-1∑_n∈ℤχ_𝕊_N(j+nN+kM);                                                                                               = ∑_k=0^p-1∑_n∈ℤχ_𝕊_N(j+np+kq/qM)    since N=pM/q;                                               =              ∑_n∈ℤχ_𝕊_N(j+M/qn)    lemma <ref>. ]

	Hence card(𝒦_j) is M/q-periodic. Then  the inequality (1) holds for all j∈ℤ and a.e θ∈ℝ if and only if it holds for all j∈ℕ_M/q and a.e θ∈ [0,1[.



The following lemma is very useful for the rest.

	Let g:={g_l}_l∈ℕ_L⊂ℓ^2(ℤ). Let f∈ℓ^2(ℤ). Then the following statements are equivalent:
	
		
  * f is orthogonal to 𝒢(g,L,M,N).
		
  * For all j∈ℕ_M, a.e. θ∈ [0,1[, Z_g(j,θ)F(j,θ)=0.
	
	Where F(j,θ):=(z_pMf(j+kM,θ))_k∈ℕ_p^t for j∈ℕ_M and a.e θ∈ [0,1[.


	We have: 

	f is orthogonal to 𝒢(g,L,M,N) ⟺ f is orthogonal to 𝒢(g_l,M,N) for all l∈ℕ_L.

	And we have:
 
	Z_g(j,θ)F(j,θ)=0 ⟺ Z_g_l(j,θ)F(j,θ)=0 for all l∈ℕ_L.

	These equivalences together with lemma 3.1 in <cit.> complete the proof.


The following proposition characterizes complete multi-window Gabor systems on 𝕊.

	Let g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊). Then the following statements are equivalent:
	
		
  * 𝒢(g,L,M,N) is complete in ℓ^2(𝕊).
		
  * For all j∈ℕ_M/q, a.e θ∈ [0,1[, rank(Z_g(j,θ))=card(𝒦_j).
		
  * For all j∈ℤ, a.e θ∈ℝ, rank(Z_g(j,θ))=card(𝒦_j).
	


	By proposition <ref>, rank(Z_g(j,θ)) is M/q-periodic with repect to j and 1-periodic with respect to θ.  And by remark <ref>, card(𝒦_j) is M/q-periodic. Hence (2)⟺ (3).

	We will use the notations in lemma <ref>.  It is obvious that for f∈ℓ^2(ℤ), F(j,θ)=0 for all j∈ℕ_M and  a.e θ∈ [0,1[ if and only if f=0. Then by lemma <ref>, (1) is equivalent to the fact that: for f∈ℓ^2(𝕊),
 (∀ j∈ℕ_M, a.e θ∈ [0,1[, Z_g(j,θ)F(j,θ)=0) ⟹  (∀ j∈ℕ_M, a.e θ∈ [0,1[,  F(j,θ)=0).
	
		
  (1)⟹(3) Assume that 𝒢(g,L,M,N) is complete in ℓ^2(𝕊) and suppose, by contradiction, that (3) fails. Then by proposition <ref>, there exist j_0∈ℕ_M and E_0⊂ [0,1[ with positive measure such that for all θ∈ E_0:
		
    rank(Z_g(j_0,θ))< card(𝒦_j_0).

		For a.e θ∈ [0,1[, denote ℙ(j_0,θ):ℂ^p→ℂ^p the orthogonal projection onto the kernel of Z_g(j_0,θ). Let {e_k}_k∈ℕ_p be the standard orthonormal basis of ℂ^p. 
		Suppose that F=span{e_k: k∈𝒦_j}⊂ N(ℙ(j_0,θ)) for a.e θ∈ [0,1[. Then F⊕ N(Z_g(j_0,θ)) is an orthogonal sum. Thus:
		
    [                                p                                ⩾           dim(F⊕ N(Z_g(j_0,θ)) );                                                                 =       dim F+dim (N(Z_g(j_0,θ)) );                                                                 = card(𝒦_j)+(p-rank(Z_g(j_0,θ)) ). ]

		Hence rank(Z_g(j_0,θ) )⩾ card(𝒦_j). Contradiction. Then there exist k_0∈𝒦_j_0 and E_0'⊂ [0,1[ with positive measure such that e_k_0∉ N(ℙ(j_0,θ) ) for a.e θ∈ E_0'. i.e. ℙ(j_0,θ)e_k_0≠ 0 for a.e θ∈ E_0'. Define for all j∈ℕ_M, a.e θ∈ [0,1[, F(j,θ)=δ_j,j_0.ℙ(j_0,θ)e_k_0. Observe that if k∈ℕ_p-𝒦_j_0, then e_k∈ N(Z_g(j_0,θ) ) for a.e θ∈ [0,1[. Then for all k∈ℕ_p-𝒦_j_0, a.e θ∈ [0,1[, ℙ(j_0,θ)e_k=e_k. Thus for k∈ℕ_p-𝒦_j, F(j_0,θ)_k=⟨ F(j_0,θ),e_k⟩=⟨ℙ(j_0,θ)e_k_0,e_k⟩=⟨ e_k_0,ℙ(j_0,θ) e_k⟩=⟨ e_k_0,e_k⟩=0.

		Define f∈ℓ^2(ℤ) by z_pMf(j+kM,θ)=F(j,θ)_k for all j∈ℕ_M, a.e θ∈ [0,1[. Then by lemma <ref>, f∈ℓ^2(𝕊). Since F(j_0,θ)=ℙ(j_0,θ)e_k_0≠ 0 for all θ∈ E_0' which is with positive measure, then f≠ 0.

		On the other hand, we have Z_g(j,θ)F(j,θ)=0 for all j∈ℕ_M and a.e θ∈ [0,1[. In fact, if j≠ j_0, then, by definition of F, for a.e θ∈ [0,1[, F(j,θ)=0, hence Z_g(j,θ)F(j,θ)=0. Otherwise, F(j_0,θ)=ℙ(j_0,θ)e_k_0, then Z_g(j_0,θ)F(j_0,θ)=Z_g(j_0,θ)ℙ(j_0,θ)e_k_0=0 since ℙ(j_0,θ)e_k_0∈ N(Z_g(j_0,θ) ). Then, by lemma <ref>, f is orthogonal to 𝒢(g,L,M,N) but f≠ 0. Contradiction with (1).
		
  (2)⟹ (1) Assume (3) and let f∈ℓ^2(𝕊) such that  for all j∈ℕ_M and a.e θ∈ [0,1[: 
    Z_g(j,θ)F(j,θ)=0.
Let's prove that F(j,θ)=0 for all j∈ℕ_M, a.e θ∈ [0,1[. Let j∈ℕ_M. If 𝒦_j=∅, then by the definition of z_pMf  and since pM=qN, F(j,θ)=0 for a.e θ∈ [0,1[. Otherwise, i.e. 𝒦_j≠∅. Let k∈ℕ_p-𝒦_j, then, by the definition of the Zak transform and since pM=qN, the k-th column of Z_g(j,θ) is identically zero and we also have z_pMf(j+kM,θ)=0 for a.e θ∈ [0,1[. From (3), the submatrix of Z_g(j,θ) of size qL× card(𝒦_j) obtained by removing all the columns with indices not in 𝒦_j has the same rank than Z_g(j,θ) which is  card(𝒦_j). Then this submatrix is injective, thus, by equality (2), z_pMf(j+kM,θ)=0 for a.e θ∈ [0,1[. Then for all k∈ℕ_p, z_pMf(j+kM,θ)=0 for a.e θ∈ [0,1[. Hence F(j,θ)=0 for a.e θ∈ [0,1[. Hence 𝒢(g,L,M,N) is complete in ℓ^2(𝕊).
	

Now we characterize multi-window Gabor frames for ℓ^2(𝕊) using the Zak transform.

	Given g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊). Then the following statements are equivalent:
	
		
  * 𝒢(g,L, M,N) is a frame for ℓ^2(S)with frame bounds0 < A ≤ B .
		
  * 
    A/M.𝒦(j)
    			≤∑_l∈ℕ_L Z_g_l^*(j, θ)Z_g_l(j, θ) ≤B/M.𝒦(j).

		for allj ∈M/q and a.eθ∈[0, 1[.
		
  *  The inequality(3)holds for allj ∈ℕ_Mand a.eθ∈[0, 1[.

For the proof, we will need the following lemma.

	Denote L^∞(𝕊_pM× [0,1[) the set of functions F on 𝕊_pM× [0,1[ such that for all j∈𝕊_pM, F(j,.)∈ L^∞([0,1[), and  Δ := z_pM |ℓ^2(𝕊)^-1(L^∞(𝕊_pM× [0,1[)). Let g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊). Then the following statements are equivalent:
	
  * 𝒢(g,L,M,N) is a frame for ℓ^2(𝕊) with frame bounds A≤ B.
		
  *  For all  f ∈Δ, we have: 
    A/M∑_j=0^M-1∫_0^1F(j, θ)^2   dθ≤∑_l=0^l-1∑_j=0^M-1∫_0^1 Z_g_l(j, θ) F(j, θ)^2   dθ≤B/M∑_j=0^M-1∫_0^1F(j, θ)^2   dθ.

		Where F(j,θ), for all j∈ℕ_M and a.e θ∈ [0,1[, is as defined in lemma <ref>.
	
	By density of L^∞(𝕊_pM× [0,1[) in L^2(𝕊_pM× [0,1[) and by the unitarity of z_pM from ℓ^2(𝕊) onto L^2(𝕊_pM× [0,1[) (Since pM=qN, then 𝕊 is qNℤ-periodic in ℤ), Δ is dense in ℓ^2(𝕊). Hence 𝒢(g,L,M,N) is a frame for ℓ^2(𝕊) with frame bounds A≤ B if and only if for all f∈Δ, Af^2≤∑_l=0^L-1∑_n∈ℤ∑_m=0^M-1|⟨ f, e^2π i m/M.g_l(.-nN)⟩|^2≤ Bf^2.
	Let f∈Δ, it is clear that  for all r∈ℕ_q, Z_g_l(j,.)F(j,.)_r∈ L^2([0,1[). 
	We have: [                                                                                                                                                       ∑_l=0^L-1∑_n∈ℤ∑_m=0^M-1|⟨ f, e^2π i m/M.g_l(.-nN)⟩|^2;                                                                                                                                                                                                           =                                   ∑_l=0^L-1∑_r=0^q-1∑_n∈ℤ∑_m=0^M-1|⟨ f, e^2π i m/M.g_l(.-(r+nq)N)⟩|^2;                                                                                                                                                                                                           = ∑_l=0^L-1∑_r=0^q-1∑_n∈ℤ∑_m=0^M-1|⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-(r+nq)N))⟩|^2    by unitarity of z_pM;                                                                                                                                                                                                           =          ∑_l=0^L-1∑_r=0^q-1∑_n∈ℤ∑_m=0^M-1|⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-rN-npM))⟩|^2    since qN=pM. ]
	By a simple calculation, we obtain that for all j∈ℤ and a.e θ∈ℝ: 
	z_pM(e^2π i m/M.g_l(.-rN-npM))(j,θ)=z_pMg_l(j-rN,θ)  e^2π i nθ e^2π i m/Mj.
	Then: [                                                                                                                                  ⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-rN-npM))⟩;                                                                                                                                                                         =               ∑_j=0^pM-1∫_0^1 z_pMf(j,θ) z_pMg_l(j-rN,θ)e^-2π i nθ  dθ. e^-2π i m/Mj;                                                                                                                                                                         = ∑_j=0^M-1∑_k=0^p-1∫_0^1 z_pMf(j+kM,θ) z_pMg_l(j+kM-rN,θ)e^-2π i nθ  dθ. e^-2π i m/Mj;                                                                                                                                                                         =                         ∑_j=0^M-1∫_0^1 Z_g_l(j,θ)F(j,θ)_r e^-2π i nθdθ.  e^-2π im/Mj;                                                                                                                                                                         =     ∑_j=0^M-1T(j).  e^-2π im/Mj    where  T(j)=∫_0^1 Z_g_l(j,θ)F(j,θ)_r e^-2π i nθdθ;                                                                                      ]
	Observe that T is M-periodic. Since {1/√(M)e^2π i m/M.}_m∈ℕ_M is an orthonormal basis for ℓ^2(ℕ_M); the space of M-periodic sequences, then we have:[                                                       ∑_m=0^M-1|⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-rN-npM))⟩|^2;                                                                                                           =                        ∑_m=0^M-1|⟨ T, e^2π i m/M.⟩|^2;                                                                                                           =                                                  MT^2;                                                                                                           =  M∑_j=0^M-1|∫_0^1 Z_g_l(j,θ)F(j,θ)_r e^-2π i nθdθ|^2.;                                                       ]
	Since {e^2π i nθ}_n∈ℤ is an orthonormal basis for L^2([0,1[), then we have: 
	[                                                            ∑_n∈ℤ∑_m=0^M-1|⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-rN-npM))⟩|^2;                                                                                                                     =  M∑_j=0^M-1∑_n∈ℤ|∫_0^1 Z_g_l(j,θ)F(j,θ)_r e^-2π i nθdθ|^2.;                                                                                                                     =         M∑_j=0^M-1∑_n∈ℤ|⟨Z_g_l(j,.)F(j,.)_r,e^2π i n.⟩|^2.;                                                                                                                     =                             M∑_j=0^M-1Z_g_l(j,.)F(j,.)_r^2;                                                                                                                     =                 M∑_j=0^M-1∫_0^1 | Z_g_l(j,θ)F(j,θ)_r|^2 dθ;                                                            ]
	Hence: [                                                                     ∑_r=0^q-1∑_n∈ℤ∑_m=0^M-1|⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-rN-npM))⟩|^2;                                                                                                                                       =                 M∑_j=0^M-1∫_0^1 ∑_r=0^q-1| Z_g_l(j,θ)F(j,θ)_r|^2 dθ;                                                                                                                                       =                             M∑_j=0^M-1∫_0^1  Z_g_l(j,θ)F(j,θ)^2 dθ.;                                                                     ]
	The norm in the last line is the 2-norm in ℂ^q. Thus: ∑_r=0^q-1∑_n∈ℤ∑_m=0^M-1|⟨ z_pMf, z_pM(e^2π i m/M.g_l(.-rN-npM))⟩|^2=M∑_l=0^L-1∑_j=0^M-1∫_0^1  Z_g_l(j,θ)F(j,θ)^2 dθ.
	Hence: 
    ∑_l=0^L-1∑_n∈ℤ∑_m=0^M-1|⟨ f, e^2π i m/M.g_l(.-nN)⟩|^2=M∑_l=0^L-1∑_j=0^M-1∫_0^1  Z_g_l(j,θ)F(j,θ)^2 dθ.

	On the other hand, we have by unitarity of z_pM:[                                           f^2                                             =                                       z_pMf^2;                                                                                           =            ∑_j=0^pM-1∫_0^1| z_pMf(j,θ)|^2  dθ;                                                                                           = ∑_j=0^M-1∑_k=0^p-1∫_0^1 | z_pM(j+kM,θ)|^2  dθ;                                                                                           =     ∑_j=0^M-1∫_0^1 ∑_k=0^p-1| F(j,θ)_k|^2  dθ;                                                                                           =                  ∑_j=0^M-1∫_0^1  F(j,θ)^2  dθ;                                               ]
	The norm in the last line is the 2-norm in ℂ^p. Thus: 
    f^2=∑_j=0^M-1∫_0^1  F(j,θ)^2  dθ.

	Then, combining (5) and (6), the proof is completed.

  (1)⟹ (3) Assume that 𝒢(g,L,M,N) is a frame for ℓ^2(𝕊). Then for all f∈Δ, 
    A/M∑_j=0^M-1∫_0^1F(j, θ)^2   dθ≤∑_l=0^l-1∑_j=0^M-1∫_0^1 Z_g_l(j, θ) F(j, θ)^2   dθ≤B/M∑_j=0^M-1∫_0^1F(j, θ)^2   dθ.

		Fix x:={x_k}_k∈ℕ_p∈ℂ^p, j_0∈ℕ_M and h∈ L^∞([0,1[), and define for all j∈ℕ_M and a.e θ∈ [0,1[, F(j,θ):={δ_j,j_0 χ_𝒦_j(k) x_k  h(θ) }_k∈ℕ_p. 
Then A/M∑_j=0^M-1∫_0^1F(j, θ)^2   dθ=A/M𝒦(j_0)x^2∫_0^1 | f(θ)|^2  dθ=A/M⟨𝒦(j_0)x,x⟩∫_0^1 | h(θ)|^2 dθ (lemma <ref>). On the other hand, we have: [                                   ∑_l=0^l-1∑_j=0^M-1∫_0^1 Z_g_l(j, θ) F(j, θ)^2   dθ                                                                                    =                                        ∑_l=0^l-1∫_0^1 Z_g_l(j_0, θ) F(j_0, θ)^2   dθ;                                                                                                                                                                         =                        ∑_l=0^l-1∫_0^1∑_r=0^q-1|(Z_g_l(j_0, θ) F(j_0, θ))_r  |^2   dθ;                                                                                                                                                                         =            ∑_l=0^l-1∫_0^1∑_r=0^q-1|∑_k=0^p-1 Z_g_l(j_0, θ)_r,k F(j_0, θ)_k  |^2   dθ;                                                                                                                                                                         =    ∑_l=0^l-1∫_0^1∑_r=0^q-1|∑_k=0^p-1 Z_g_l(j_0, θ)_r,k χ_𝒦_j_0(k) x_k  h(θ) |^2   dθ;                                                                                                                                                                         = ∑_l=0^l-1∫_0^1∑_r=0^q-1|∑_k=0^p-1 Z_g_l(j_0, θ)_r,k (𝒦(j_0)x)_k  |^2  | h(θ)|^2   dθ;                                                                                                                                                                         =                 ∑_l=0^l-1∫_0^1∑_r=0^q-1|(Z_g_l(j_0, θ)𝒦(j_0)x)_r  |^2  | h(θ)|^2  dθ;                                                                                                                                                                         =                                ∑_l=0^l-1∫_0^1Z_g_l(j_0,θ)𝒦(j_0)x  ^2  | h(θ)|^2   dθ;                                                                                                                                                                         =                    ∑_l=0^l-1∫_0^1Z_g_l(j_0,θ)x  ^2  | h(θ)|^2   dθ    by lemma <ref>;                                                                                                                                                                         =                      ∫_0^1⟨∑_l=0^l-1 Z_g_l(j_0,θ)^*Z_g_l(j_0,θ)x,x⟩ | h(θ)|^2    dθ.;                                                                                      ]
		Then for all j∈ℕ_M,  x∈ℂ^p and f∈ L^∞([0,1[), we have: 
		
    A/M.⟨𝒦(j)x,x⟩ ∫_0^1| h(θ)|^2  dθ≤∫_0^1⟨∑_l=0^l-1 Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩ | h(θ)|^2   dθ≤B/M.⟨𝒦(j)x,x⟩∫_0^1| h(θ)|^2  dθ.

		For j∈ℕ_M and x∈ℂ^p fixed, denote C=A/M.⟨𝒦(j)x,x⟩ and D=B/M.⟨𝒦(j)x,x⟩. Assume, by contradiction, that 
    C>⟨∑_l=0^L-1Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩ ,
 on a subset of [0,1[ with a positive measure. Denote 
		D={θ∈ [0,1[:  (8)  holds}.

		For all k∈ℕ, denote D_k:={θ∈ [0,1[:  C-C/k≤⟨∑_l=0^L-1Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩≤ C-C/k-1}.

		It is clear that {D_k}_k∈ℕ forms a partition for D. Since mes(D)>0, then there exists k∈ℕ such that mes(D_k)>0. Let h:=χ_D_k, we have: [ ∫_0^1⟨∑_l=0^l-1 Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩ | f(θ)|^2   dθ                                                         =            ∫_D_k⟨∑_l=0^l-1 Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩  dθ;                                                                                                                   ≤                                          (C-C/K. mes(D_k);                                                                                                                   <                                                C.mes(D_k);                                                                                                                   =             C∫_0^1 | h(θ)|^2 dθ   Contradiction with (7). ]
		Suppose, again by contradiction, that 
    D<⟨∑_l=0^L-1Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩,
 on a subset of [0,1[ with a positive measure. Denote 
		D'={θ∈ [0,1[:  (9)  holds}.

		For all k∈ℕ, m∈ℕ, denote D_k,m':={θ∈ [0,1[:  D(k+1/m+1)≤⟨∑_l=0^L-1Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩≤ D(k+1/m)}.
		It is clear that {D_k,m'}_k∈ℕ forms a partition for D'. Since mes(D')>0, then there exist k∈ℕ and m∈ℕ such that mes(D_k,m')>0. Let h:=χ_D_k,m', we have: [ ∫_0^1⟨∑_l=0^l-1 Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩ | h(θ)|^2   dθ                                                         =         ∫_D_k,m'⟨∑_l=0^l-1 Z_g_l(j,θ)^*Z_g_l(j,θ)x,x⟩  dθ;                                                                                                                   ⩾                                   D(k+1/m+1). mes(D_k,m');                                                                                                                   >                                             D.mes(D_k,m');                                                                                                                   =             D∫_0^1 | h(θ)|^2 dθ   Contradiction with (7). ]
		Hence for all j∈ℕ_M, and a.e θ∈ [0,1[, we have: 
		
    A/M.𝒦(j)
    			≤∑_l∈ℕ_L Z_g_l^*(j, θ)Z_g_l(j, θ) ≤B/M.𝒦(j).

  (3)⟹(1) Assume (3). For f∈Δ, j∈ℕ_M and a.e θ∈ [0,1[,  the k-th component of F(j,θ) is zero if k∉𝒦_j. Then 𝒦(j)F(j,θ)=F(j,θ). Then: 
    A/MF(j,θ)^2≤⟨∑_l=0^L-1 Z_g_l(j,θ)^*Z_g_l(j,θ)F(j,θ),F(j,θ)⟩≤B/MF(j,θ)^2.

		Hence: 
    A/M∑_j=0^M-1∫_0^1F(j,θ)^2≤∑_l=0^L-1∑_j=0^M-1∫_0^1  Z_g_l(j,θ)F(j,θ)^2≤B/M∑_j=0^M-1∫_0^1F(j,θ)^2.

		Then lemma <ref> implies (1).
		
  (3)⟹ (2) Since ℕ_M/q⊂ℕ_M.
		
  (2)⟹ (3) Assume (2). Then the inequality (3) holds for all j ∈ℕ_M/q and a.e. θ∈ [0, 1[. Let's prove that it  holds for all j ∈ℕ_M and a.e. θ∈ [0, 1[. Let j ∈ℕ_M. By Lemma <ref>, there exists a unique (j', r', k', ℓ') ∈ℕ_M/q×ℕ_q ×ℕ_p ×ℤ such that j = j' + k'M - r'N + ℓ'qN. Then, by the
		quasiperiodicity of the discrete Zak transform, we have, for all l∈ℕ_L, after a simple calculation:
		
		(Z_g_l(j, θ)^*Z_g_l(j, θ))_k_1, k_2 =
		∑_r=0^q-1
		(Z_qN g_l) (j' + (k_1 + k')M - (r + r')N, θ) (Z_qNg) (j' + (k_2 + k')M - (r + r')N, θ)
		
    =
    			 
    				(Z_g_l(j', θ)^*Z_g_l(j', θ))_k_1+k', k_2+k',    if  k_1 + k' < p,   k_2 + k' < p 
    
    				e^-2π iθ (Z_g_l(j', θ)^*Z_g_l(j', θ))_k_1+k', k_2+k'-p,    if  k_1 + k' < p,   k_2 + k' ≥ p 
    
    				e^2π iθ (Z_g_l(j', θ)^*Z_g_l(j', θ))_k_1+k'-p, k_2+k',    if  k_1 + k' ≥ p,   k_2 + k' < p 
    
    				(Z_g_l(j', θ)^*Z_g_l(j', θ))_k_1+k'-p, k_2+k'-p,    if  k_1 + k' ≥ p,   k_2 + k' ≥ p.

		for k_1, k_2 ∈ℕ_p and a.e. θ∈ [0, 1[. Define V: ℂ^p →ℂ^p by Vx = y = (y_0, y_1, …, y_p-1)^t:
		
		
		y_k =
		 
			e^-2π i θx_k-k'+p,    if  0 ≤ k < k' 

			x_k-k',    if  k' ≤ k < p
		
		for x ∈ℂ^p. Then V is a unitary operator, and
		
		⟨ Z_g_l(j, θ)^*Z_g_l(j, θ)x, x ⟩ = ⟨ Z_g_l(j', θ)^*Z_g_l(j', θ)Vx, Vx ⟩,
		
		for a.e. θ∈ [0, 1[ and all x ∈ℂ^p. Then, by (2), we have:
		
		
    A/M⟨ V^*𝒦(j') Vx, x ⟩≤⟨∑_l=0^L-1Z_g_l(j, θ)^*Z_g_l(j, θ)x, x ⟩≤B/M⟨ V^*𝒦(j') Vx, x ⟩,

		for a.e. θ∈ [0, 1) and each x ∈ℂ^p. When k+k' < p, k+k' ∈𝒦_j' if and only if j'+(k+k')M ∈𝕊,  equivalently, j' + k'M - r'N + ℓ'qN + kM ∈𝕊, i.e. j + kM ∈𝕊. Therefore, k + k' ∈𝒦_j if and only if k ∈𝒦_j when k + k' < p. Similarly, k + k' - p ∈𝒦_j' if and only if k ∈𝒦_j when
		k + k' ≥ p. It follows that:
		
		V^*𝒦(j')V = 𝒦(j).
		
		Which together with (10) gives (3). The proof is completed.
	
	In the case of 𝕊=ℤ. For all j∈ℕ_M/q, 𝒦_j=ℕ_p. Then the condition (2) in the proposition <ref> is equivalent to rank(Z_g(j,θ) )=p for all j∈ℕ_M/q and a.e θ∈ [0,1[. And the condition (2) in the proposition <ref> is equivalent to: For all j∈ℕ_M/q and a.e θ∈ [0,1[, 
	
    A/M.I_p,p≤∑_l∈ℕ_L Z_g_l^*(j, θ)Z_g_l(j, θ) ≤B/M.I_p,p.

	
	Where I_p,p is the identity matrix in ℳ_p,p.
	
	
	
	
	
	


§ ADMISSIBILITY CONDITIONS FOR A COMPLETE MULTIWINDOW GABOR SYSTEM AND A MULTIWINDOW GABOR FRAME

Note that, in what follows, we will use all the notations already introduced Without introducing them again. In this section, we study conditions for a periodic set𝕊to admit a complete multi-window Gabor system, and a multi-window Gabor frame. LetL,M,N∈ℕandp,q∈ℕsuch thatN/M = p/qandp∧q=1.

In what follows, we give some useful lemmas for the rest.

  * card(𝒦_j)≤ qL for all j∈ℕ_M/q⟹card(𝕊_N)≤  LM.
		
  *  Assume that card(𝒦_j)≤ qL for all j∈ℕ_M/q. Then:
		card(𝒦_j)=qL  for all j∈ℕ_M/q ⟺ card(𝕊_N)=  LM.
  *  Assume  that card(𝒦_j)≤ qL for all j∈ℕ_M/q.  We have: 
		[                             card(𝕊_N)                                     =                         ∑_j∈ℤχ_𝕊_N(j);                                                                           =           ∑_j∈ℕ_M/q∑_n∈ℤχ_𝕊_N(j+M/qn);                                                                           = ∑_j∈ℕ_M/qcard(𝒦_j)    by remark <ref>;                                                                           ≤                            M/q.qL=LM. ]
  *  Assume that card(𝒦_j)≤ qL for all j∈ℕ_M/q.

		Assume that card(𝒦_j)=qL for all j∈ℕ_M/q. Then by the proof of (1), we have: [          card(𝕊_N)                  = ∑_j∈ℕ_M/qcard(𝒦_j);                                     =         M/q.qL=LM. ]
		Conversely, assume that card(𝕊_N)=  LM. Again by the proof of (1), we have ∑_j∈ℕ_M/qcard(𝒦_j)=card(𝕊_N)=LM=M/q.qL. Since 
		card(𝒦_j)≤ qL for all j∈ℕ_M/q, then card(𝒦_j)= qL for all j∈ℕ_M/q.
	
	Let L,M∈ℕ and E_0, E_1, …, E_L-1⊂ℤ be mutually disjoint. Denote E=⋃_l∈ℕ_LE_l. Then the following statements are equivalent:
	
  * {e^2π i m/M.χ_E_l}_m∈ℕ_M,   l∈ℕ_L is complete in  ℓ^2(E).
		
  *  For all l∈ℕ_L, {e^2π i m/M.χ_E_l}_m∈ℕ_M is complete in ℓ^2(E_l).
	
  (1)⟹ (2) Assume (1). Fix l_0∈ℕ_L and let f∈ℓ^2(E_l_0) be orthogonal to  {e^2π i m/M.χ_E_l_0}_m∈ℕ_M. 
		Define f∈ℓ^2(E) by f(j)=f(j) if j∈ E_l_0 and 0 otherwise. It is clear that if l≠ l_0, f is orthogonal to  {e^2π i m/M.χ_E_l}_m∈ℕ_M. And we have: 
		[                                     ⟨f,e^2π i m/M.χ_E_l_0⟩                                                          =                           ∑_j∈ Ef(j)e^-2π i m/Mjχ_E_l_0(j);                                                                                                                     =                       ∑_j∈ E_l_0f(j)e^-2π i m/Mjχ_E_l_0(j);                                                                                                                     = 0    since  f is orthogonal to {e^2π i m/M.χ_E_l_0}_m∈ℕ_M. ]
		Then f is orthogonal to {e^2π i m/M.χ_E_l}_m∈ℕ_M, l∈ℕ_L which is complete in ℓ^2(E), thus f=0 on E, and then f=0 on E_l.
		
  (2)⟹ (1) Assume (3) and let h∈ℓ^2(E) be orthogonal to {e^2π i m/M.χ_E_l}_m∈ℕ_M, l∈ℕ_L. For all l∈ℕ_L, define h_l∈ℓ^2(E_l) as the restriction of h on E_l, i.e.  h_l:=h|_E_l. Let l∈ℕ_L. Fix l∈ℕ_L. Since h is orthogonal to {e^2π i m/M.χ_E_l}_m∈ℕ_M, then ∑_j∈ Eh(j) e^-2π i m/M.χ_E_l(j)=0, then ∑_j∈ E_lh(j) e^-2π i m/M.χ_E_l(j)=0, thus ∑_j∈ E_lh_l(j) e^-2π i m/M.χ_E_l(j)=⟨ h_l,e^2π i m/M.χ_E_l⟩ =0. Hence h_l is orthogonal to  {e^2π i m/M.χ_E_l}_m∈ℕ_M which is complete in ℓ^2(E_l). Hence h_l=0 on E_l. This for all l∈ℕ_L, therfore h=0 on E. Hence {e^2π i m/M.χ_E_l}_m∈ℕ_M,l∈ℕ_L is complete in ℓ^2(E).
		
	
	Let L,M∈ℕ and E_0, E_1, …, E_L-1⊂ℤ  be mutually disjoint. Denote E=⋃_l∈ℕ_LE_l. Then the following statements are equivalent:
	
  * {e^2π i m/M.χ_E_l}_m∈ℕ_M,   l∈ℕ_L is a tight frame for ℓ^2(E) with frame bound M.
		
  *  For all l∈ℕ_L, {e^2π i m/M.χ_E_l}_m∈ℕ_M is a tight frame for ℓ^2(E_l) with frame bound M.
	
  (1)⟹ (2) Assume (1). Fix l_0∈ℕ_L and let f∈ℓ^2(E_l_0). Define f∈ℓ^2(E) by f(j)=f(j) if j∈ E_l_0 and 0 otherwise. It is clear that ⟨f,e^2π i m/M.χ_E_l⟩=0 if l≠ l_0 and that f=f. Together with the fact that {e^2π i m/M.χ_E_l}_m∈ℕ_M,   l∈ℕ_L is a tight frame for ℓ^2(E) with frame bound M, we have: [                                              Mf^2                                                 =     ∑_m=0^M-1|∑_j∈ Ef(j)e^-2π i m/Mjχ_E_l_0(j)|^2;                                                                                                   = ∑_m=0^M-1|∑_j∈ E_l_0f(j)e^-2π i m/Mjχ_E_l_0(j)|^2;                                                                                                   =             ∑_m=0^M-1|⟨ f,e^2π i m/M.χ_E_l_0⟩|^2. ]
		Hence {e^2π i m/M.χ_E_l_0}_m∈ℕ_M is a tight frame for ℓ^2(E_l_0) with frame bound M. And this for all l_0∈ℕ_L.
		
  (2)⟹ (1) Assume (2). Let h∈ℓ^2(E). For all l∈ℕ_L, define h_l∈ℓ^2(E_l) as the restriction of h en E_l, i.e h_l=h|_E_l. We have ∑_j∈ E| h(j)|^2=∑_l∈ℕ_L∑_j∈ E_l| h_l(j)|^2, then h^2=∑_l∈ℕ_Lh_l^2. It is also clear that  ⟨ h,e^2π i m/M.χ_E_l⟩=⟨ h_l,e^2π i m/M.χ_E_l⟩. Since For all l∈ℕ_L, {e^2π i m/M.χ_E_l}_m∈ℕ_M is a tight frame for ℓ^2(E_l) with frame bound M, then for all l∈ℕ_l, we have Mh_l^2=∑_m∈ℕ_M|⟨ h_l,e^2π i m/M.χ_E_l⟩|^2. Hence: 
		Mh^2=∑_l∈ℕ_L∑_m∈ℕ_M|⟨ h,e^2π i m/M.χ_E_l⟩|^2.
		This for all h∈ℓ^2(E_l). Hence {e^2π i m/M.χ_E_l}_m∈ℕ_M,   l∈ℕ_L is a tight frame for ℓ^2(E) with frame bound M.
	
	Let L,M∈ℕ and E_0, E_1, …, E_L-1⊂ℤ be mutually disjoint. Denote E=⋃_l∈ℕ_LE_l. Then the following statements are equivalent:
	
  * {e^2π i m/M.χ_E_l}_m∈ℕ_M,   l∈ℕ_L is a tight frame for ℓ^2(E) with frame bound M.
		
  * {e^2π i m/M.χ_E_l}_m∈ℕ_M,   l∈ℕ_L is complete in  ℓ^2(E).
		
  *  For all l∈ℕ_L, E_l is Mℤ-congruent to a subset of ℕ_M.
		
  *  For all l∈ℕ_L, ∑_k∈ℤχ_E_l(.+kM)≤ 1 on ℤ.
	
	It is a direct result of lemma <ref>, lemma <ref> and lemma <ref> together.




The following proposition presents a characterization for the admissibility of𝕊to admit a complete multi-window Gabor system𝒢(g, L, M, N).

	Then the following statements are equivalent:
	
  *  There exist g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊) such that 𝒢(g,L,M,N) is complete in ℓ^2(𝕊).
		
  *  For all j∈ℕ_M/q, we have: 
		
    card(𝒦_j)≤ qL.

  * (5) holds for all j∈ℤ.
	
	By remark <ref>, card(𝒦_j) is M/q-periodic. Then (2)⟺ (3).
	
  (1)⟹ (2) Assume (1). Let j∈ℕ_M/q. Then by proposition <ref>, card(𝒦_j)=rank(Z_g(j,θ) )≤ qL since Z_g(j,θ)∈ℳ_qL,p.
		
  (2)⟹ (1) Assume (2). By proposition <ref>, it suffices to find a matrix-valued function Z:ℕ_M/q× [0,1[→ℳ_qL,p such that Z(j,.)_r,k∈ L^2([0,1[) for all (j,r,k)∈ℕ_M/q×ℕ_qL×ℕ_p and for all j∈ℕ_M/q, if k∉𝒦_j, the k-th column of Z(j,.) is identically zero, and  such that rank(Z(j,θ) )=card(𝒦_j). Indeed, in this case, for all l∈ℕ_L, define Z_l as the matrix-valued function Z_l:ℕ_M/q× [0,1[→ℳ_q,p defined for all j∈ℕ_M/q, θ∈[0,1[ by Z_l(j,θ):=Z(j,θ)_lq≤ r≤ (l+1)q-1,  0≤ k≤ p-1. Then by lemma <ref>, there exists a unique g_l∈ℓ^2(𝕊) such that Z_g_l=Z_l. Denote g={g_l}_l∈ℕ_L, then for all j∈ℕ_M/q,  θ∈ [0,1[, Z(j,θ)=Z_g(j,θ), hence by poposition <ref>, 𝒢(g,L,M,N) is complete in ℓ^2(𝕊) since rank(Z_g(j,θ) )=card(𝒦_j). 

		For the existence of a such matrix-valued function: Let j∈ℕ_M/q and a.e θ∈ [0,1[.  Define a qL× p constant matrix-valued function Z(j,.):=(Z^0(j,.),Z^1(j,.),…, Z^p-1(j,.) ) on [0,1[, where Z^k(j,.) is the k-th column of Z(j,.) for k∈ℕ_p, such that Z^k(j,.)=0 if k∉𝒦_j and {Z^k(j,.),  k∈𝒦_j} is linearly independent in ℂ^qL. This is possible since card(𝒦_j)≤ qL. Then for a.e θ∈ [0,1[,  rank(Z(j,θ) )=card(𝒦_j). Hence we obtain the desired matrix-valued function Z.
	
	The following statements are equivalent:
	
  *  There exist E_0, E_1, …, E_L-1⊂ℤ mutually disjoint such that 𝒢( {χ_E_l}_l∈ℕ_L,  L,M,N) is a tight frame for ℓ^2(𝕊) with frame bound M.
		
  *   For all j∈ℕ_M/q, we have: 
		
    card(𝒦_j)≤ qL.

  * (5) holds for all j∈ℤ.
	 By remark <ref>, card(𝒦_j) is M/q-periodic. Then (2)⟺ (3).
	
  (1)⟹ (2) By proposition <ref>.
		
  (2)⟹ (1) It suffices to find E_0,E_1, …, E_L-1⊂ℤ mutually disjoint such that for all l∈ℕ_L, E_l is Mℤ-congruent to a subset of ℕ_M and E:=⋃_l∈ℕ_LE_l is Nℤ-congruent to 𝕊_N. In fact, in this case, we have  ℓ^2(𝕊)=⊕_n∈ℤℓ^2(E+nN) and, by lemma <ref>,  {e^2π im/M.χ_E_l}_m∈ℕ_M,l∈ℕ_L is a tight frame for ℓ^2(E) with frame bound M. Then  for all n∈ℤ, {e^2π im/M.χ_E_l(.-nN)}_m∈ℕ_M,l∈ℕ_L is a tight frame for ℓ^2(E+nN) with frame bbound M. Hence, by similar arguments used in the proof of lemma <ref>, 𝒢( {χ_E_l}_l∈ℕ_L ,L,M,N):={e^2π im/M.χ_E_l(.-nN)}_n∈ℤ, m∈ℕ_M,l∈ℕ_L is a tight frame for ⊕_n∈ℤℓ^2(E+nN)=ℓ^2(𝕊) with frame bound M.
For the construction of the desired E_l:

		Let j∈ℕ_M/q. Let K be the maximal integer satisfying Kq≤ card(𝒦_j). For all l∈ℕ_K, define 𝒦_j^l as the set of the (l+1)-th q elements of 𝒦_j, 𝒦_j^K as the set of the rest elements of 𝒦_j and for l∈ℕ_L-ℕ_K+1, take 𝒦_j^l=∅. For all l∈ℕ_L such that 𝒦_j^l≠∅, write 𝒦_j^l:={k_l,j,i:  i∈ℕ_card(𝒦_j^l)} and choose       {r_l,j,i:  i∈ℕ_card(𝒦_j^l)}⊂ℕ_q such that r_l,j,i≠ r_l,j,i' if i≠ i'. This choice is guaranteed since car(𝒦_j^l)≤ q for all l∈ℕ_L. For all l∈ℕ_L, define: 
		E_j^l={[                                                                               ∅                              if 𝒦_j^l=∅;                                         {j+k_l,j,iM-r_l,j,iN:  i∈ℕ_card(𝒦_j^l)}                              otherwise. ].
		Take for all l∈ℕ_L,  E_l:=⋃_j∈ℕ_M/qE_j^l.
		
  → Let's show that for all l∈ℕ_L, E_l is Mℤ-congruent to a subset of ℕ_M. Let l∈ℕ_L. For this, it suffices to show that for all j∈ℕ_M/q,  i∈ℕ_card(𝒦_j^l), we have: M|  (j+k_l,j,iM-r_l,j,iN)-(j'+k_l,j',i'M-r_l,j',i'N) ⟹  j=j'  and i=i'.
			Let j,j'∈ℕ_M/q and i,i'∈ℕ_card(𝒦_j^l) and suppose that M|  (j+k_l,j,iM-r_l,j,iN)-(j'+k_l,j',i'M-r_l,j',i'N). Then M|  j-j'+(k_l,j,i-k_l,j',i')M-(r_l,j,i-r_l,j',i')N.

			Put s=M/q, then M=sq and N=sp. Thus sq|  j-j'+(k_l,j,i-k_l,j',i')sq-(r_l,j,i-r_l,j',i')sp, then s|j-j', hence j=j' since j,j'∈ℕ_s. On the other hand, we have sq|(r_l,j,i-r_l,j,i')sp, then q|(r_l,j,i-r_l,j,i')p, thus q|r_l,j,i-r_l,j,i' since p∧ q=1, hence r_l,j,i=r_l,j,i' since r_l,j,i,r_l,j,i'∈ℕ_q. And then i=i'.

			Hence for all l∈ℕ_L, E_l is Mℤ-congruent to a subset of ℕ_M.
			
  → Let's prove now that E=⋃_l∈ℕ_LE_l is Nℤ-congruent to 𝕊_N. We show first that E is Nℤ-congruent to a subset of ℕ_N. For this, let (l,j,i),  (l',j',i')∈ℕ_L×ℕ_M/q×ℕ_card(𝒦_j) and suppose that N| (j-j')+(k_l,j,i-k_l',j',i')M-(r_l,j,i-r_l',j',i')N. Put s=M/q, then M=sq and N=sp. Thus sp|  j-j'+(k_l,j,i-k_l',j',i')sq-(r_l,j,i-r_l',j',i')sp, then s|j-j', hence j=j' since  j,j'∈ℕ_s. On the other hand, we have sp|(k_l,j,i-k_l',j,i')sq, then p|k_l,j,i-k_l',j,i', hence k_l,j,i=k_l',j,i'
			since k_l,j,i,k_l',j,i'∈ℕ_p. Then l=l' and i=i' by definition of the elements k_l,j,i. Thus E is Nℤ-congruent to a subset of ℕ_N. Observe that E⊂𝕊, then E is Nℤ-congruent to a subset of 𝕊_N. By what above, we have, in particular, that the E_j^l are mutually disjoint (and also the E_l are mutually disjoint). Then  [                                    card(E)                                          =                ∑_l∈ℕ_L∑_j∈ℕ_M/qcard(𝒦_j^l);                                                                                     =                        ∑_j∈ℕ_M/q card(𝒦_j);                                                                                     = ∑_j∈ℕ_M/q∑_n∈ℤχ_𝕊_N(j+M/qn)   remark <ref>;                                                                                     =                              ∑_j∈ℤχ_𝕊_N(j);                                                                                     =                                 card(𝕊_N). ]
			Hence E is Nℤ-congruent to 𝕊_N.
		

The following result presents an admissibility characterization for𝕊to admit a  multi-window Gabor (Parseval) frame𝒢(g,L,M,N).

	The following statements are equivalent:
	
  *  There exist g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊) such that 𝒢(g,L,M,N) is a Parseval frame for ℓ^2(𝕊).
		
  *  There exist g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊) such that 𝒢(g,L,M,N) is a frame for ℓ^2(𝕊).
		
  *  For all j∈ℕ_M/q (for all j∈ℤ), we have: 
		
    card(𝒦_j)≤ qL.

	We have (1) impies (2). And since a frame is in particular a complete sequence, then (2) implies (3) by proposition <ref>. And by proposition <ref>, (3) implies the existence of ∅≠ E_0,E_1,…,E_L-1⊂ℤ such that 𝒢( {χ_E_l}_l∈ℕ_L,  L,M,N) is a tight frame for ℓ^2(𝕊) with frame bound M. Hence 𝒢( {1/√(M).χ_E_l}_l∈ℕ_L,  L,M,N) is a Parseval frame for ℓ^2(𝕊).
	

The following proposition presents a characterization for the admissibility of𝕊to admit aL-window Gabor basis andL-window Gabor orthonormal basis𝒢(g, L, M, N).
 The following statements are equivalent:
	
  *  There exist g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊) such that 𝒢(g,L,M,N) is an orthonormal basis for ℓ^2(𝕊).
		
  *  There exist g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊) such that 𝒢(g,L,M,N) is a Riesz basis for ℓ^2(𝕊).
		
  *  For all j∈ℕ_M/q (for all j∈ℤ), we have: 
		
    card(𝒦_j)= qL.

	It is well known that (1) implies (2). Assume that  𝒢(g,L,M,N) is a Riesz basis for ℓ^2(𝕊), then by corollary <ref> we have for all j∈ℕ_M/q (∀ j∈ℤ) card(𝒦_j)≤ qL. And by proposition <ref>, we have card(𝕊_N)=LM. Then by lemma <ref>, we have card(𝒦_j)= qL. Hence (2) implies (3). Assume that card(𝒦_j)= qL. Then by corollary <ref>, There exists g:={g_l}_l∈ℕ_L⊂ℓ^2(𝕊) such that 𝒢(g,L,M,N) is Parseval frame for ℓ^2(𝕊). By lemma <ref>, we have that card(𝕊_N)=LM and then by proposition <ref>, 𝒢(g,L,M,N) is a Riesz basis for ℓ^2(𝕊), then is an orthonormal basis for ℓ^2(𝕊) (lemma <ref>). Hence (3) implies (1).

	In the case of 𝕊=ℤ, we hace for all j∈ℕ_M/q, 𝒦_j=ℕ_p. Then the condition (3) in the corollary <ref> is equivalent to p≤ Lq which is equivalent to N≤ LM. Then we obtain the proposition 3.5 in <cit.>. And also the condition (3) in the proposition <ref> is equivalent to N=LM. Then we obtain the proposition 3.11 in <cit.>.

We finish this work by the following example:

	In this example, we use the notations already introduced in what above.

	Let M=3 and N=5. Let 𝕊={0,1,2,4}+5ℤ. It is clear that p=5 and q=3. Then M/q=1, then ℕ_M/q={0}. We have clearly 𝒦_0={0,2,3,4}. Then card(𝒦_0)=4>q. Then, by corollary <ref>,  there does not exist a Gabor frame with a signe window for ℓ^2(𝕊), but by the same corollary, we can always find a Multiwindow Gabor frame for ℓ^2(𝕊) with L-window for all L⩾ 2 since card(𝒦_j)=4≤3×2=6. Here is an example of 2-window Gabor frame for ℓ^2(𝕊).

	Define g_0:=χ_{-1,0,1} and g_1:=χ_{-4,4,12}, since -1,0,1,-4,4,12∈𝕊, then g_0,g_1∈ℓ^2(𝕊). Observe, also, that 𝕊={0,1,2,4,5,-,7,9,10,11,12,14}+15ℤ. Then we have g_0 vanishes on {-10, -5, -4, 2, 3, 4, 6, 7, 8, 9, 12, 13}+15ℤ⋃{-1,0,1}+15(ℤ-{0}), and g_1 vanishes on {-10, -5, -1, 0, 1, 2, 3, 6, 7, 8, 9, 13}+15ℤ⋃{-4,4,12}+15(ℤ-{0}). Then, after a simple computation, we have for a.e θ∈ [0,1[: 
	[ Z_g_0(0,θ)=[ 1 0 0 0 0; 0 0 1 0 0; 0 0 0 1 0 ], 
		Z_g_1(0,θ)=[ 0 0 0 0 1; 0 0 0 1 0; 0 0 1 0 0 ], ]
	Then for all x:={x_k}_k∈5, we have: ⟨ Z_g_0(0,θ)^*Z_g_0(0,θ)x,x⟩ =| x_0|^2+| x_2|^2+| x_3|^2 and  ⟨ Z_g_1(0,θ)^*Z_g_1(0,θ)x,x⟩ =| x_2|^2+| x_3|^2+| x_4|^2. Then ⟨ Z_g_0(0,θ)^*Z_g_0(0,θ)x,x⟩ +⟨ Z_g_1(0,θ)^*Z_g_1(0,θ)x,x⟩
=| x_0|^2+2| x_2|^2+2| x_3|^2+| x_4|^2.  Since ⟨𝒦(0)x,x⟩=| x_0|^2+| x_2|^2+| x_3|^2+| x_4|^2, then we obtain: 
	⟨𝒦(0)x,x⟩≤⟨ Z_g_0(0,θ)^*Z_g_0(0,θ)x,x⟩ +⟨ Z_g_1(0,θ)^*Z_g_1(0,θ)x,x⟩≤ 2⟨𝒦(0)x,x⟩.
	Hence, by proposition <ref>, 𝒢({g_0,g_1}, 2,3,5) is a 2-window Gabor frame for ℓ^2(𝕊) with frame bounds 3 and 6.


§ ACKNOWLEDGMENTS

	It is our great pleasure to thank the referee for his careful reading of the paper and for several helpful suggestions.
	
	

§ ETHICS DECLARATIONS


 §.§ Availablity of data and materials

	Not applicable.
	

 §.§ Conflict of interest

	The authors declare that they have no competing interests.
	

 §.§ Fundings

	Not applicable.
	
	991 O. Christensen, An Introduction to Frames and Riesz Bases, 2nd ed., Birkhäuser, 2016.
2 C. Heil,   A Discrete Zak Transform, Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 1999, pp. 1465-1468.
3 N. Khachiaa, M. Rossafi and S. Kabbaj, Multi-window Gabor frames on discrete periodic sets, arXiv:2407.05495v1 [math.FA] 07 Jul 2024.
4 Y.-Z. Li and Q.-F. Lian, Gabor systems on discrete periodic sets, Sci. China Ser. A,52(2009), 1639-1660.
5 Y.-Z. Li and Q.-F. Lian, Tight Gabor sets on discrete periodic sets, Acta Appl. Math., 107 (2009), 105-119.
6 Y.-Z. Li and Q.-F. Lian, Multiwindow Gabor frames and oblique Gabor duals on discrete periodic sets, Sci. China Math., 54(2011), 987-1010.
7 Q.-F. Lian, J. Gong and M.-H. You, Time domain characterization of multiwindow Gabor systems on discrete periodic sets, Indian J. Pure Appl. Math., 44(1):47-76, February 2013.
8 Q.-F. Lian and Y.-Z. Li, The duals of Gabor frames on discrete periodic sets, J. Math. Phys., 50(2009), 013534, 22pp.