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TIME-FREQUENCY ANALYSIS ON FLAT TORI AND GABOR
FRAMES IN FINITE DIMENSIONS
L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
Abstract. We provide the foundations of a Hilbert space theory for the short-
time Fourier transform (STFT) where the flat tori
T2
N=R2/(Z×NZ) = [0,1] ×[0, N ]
act as phase spaces. We work on an N-dimensional subspace SNof distri-
butions periodic in time and frequency in the dual S0
0(R) of the Feichtinger
algebra S0(R) and equip it with an inner product. To construct the Hilbert
space SNwe apply a suitable double periodization operator to S0(R). On SN,
the STFT is applied as the usual STFT defined on S0
0(R). This STFT is a
continuous extension of the finite discrete Gabor transform from the lattice
onto the entire flat torus. As such, sampling theorems on flat tori lead to Ga-
bor frames in finite dimensions. For Gaussian windows, one is lead to spaces
of analytic functions and the construction allows to prove a necessary and suf-
ficient Nyquist rate type result, which is the analogue, for Gabor frames in
finite dimensions, of a well known result of Lyubarskii and Seip-Wallst´en for
Gabor frames with Gaussian windows.
1. Introduction
The short-time Fourier transform (STFT) is the central instrument of time-
frequency analysis. The most classical setting considers the analysis of functions f
with respect to windows g, both contained in L2(R), defined as
(1) Vgf(x, ξ) = ZR
f(t)g(t−x)e−2πiξt dt =hf, MξTxgi=hf , π(x, ξ)gi,
where Txf(t) = f(t−x), Mξf(t) = e2πiξt f(t), and π(x, ξ) = MξTxdefine the
translation,modulation and time-frequency shift operators, respectively. By inter-
preting the brackets as a duality pairing, this definition also holds for pairs of test
function and distribution spaces, like the Schwartz space and tempered distribu-
tions S(R),S0(R) [11] and, in particular, the Feichtinger algebra S0(R) and its dual
S0
0(R) [7].
In this paper, we consider the STFT acting on the N-dimensional space SNof
time and frequency periodic distributions in S0
0(R), see definitions in Section 2 and
[10, Chapter 16.3] or [4, Chapter 6]. This will lead to new phase spaces for the
joint time and frequency values: the flat tori T2
N= [0,1] ×[0, N ], providing, as
2010 Mathematics Subject Classification. 42C40, 46E15, 42C30, 46E22, 42C15.
Key words and phrases. short-time Fourier transform, flat torus, finite Gabor frames, Fe-
ichtinger algebra, sampling theory.
The authors would like to thank Hans Georg Feichtinger for valuable discussions and comments,
and Antti Haimi for his input during the early stages of this work. This research was supported
by the Austrian Science Fund (FWF) through the projects P-31225-N32 (L.D.A.), P 34624 (P.B.)
Y-1199, J-4254 (M.S.), as well as I 3067-N30 (N.H.).
1
2 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
we will show, a continuous extension of the coefficient space of the finite Gabor
transform. The space SNis isometrically isomorphic to CNequipped with the
Euclidean norm and can similarly be obtained by sampling and periodization of
S0(R) [15]. This connection implies that results for the STFT on SNhave impli-
cations for the discrete Gabor transform (DGT) on CNand vice-versa. However,
we will demonstrate that the STFT on the distribution space SN, embedded into
S0
0(R), has much stronger structural properties, similar to those enjoyed by the
STFT on L2(R). As a continuous phase space extension of the DGT, the STFT on
flat tori provides a natural way of defining off-the-grid values, offering flexibility in
applications and the chance of using continuous variable methods in finite Gabor
analysis. In the case of Gaussian windows, we obtain spaces of analytic functions.
The resulting possibility of using analytic complex variable tools will allow us to
prove a necessary and sufficient Nyquist rate type result for Gabor frames with
Gaussian windows in finite dimensions, which can be seen as the finite-dimensional
analogue of the celebrated result of Seip-Wallst´en [22] and Lyubarskii [17] for Ga-
bor frames with Gaussian windows. The sufficient condition provides theoretical
support to numerical procedures for increasing grid resolution, due to the principle
of stable reconstruction using frames above the Nyquist rate. As a step in the
proof of the sufficient Nyquist rate, we show that the STFT of any signal in SN
with Gaussian window has exactly Nzeros, thereby making precise and proving
the claim in [9].
Our methods are innovative in the sense that they allow to obtain results for finite
sequences merely as a byproduct of the theory on SN. But it must be noted that
the relation between the continuous STFT and the discrete Gabor transform has
been studied by several authors over the last 30 years, in particular by Janssen [14],
and later by Kaiblinger [15] and Søndergaard [23, 24]. Where the works of Janssen
and Søndergaard are concerned with the construction of discrete Gabor frames
and dual windows from Gabor systems on S0(R), Kaiblinger’s work is concerned
with finite dimensional approximation of dual windows for Gabor frames on S0(R).
The sampling-periodization duality of the Fourier transform, succintly expressed in
(generalizations of) Poisson’s summation formula and considered in many works,
including [2, 5, 13, 15], is central to these contributions. In essence, the transition
between S0(R) and CNis achieved by studying a composition of periodization op-
erators P(1),P(2) on certain intervals with forward and inverse Fourier transforms
F(1),F−1
(2) as F−1
(2) P(2)F(1) P(1). Here, F(1) is the Fourier transform of L2(R) of a
finite interval and F−1
(2) is the inverse discrete Fourier transform. From this an-
gle, the central deviation of the present paper from these prior works is that we
consider F(1) and F−1
(2) to be distributional Fourier transforms on S0
0(R), such that
F−1
(2) P(2)F(1) P(1)f,f∈S0(R), yields a doubly periodic distribution in S0
0(R) in-
stead of a finite sequence in CN, enabling the subsequent application of the STFT
on S0
0(R) instead of the finite Gabor transform.
2. Overview
We consider functions, distributions, and finite sequences, denoted by lower case
latin letters f, g, greek letters φ, ψ, and sans font latin letters f,g, respectively.
For the latter, the discrete nature of the domain of f,gis emphasized by using
square brackets for indexing, e.g. f[l]. Operators are denoted by upper case letters
TIME-FREQUENCY ANALYSIS ON FLAT TORI 3
V,Σ. Exceptions from this convention are time-frequency shifts π, the Jacobi theta
function ϑ, and the Fourier transform F, for which we adopt established notation.
With the Gaussian window h0(t) = e−πt2, the Feichtinger algebra S0(R) [6, 12]
is the space
S0(R) := f∈L2(R): Vh0f∈L1(R2),equipped with the norm
kfkS0:= ZR2|Vh0f(x, ξ)|dxdξ =kVh0fkL1(R2).
We define the space SNas the span of {n}N−1
n=0 , the sequence of periodic delta
trains [10]
(2) n:= X
k∈Z
δn
N+k⊂S0
0(R), n = 0, ..., N −1,
and will show that SNcan be characterized as the image of S0(R) under the double
periodization operator
(3) ΣNf:= X
k1,k2∈Z
MNk2Tk1f=X
k1,k2∈Z
e2πiN k2·f(· − k1).
It can be directly observed that Vg(ΣNf) is quasiperiodic, i.e.
(4) Vg(ΣNf)(x+ 1, ξ) = e−2πiξ Vg(ΣNf)(x, ξ ),
Vg(ΣNf)(x, ξ +N) = Vg(ΣNf)(x, ξ).
Thus, the phase spaces of Vg◦ΣNare the flat tori T2
N= [0,1] ×[0, N ]. As we
will see, Vg◦ΣN:S0(R)→L2(T2
N) and Vg:SN→L2(T2
N) have the same
range in phase-space. It will often be convenient to jump from one to the other
representation to simplify proofs.
The STFT on SNnaturally introduces the compact phase space T2
Nfor time-
frequency analysis on finite, N-dimensional Hilbert spaces. Thereby, it provides a
continuous model that, by construction, eliminates the truncation, or alternatively
aliasing, errors usually associated with the transition from the STFT on L2(R) to
numerical implementations by means of the finite Gabor transform. That is not to
say that these errors are removed: They are instead separated from the continuous
model to the double periodization operator ΣN, i.e., the mapping from S0(R) onto
SN⊂S0
0(R).
As discussed in [14, 15, 24] in a slightly different formal framework, the compo-
sition Vg◦ΣNrelates to the finite Gabor transform on CN, defined as
Vgf[k, l] =
N−1
X
m=0
f[m]g[m−k]e−2πilm
N,f,g∈CN,
We will show that Vgmaps SNinto L2(T2
N) and that Vgcan be viewed as a
continuous extension of Vg:CN→CN×Nto T2
Nin the sense of the following
result.
Theorem 1. Let f, g ∈S0(R)and let fN=PNf , gN=PNg, with the periodization
operator
PN:S0(R)→CN,defined by f7→ X
j∈Z
f(n/N −j)N−1
n=0 .
4 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
Then, for k, l ∈0, ...., N −1,
Vg(ΣNf)k
N, l=N−1·VgNfN[k, l].
Specifically, the finite discrete STFT can be obtained by sampling the phase space
T2
Nof the continuous STFT restricted to SNon the grid points (k/N, l), providing
a direct link between continuous and finite discrete time-frequency analysis.
Remark 2. We chose the periods (1, N )in equation (3) for notational convenience.
Any other pair (c, d)∈R2
+, with cd =N, leads to equivalent results on the phase
space f
T2
N= [0, c]×[0, d]. When studying the approximation of the STFT by finite
Gabor transforms, as in [15], it is usually more convenient to consider the sym-
metric convention (c, d)=(√N , √N), such that an increase in Nsymmetrically
expands the considered phase space area and the sampling density within.
We will study the Hilbert space properties of the map
Vg:SN→L2(T2
N),
and derive the Moyal-type orthogonality relation
ZT2
N
Vg1ϕ1(x, ξ)Vg2ϕ2(x, ξ)dxdξ =Nhϕ1, ϕ2iSNhg2, g1iL2,
as well as inversion and reproducing formulas similar to those of the continuous
STFT.
For the STFT with dilated Gaussian windows hλ
0(t) = e−πλt2we will obtain a
sampling theorem on the torus which leads to a full description of the frame set for
finite Gabor frames with Gaussian windows in CN. The proof uses a Bargmann-
type transform, (which up to a weight is the STFT with hλ
0) whose action on
the space SNhas previously been considered in a slightly different form in [16].
Finally, combining the sampling theorem on the torus with Theorem 1, we are lead
to the following full description of the frame set for finite Gabor expansions using a
periodized, dilated Gaussian window. As far as we could check, this is a completely
new result.
Theorem 3. Let λ > 0,hλ
N=PNhλ
0, and {(jk, lk)}k=1,...,K be a collection of
distinct pairs of integers jk, lk∈0, ...., N −1. The following are equivalent:
1). The set {(jk, lk)}k=1,...,K gives rise to a finite Gabor frame with window
hλ
N, i.e., there are constants A, B > 0such that, for every f∈CN,
Akfk2
CN≤
K
X
k=1 Vhλ
Nf[jk, lk]
2≤Bkfk2
CN.
2). One of the three following conditions is satisfied:
(i) N2≥K≥N+ 1,
(ii) K=Nis odd,
(iii) K=Nis even and PN
k=1(jk, lk)/∈NN2.
We emphasize that this result has been possible to prove only thanks to our
Hilbert space theory for the STFT on flat tori, and that it strongly depends on
the use of complex variable methods for almost periodic analytic functions. This
reinforces the suggestion that time-frequency analysis on the torus provides a rich
TIME-FREQUENCY ANALYSIS ON FLAT TORI 5
theory which encompasses the theory of finite Gabor frames and leads to new
insights, potential in applications and proof of results which were out of reach
without the toric phase space.
The paper is organized as follows. Some required properties of the Hilbert space
SNand the operator ΣNare presented in Section 3. Section 4 contains the proof
of Theorem 1 above, explicit computations with dilated Gaussian windows hλ
0, and
derivations of the Moyal-type formula, together with the inversion and reproducing
kernel formulas. In the last section, the window is specialized to be the Gaussian.
The resulting Bargmann-type transform is defined, and several properties of its
range space of entire functions with periodic constraints (a toric analogue of the
Fock space) are studied in detail. All these properties are then used in the proof
of the main result of the section: the sampling theorem on the torus. Finally,
combining this result with Theorem 1, we derive a full characterization of finite
Gabor frames with periodized and sampled Gaussian windows.
3. Properties of SNand ΣN
3.1. The Hilbert space SNof time-frequency periodic distributions. The
space SNappears in theoretical physics in coherent state approaches [4, 10]. By
definition of SNit is clear that the family {n}N−1
n=0 defined in (2) forms a basis.
Therefore, expanding ϕ, ψ ∈SNwith respect to this basis
ϕ=
N−1
X
n=0
ann, ψ =
N−1
X
n=0
bnn,
we can define an inner product on SNby
hϕ, ψiSN=
N−1
X
n=0
anbn.
Clearly, SNcan be identified with CNequipped with the standard inner product,
and {n}N−1
n=0 forms an orthonormal basis of SNas
hn, miSN=δn,m.
Note that for every ϕ∈SN
(5) T1ϕ=ϕ, and MNϕ=ϕ.
Therefore, SNis a space of distributions that are periodic in time and frequency.
Actually, SNcontains all distributions in S0
0(R) that satisfy (5), see [10, page 262,
(16.12)].
3.2. The double periodization operator ΣN.We formally define the double
periodization operator as
f7→ ΣNf=X
k1,k2∈Z
MNk2Tk1f .
The next lemma shows when and in which sense this object is well-defined.
Lemma 4. The operator ΣNis well-defined from S0(R)into S0
0(R)with uncondi-
tional weak-∗convergence in S0
0(R).
6 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
Proof: If f, g ∈S0(R), then
|Vg(ΣNf)(x, ξ)| ≤ X
k1,k2∈Z|e−2πik1ξVgf(x−k1, ξ −Nk2)|
≤X
k1,k2∈Z|Vg(T−xM−ξf)(k1, N k2)|
≤CkT−xM−ξfkS0kgkS0
≤CkfkS0kgkS0≤ ∞,
by [11, Lemma 3.1.3 and Corollary 12.1.12]. This implies that ΣNf∈S0
0(R) is
well-defined. If we choose x=ξ= 0, then absolute weak-∗convergence of the
series in S0
0(R) follows which in turn implies unconditional convergence.
As ΣNfis periodic in time and frequency, we can expand it with respect to the
orthonormal basis {n}N−1
n=0 . In the next lemma, this expansion is obtained explic-
itly. We will also show that ΣNis surjective as a mapping from S0(R) to SN. To
do so, we need to define the periodization operator Pf(t) = Pk∈Zf(t−k).
Lemma 5. For every f∈S0(R)
(6) ΣNf=1
NPf·X
k∈Z
δk
N=1
N
N−1
X
n=0
Pfn
Nn∈SN,
and
(7) hΣNf, giS0
0×S0=1
N
N−1
X
n=0
Pfn
NPgn
N=N· hΣNf, ΣNgiSN.
Moreover, ΣN:S0(R)→SNis surjective.
Proof: Let f, g ∈S0(R). Since the Poisson summation formula holds for functions
in S0(R) (see e.g. [11, Corollary 12.1.5]), we have that the following equality holds
in the distributional sense
X
k∈Z
e2πiktN =1
NX
k∈Z
δk
N(t)∈SN.
This shows that
ΣNf=X
k∈Z
MNk X
l∈Z
Tlf=X
k∈Z
MNk Pf=1
NX
k∈Z
δk
NPf
with unconditional weak-∗convergence in S0
0(R). Hence,
hΣNf, giS0
0×S0=1
N*X
k∈Z
δk
NPf, g+S0
0×S0
.
Let us write
X
k∈Z
δk
N=
N−1
X
n=0 X
k∈Z
δn
N+k=
N−1
X
n=0
n
TIME-FREQUENCY ANALYSIS ON FLAT TORI 7
where the change of summation order is justified by e.g. [11, Corollary 12.1.5].
Using the periodicity of Pfthen yields
hΣNf, giS0
0×S0=1
N
N−1
X
n=0 *X
k∈Z
δn
N+kPf, g+S0
0×S0
=1
N
N−1
X
n=0 X
k∈Z
Pfn
N+kgn
N+k
=1
N
N−1
X
n=0
Pfn
NX
k∈Z
gn
N+k
=1
N
N−1
X
n=0
Pfn
Nhn, giS0
0×S0.
Hence, (6) holds. The first equality of (7) follows from the second to last equality
above. Finally, the second equality of (7) results from combining (6) and the first
equality of (7).
It thus remains to show that Σn:S0(R)→SNis surjective. By (6) it suffices
to show that there exists a family of function fn∈S0(R), n= 0, . . . , N −1, sat-
isfying Pfnk
N=δn(k), k = 0, . . . , N −1. Such functions obviously exists. Take
for instance fn(t) := sinc(Nt−n)·e−π(t−n/N)2which is even a Schwartz function.
4. Time-frequency analysis on flat tori
4.1. Basic properties of Vgon SN.The STFT defined on SNis, as we we
subsequently show, closely connected to the Zak transform which is defined as
Zf(x, ξ) = X
k∈Z
f(x−k)e2πikξ .
For later reference we state here some elementary facts about the Zak transform
(see e.g. [11]):
Quasiperiodicity:
(8) Zf(x, ξ +k) = Zf(x, ξ),and Zf(x+k, ξ) = e2πikξZ(x, ξ),
Action on time-frequency shifts:
(9) Z(MωTyf)(x, ξ) = e2πiω xZf(x−y, ξ −ω),
Unitarity: for f1, f2∈L2(R) it holds
(10) Z1
0Z1
0
Zf1(x, ξ)Zf2(x, ξ)dxdξ =hf1, f2iL2.
Lemma 6. Let f, g ∈S0(R)and ϕ∈SN. Then
(11) Vg(ΣNf)(x, ξ) =
N−1
X
n=0
Pfn
Ne−2πiξ n
NZgn
N−x, ξ,
8 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
and for ϕ=PN
n=1 ann
(12) Vgϕ(x, ξ) =
N−1
X
n=0
ane−2πiξ n
NZgn
N−x, ξ.
Proof: First, let us compute the STFT of the basis functions n
Vgn(x, ξ) = hn,MξTxgiS0
0×S0=X
k∈Zδn
N+k,MξTxgS0
0×S0
=X
k∈Z
gn
N+k−xe−2πiξ(n
N+k)=e−2πiξ n
NZgn
N−x, ξ,(13)
For general ϕ=PN−1
n=0 ann∈SNone thus gets (12). Applying Vgto (6) from
Lemma 5 gives
Vg(ΣNf)(x, ξ) = 1
N
N−1
X
n=0
Pfn
NVgn(x, ξ)
which combined with (13) yields (11).
With these basic observations, it is now straightforward to show Theorem 1. For
convenience, we repeat the statement here.
Theorem 1. Let f, g ∈S0(R)and let fN=PNf, gN=PNg. Then, for l, k ∈
0, ...., N −1,
Vg(ΣNf)k
N, l=N−1·VgNfN[k, l].
Proof: Setting ξ=l∈0, ...., N −1 and x=k
N, k ∈0, . . . , N −1,yields
Vg(ΣNf)k
N, l=1
N
N−1
X
n=0
Pfn
Ne−2πi nl
NZgn−k
N, l
=1
N
N−1
X
n=0
Pfn
Ne−2πi nl
NPgn−k
N
This shows that by definition of the finite Gabor transform
Vg(ΣNf)k
N, l=1
NVgNfN[k, l].
4.2. Moyal-type and inversion formulas for the STFT on flat tori. One
of the most fundamental identities in time-frequency analysis is Moyal’s formula
ensuring that the STFT is an isometry from L2(R) to L2(R2)
ZR2
Vg1f1(x, ξ)Vg2f2(x, ξ)dxdξ =hf1, f2iL2hg2, g1iL2, f1, f2, g1, g2∈L2(R).
We will now show the toric equivalent of this identity ensuring that the STFT on
the flat tori is a multiple of an isometry.
TIME-FREQUENCY ANALYSIS ON FLAT TORI 9
Theorem 7. If ϕ1, ϕ2∈SN, and g1, g2∈S0(R), then
(14) ZT2
N
Vg1ϕ1(x, ξ)Vg2ϕ2(x, ξ)dxdξ =Nhϕ1, ϕ2iSNhg2, g1iL2.
Proof: For ϕ1=PN−1
n=0 ann, and ϕ2=PN−1
n=0 bnn, we use Lemma 6 and the
periodicity of the Zak transform in the frequency variable (8) to obtain
ZT2
N
Vg1ϕ1(x, ξ)Vg2ϕ2(x, ξ)dxdξ
=ZN
0Z1
0
N−1
X
n,k=0
anbke−2πiξ n−k
NZg1n
N−x, ξZg2k
N−x, ξdxdξ
=
N−1
X
l=0 Z1
0Z1
0
N−1
X
n,k=0
anbke−2πi(ξ+l)n−k
N·
Zg1n
N−x, ξ +lZg2k
N−x, ξ +ldxdξ
=Z1
0Z1
0
N−1
X
n,k=0
anbk N−1
X
l=0
e−2πil n−k
N!e−2πiξ n−k
N·
Zg1n
N−x, ξZg2k
N−x, ξdxdξ
=N
N−1
X
n=0
anbnZ1
0Z1
0
Zg1n
N−x, ξZg2n
N−x, ξdxdξ,
where we used the basic fact PN−1
l=0 e−2πil k
N=Nδk,0to obtain the final equality.
Applying consecutively (9), (8) and (10) to the integral above yields
Z1
0Z1
0
Zg1n
N−x, ξZg2n
N−x, ξdxdξ
=Z1
0Z1
0
Z(T−n
Ng1) (−x, ξ)Z(T−n
Ng2) (−x, ξ)dxdξ
=Z1
0Z1
0
e−2πiξ Z(T−n
Ng1) (1 −x, ξ )e−2πiξZ(T−n
Ng2) (1 −x, ξ )dxdξ
=Z1
0Z1
0
Z(T−n
Ng1) (x, ξ)Z(T−n
Ng2) (x, ξ)dxdξ
=hT−n
Ng1,T−n
Ng2iL2=hg2, g1iL2,
which concludes the proof.
Remark 8. Using (7) we thus have shown that ΣN(π(x, ξ)g)(x,ξ)∈T2
N
is a con-
tinuous tight frame for SN.
From Theorem 7 and (7) we can now derive two inversion formulas.
10 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
Theorem 9. Let g∈S0(R)\{0}. For every ϕ∈SNand every f∈S0(R), it holds
(15) ϕ=1
Nkgk2
2
N−1
X
n=0 ZT2
N
Vgϕ(x, ξ)e2πiξ n
NZgn
N−x, −ξdxdξ!n,
and
(16) ΣNf=1
kgk2
2ZT2
N
Vg(ΣNf)(x, ξ)ΣNπ(x, ξ)gdxdξ,
where the integral is understood in the weak-∗sense.
Proof: Let ϕ=PN−1
n=0 ann. Then by Theorem 7 we know
an=hϕ, niSN=1
Nkgk2
2ZT2
N
Vgϕ(x, ξ)Vgn(x, ξ)dxdξ
=1
Nkgk2
2ZT2
N
Vgϕ(x, ξ)e2πiξ n
NZgn
N−x, ξdxdξ
=1
Nkgk2
2ZT2
N
Vgϕ(x, ξ)e2πiξ n
NZgn
N−x, −ξdxdξ,
where we used (12) and the fact that Zg(x, ξ) = Zg(x, −ξ). To show the second
identity we first observe that ΣNsatisfies
hΣNf, giS0
0×S0=hf, ΣNgiS0×S0
0, f, g ∈S0(R).
Let f, h ∈ S0(R). The result thus follows from (7) and (14) as
hΣNf, hiS0
0×S0=1
kgk2
2ZTN2
Vg(ΣNf)(x, ξ)hπ(x, ξ)g, ΣNhiS0×S0
0dxdξ
=1
kgk2
2ZTN2
Vg(ΣNf)(x, ξ)hΣNπ(x, ξ)g, hiS0
0×S0dxdξ.
Remark 10. Note that coorbit theory [7, 8] also guarantees an inversion of the
short-time Fourier transform on S0
0(R)as V∗
gVg=kgk2IS0
0. The benefit of our
point of view is however that the coefficients in (15) can directly be calculated without
resorting to weakly defined integrals.
4.3. Reproducing kernel. We now prove that the range of the STFT restricted
to SNis an N-dimensional reproducing kernel Hilbert space (RKHS) of L2(T2
N)
and give several expressions for its reproducing kernel.
Proposition 11. The space Vg(SN)⊂L2(T2
N)is a RKHS and its kernel is given
by
Kg(x0, ξ0),(x, ξ)=1
Nkgk2
2
N−1
X
n=0
e2πi(ξ−ξ0)n
NZgn
N−x, ξZgn
N−x0, ξ0
=N
kgk2
2hΣNπ(x, ξ)g,ΣNπ(x0, ξ0)giSN
(17)
=1
kgk2
2X
k1,k2∈Z
e−2πik1ξhπ(x+k1, ξ +N k2)g, π(x0, ξ0)giL2.
TIME-FREQUENCY ANALYSIS ON FLAT TORI 11
Proof: Using consecutively (12), Cauchy-Schwarz inequality, [11, Lemma 8.2.1]
and (14) yields
|Vgϕ(x, ξ)|≤kϕkSN N−1
X
n=0 Zgn
N−x, ξ
2!1/2
≤ kϕkSN√NkZgk∞=kZgk∞
kgk2kVgϕkL2(TN),
meaning that point evaluation is continuous and Vg(SN) is a RKHS.
By Theorem 9 and Lemma 6, it follows
Vgϕ(x0, ξ0) = 1
Nkgk2
2
N−1
X
n=0 ZT2
N
Vgϕ(x, ξ)e2πiξ n
NZgn
N−x, −ξVgn(x0, ξ0)dxdξ
=ZT2
N
Vgϕ(x, ξ)1
Nkgk2
2
N−1
X
n=0
e2πi(ξ−ξ0)n
NZgn
N−x, ξZgn
N−x0, ξ0dxdξ,
as well as by (16) and (7)
Vg(ΣNf)(x0, ξ0)
=1
kgk2
2ZT2
N
VgΣNf(x, ξ)hΣNπ(x, ξ)g, π(x0, ξ0)giS0
0(R)×S(R)dxdξ
=N
kgk2
2ZT2
N
VgΣNf(x, ξ)hΣNπ(x, ξ)g,ΣNπ(x0, ξ0)giSNdxdξ.
Therefore, the first two identities of (17) hold. Moreover,
Kg((x0, ξ0),(x, ξ)) = N
kgk2
2hΣN(π(x, ξ)g),ΣN(π(x0, ξ0)g)iSN
=1
kgk2
2hΣN(π(x, ξ)g), π(x0, ξ0)giS0
0×S0
=1
kgk2
2X
k1,k2∈Z
e−2πik1ξhπ(x+k1, ξ +N k2)g, π(x0, ξ0)giL2.
4.4. Examples of the STFT on flat tori. In this section, we consider ex-
plicit calculations of the objects discussed in the previous sections using the non-
normalized dilated Gaussian
hλ
0(t) := e−πλt2, λ > 0.
The reason we introduce the additional dilation is that, having the connection to
finite Gabor systems (Theorem 1) in mind, we would like to impose a certain degree
of localization of Phλ
0. This is only guaranteed if λ > 0 is chosen large enough, see
Figure 1.
Following (12), the STFT of a basis function nis given by
Vhλ
0n(x, ξ) = e−2πi n
NξZhλ
0n
N−x, ξ
=e−2πi n
NξX
k∈Z
e−πλ(x−n
N+k)2e2πiξk
12 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
Figure 1. Phλ
0, for λ= 1, (left) λ= 16 (middle), and λ= 64 (right).
=e−πλ(x−n
N)2−2πiξ n
NX
k∈Z
e−πλk2e2πik(ξ+iλ(x−n
N))
=e−πλx2e−πλ(n
N)2
e2πzλn
Nϑizλ−λn
N, iλ,(18)
where zλ:= λx +iξ and the Jacobi theta function ϑis defined as
ϑ(z, τ ) := X
k∈Z
eπik2τ+2πikz,Im(τ)>0.
Moreover, we explicitly calculate the reproducing kernel Khλ
0,T2
N. For this purpose,
the third equation of (17) is the most convenient representation of the kernel. Let
us write wλ=λx0+iξ0. By [11, Lemma 1.5.2]
hπ(x, ξ)hλ
0, π(x0, ξ0)hλ
0i= (2λ)−1
2eπi(ξ−ξ0)(x+x0)e−π
2[λ(x−x0)2+1
λ(ξ−ξ0)2].
from which we deduce
Khλ
0(wλ, zλ) = 1
khλ
0k2X
k1,k2∈Z
e−2πik1ξhπ(x+k1, ξ +N k2)hλ
0, π(x0, ξ0)hλ
0i
=eπi(ξ−ξ0)(x+x0)X
k1,k2∈Z
eπ
2[−2i(ξ+ξ0)k1+2i(x+x0)Nk2−λ(x−x0+k1)2−1
λ(ξ−ξ0+Nk2)2]
=eπ[i(ξ−ξ0)(x+x0)−λ
2(x−x0)2−1
2λ(ξ−ξ0)2]X
k1∈Z
e−πk1[λ(x−x0)+i(ξ+ξ0)]−πλk2
1
2
X
k2∈Z
eπi N k2
λ[λ(x+x0)+i(ξ−ξ0)]−πN2k2
2
2λ
=e−πλ(x2+(x0)2)eπ
2λ(zλ+wλ)2ϑizλ−wλ
2,iλ
2ϑ(zλ+wλ)N
2λ,iN2
2λ.
5. A Bargmann-type transform and finite Gabor Frames
Given ϕ=PN−1
n=0 ann∈SNwe can define a Bargmann-type transform by
B(λ,N)ϕ(z) = Vhλ
0ϕx
λ,−ξeπx2/λ,z=x+iξ, or equivalently using (18)
B(λ,N)ϕ(z) =
N−1
X
n=0
ane−πλ(n
N)2
e2πz n
Nϑiz−λn
N, iλ.
TIME-FREQUENCY ANALYSIS ON FLAT TORI 13
By the quasi-periodicity of Vhλ
0ϕ(4), we then get that
(19) B(λ,N)ϕ(z+λn) = eπλn2+2πz nB(λ,N )ϕ(z), n ∈Z,
as well as
(20) B(λ,N)ϕ(z+iN m) = B(λ,N)ϕ(z), m ∈Z.
This can also be derived using the quasi-periodicity of the Jacobi theta function
(21) ϑ(z+n+τ m, τ) = e−πiτm2e−2π imzϑ(z , τ), n, m ∈Z.
We will now show that the range of the Bargmann-type transform B(λ,N)is precisely
the space of analytic functions satisfying (19) and (20). This follows from the
following result.
Lemma 12. The space of entire functions satisfying the periodicity conditions (19)
and (20) is N-dimensional. Thus, any such function Fcan be written as the linear
combination of Northogonal functions
F(z) =
N−1
X
n=0
ane−πλ(n
N)2
e2πz n
Nϑiz−λn
N, iλ
and F(z) = B(λ,N)ϕ(z), for some ϕ=PN−1
n=0 ann.
Proof: Let Fbe analytic and satisfy (19) and (20). Since Fis N-periodic with
respect to purely imaginary shifts, we can write Fas a Fourier series
F(z) = X
k∈Z
cke2πkz/N .
Plugging this expression into (19) yields
X
k∈Z
cke2πk(z+λn)/N =eπλn2+2πzn X
k∈Z
cke2πkz/N =X
k∈Z
ckeπλn2e2π(k+nN )z/N
=X
k∈Z
ck−nN eπλn2e2πkz /N,
which shows that the coefficients cksatisfy
(22) ck+nN =cke−πλ(n2+2kn/N), k ∈ {0, ..., N −1}, n ∈Z.
There are therefore exactly Ncoefficients c0, ..., cN−1that can be chosen freely and
the other coefficients are determined by (22). The orthogonality of the functions
e−πλ(n
N)2
e2πz n
Nϑiz−λn
N, iλ=Vhλ
0nx
λ,−ξeπx2/λ
follows from hn, miSN=δn,m and Moyal’s formula (14).
The periodicity conditions (19) and (20) will also allow us to count the number
of zeros of B(λ,N)ϕin the torus T2
λ,N := R2/(λZ×NZ), which coincides with the
number or zeros of Vhλ
0ϕin T2
N, and to obtain a constraint these zeros must satisfy.
14 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
Proposition 13. Let ϕ∈SN\{0}. The function B(λ,N )ϕhas exactly Nze-
ros (counted with their multiplicities) on the torus T2
λ,N . Moreover, the Nzeros
z1, ..., zN∈T2
λ,N satisfy
N
X
k=1
zk=Nλ
2+λn +iN2
2+Nm,for some n, m ∈Z.(23)
Proof: Let the curve Γ = ∂T2
λ,N be positively oriented and assume for now that
Γ contains no zero of B(λ,N)ϕ. As B(λ,N)ϕis an analytic function on C, it follows
by Cauchy’s argument principle (see, e.g. [1, Sec. 5.2]) that the number of zeros of
B(λ,N)ϕ(counted with their multiplicities) is given by
1
2πi ZΓ
(B(λ,N)ϕ)0
B(λ,N)ϕdγ =1
2πi "Zλ
0
(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt +iZN
0
(B(λ,N)ϕ)0(λ+it)
B(λ,N)ϕ(λ+it)dt
−Zλ
0
(B(λ,N)ϕ)0(t+iN )
B(λ,N)ϕ(t+iN )dt −iZN
0
(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt#
=1
2π"ZN
0
(B(λ,N)ϕ)0(λ+it)
B(λ,N)ϕ(λ+it)dt −ZN
0
(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt#,
where we used (20). By (19) we have
(B(λ,N)ϕ)0(z+λ)=2πeπ λ+2πz B(λ,N)ϕ(z) + eπ λ+2πz (B(λ,N)ϕ)0(z),
and consequently
1
2πi ZΓ
(B(λ,N)ϕ)0
B(λ,N)ϕdγ =1
2π"ZN
0
eπλ+2πit 2πB(λ,N)ϕ(it)+(B(λ,N)ϕ)0(it)
eπλ+2πit B(λ,N)ϕ(it)dt
−ZN
0
(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt#=1
2πZN
0
2πdt =N.
Let z1, ..., zNbe the Nzeros of B(λ,N )ϕ. Then, using again (19), (20) and Cauchy’s
argument principle (see, e.g. [1, Sec. 5.2, (49)]) , we get with id: C→C, z 7→ z
N
X
k=1
zk=1
2πi ZΓ
id ·(B(λ,N)ϕ)0
B(λ,N)ϕdγ
=1
2πi "Zλ
0
t(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt +iZN
0
(λ+it)(B(λ,N)ϕ)0(λ+it)
B(λ,N)ϕ(λ+it)dt
−Zλ
0
(t+iN)(B(λ,N)ϕ)0(t+iN)
B(λ,N)ϕ(t+iN )dt −iZN
0
it(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt#
=1
2π"−NZλ
0
(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt −ZN
0
it(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt
+ZN
0
(λ+it)(B(λ,N)ϕ)0(λ+it)
B(λ,N)ϕ(λ+it)dt#
=1
2π"−NZλ
0
(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt −ZN
0
it(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt
TIME-FREQUENCY ANALYSIS ON FLAT TORI 15
+ZN
0
(λ+it)2πB(λ,N)ϕ(it)+(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt#
=1
2π"λZN
0
(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt −NZλ
0
(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt + 2πZN
0
(λ+it)dt#.
Let us now think of t7→ B(λ,N)ϕ(t) as a parametrization of the curve Γ1=
{B(λ,N)ϕ(t)}t∈[0,λ], i.e.
Zλ
0
(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt =ZΓ1
1
zdγ.
Let Γ2=B(λ,N)ϕ(0)(eπ λ(1 −t) + t)t∈[0,1] be the line segment that connects the
endpoints of Γ1. Therefore, Γ0= Γ1∪Γ2is a closed continuous curve. By the
residue theorem, it follows that
ZΓ0
1
zdγ = 2πik,
for some k∈Z. Therefore
Zλ
0
(B(λ,N)ϕ)0(t)
B(λ,N)ϕ(t)dt =−ZΓ2
1
zdγ + 2πik =Z1
0
eπλ −1
eπλ −t(eπλ −1) dt + 2πik
=Zeπλ−1
0
1
eπλ −tdt + 2πik =Zeπ λ
1
1
tdt + 2πik =πλ + 2πik.
Similarly, Γ3={B(λ,N )ϕ(it)}t∈[0,N]is a closed and continuous curve. Let n∈Z
be the winding number of Γ3around the origin. Then
ZN
0
(B(λ,N)ϕ)0(it)
B(λ,N)ϕ(it)dt =−iZΓ3
1
zdγ = 2πn.
Thus,
N
X
k=1
zk=1
2π2πλn −πλN −2πiNk + 2πλN +πiN2
=Nλ
2+λn +iN2
2+Nk, k, n ∈Z.
Finally, if Γ contains at least one zero of B(λ,N)ϕ, then there exists z∗∈Csuch
that z∗+ Γ does not contain any zero as every nonzero analytic function can only
have finitely many zeros on any compact set. The previous arguments can then
be repeated for the curve z∗+ Γ yielding the same number of zeros which are con-
strained by the same condition.
The next result is a full characterization of frames obtained from the STFT on SN
with Gaussian windows via sampling points in T2
N.
Theorem 14. Let λ > 0, and z1, ..., zK∈T2
Nbe a collection of Kdistinct points.
The following are equivalent:
16 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
1). The points z1, ..., zK∈T2
Ngive rise to a frame for SN, i.e., there are
constants A, B > 0such that, for every ϕ∈SN,
Akϕk2
SN≤
K
X
k=1 |Vhλ
0ϕ(zk)|2≤Bkϕk2
SN.
2). One of the following two conditions is satisfied:
(i) K≥N+ 1,
(ii) K=Nand 1
NPN
k=1 zk6= (1/2 + n/N, N/2 + m),for every n, m ∈Z.
Proof: If the family is a frame for SN, then it has to include at least Nvectors,
i.e., K≥N.
Let K≥N+ 1. It follows from Proposition 13 and the relation between B(λ,N)
and Vhλ
0that Vhλ
0ϕhas at most Ndistinct zeros. Consequently, PK
k=1 |Vhλ
0ϕ(zk)|2
is always positive. Moreover, this expression depends continuously on the basis
coefficients anof ϕby (12). Therefore, as the unit sphere in CNis compact, it
follows that
A= inf
kϕk2
SN=1
K
X
k=1 |Vhλ
0ϕ(zk)|2>0.
Now, let K=N, and suppose we are given a collection of Ndistinct points
z1, ..., zN∈T2
N. Moreover, we set zλ,k =λxk−iξk∈T2
λ,N ,k= 1, ..., N . Let
z0=λ/2 + iN/2 be the single zero of ϑ(iz /N, iλ/N) in [0, λ]×i[0, N] (see [19,
20.2(iv)]) and define the function F(z) as
F(z) =
N
Y
k=1
e2πRe(zλ,k−z0)z /λN ϑi(z−zλ,k +z0)/N, iλ/N .
Using (21) one can directly show that Fsatisfies the periodicity conditions
F(z+λn) =
N
Y
k=1
e2πRe(zλ,k−z0)(z+λn)/λN ϑi(z−zλ,k +z0)/N +iλn/N, iλ/N
=eπλn2F(z)
N
Y
k=1
e2πRe(zλ,k−z0)n/N e2πn(z−zλ,k +z0)/N
=eπλn2e2πnz e−2πinImN z0−PN
k=1 zλ,k/N F(z),
and
F(z+iNm) =
N
Y
k=1
e2πRe(zλ,k−z0)(z+N m)/λN ϑi(z−zλ,k +z0)/N −m, iλ/N
=e−2πimReN z0−PN
k=1 zλ,k/λ F(z).
Therefore, if PN
k=1 zλ,k =Nz0+λm +iN n =λN/2 +λm +i(N2/2 +N n),for some
n, m ∈Z,it follows that Fsatisfies (19) and (20) and is analytic in T2
λ,N . Thus,
by Lemma 12 there exists ϕ∈SNsuch that Vhλ
0ϕx
λ,−ξ=F(z)e−πx2/λ and, by
construction, Vhλ
0ϕ(z) = 0, for z=zλ,1, ..., zλ,K . Consequently,
K
X
k=1 |Vhλ
0ϕ(zλ,k)|2= 0,
TIME-FREQUENCY ANALYSIS ON FLAT TORI 17
and the lower frame bound is violated.
If PN
k=1 zλ,k 6=Nz0+λm +Nn, for every n, m ∈Z,it follows by Proposition 13
that {zλ,1, ..., zλ,N }is a uniqueness set for the space of entire functions satisfying
the periodicity conditions (19) and (20). It thus follows again by compactness of
the unit ball in SN, that the points z1, ..., zNgenerate a frame for SN
Theorem 14 implies that if one chooses Npoints in T2
Nuniformly at random, then
one obtains a frame with probability 1. While the frame set for the Gaussian
window STFT on L2(R) is known from [22] and [17], it seems to be a folklore
result, backed up by substantial experience through numerical computations, that
the finite Gabor transform with sampled, periodized Gaussian yields a frame when
sampled on any lattice within ZN×ZNwith cardinality larger than N, but we
found no proof in the literature. On the other hand, a result by Søndergaard [23],
adapted from a remarkable observation by Benedetto et al [3] for the STFT on
`2(Z), demonstrates that half-point shifts of sampled, periodized Gaussians yield
frames on lattices of cardinality N. The result below, a direct consequence of
Theorem 14, demonstrates that, in fact, any K > N distinct points from the set
{(j, k) : j, k ∈0, ..., N }yield a finite Gabor frame for CNwith the sampled,
periodized Gaussian window hλ
N:= PNhλ
0. For K=N, additional arithmetic
conditions ensure the frame property. This is the result for Gabor frames in finite
dimensions stated in the introduction as one of the main achievements of our theory.
We state it again for convenience.
Theorem 3. Let λ > 0, and {(jk, lk)}k=1,...,K a collection of distinct pairs of
integers jk, lk∈0, ...., N −1. The following are equivalent:
1). The set {(jk, lk)}k=1,...,K gives rise to a finite Gabor frame with window
hλ
N, i.e., there are constants A, B > 0such that, for every f∈CN,
Akfk2
CN≤
K
X
k=1 Vhλ
Nf[jk, lk]
2≤Bkfk2
CN.
2). One of the three following conditions is satisfied:
(i) N2≥K≥N+ 1,
(ii) K=Nis odd,
(iii) K=Nis even and PN
k=1(jk, lk)/∈NN2.
Proof: Select a collection of Kdistinct points z1, ..., zK∈TNof the form jk
N, lk∈
TN,jk, lk∈0, ...., N −1 and rewrite Corollary 14 as
AkΣNfk2
SN≤
K
X
k=1 Vhλ
0ΣNfjk
N, lk
2
≤BkΣNfk2
SN.
Lemma 4 ensures that for every f∈CNthere exists f∈S0(R) such that PNf=f.
By (6)
kΣNfkSN=
1
NPNf
CN
=1
NkfkCN.
Finally, by Theorem 1,
Vhλ
0ΣNfjk
N, lk=N−1Vhλ
Nf[jk, lk].
18 L. D. ABREU, P. BALAZS, N. HOLIGHAUS, F. LUEF, AND M. SPECKBACHER
By Theorem 14, it follows that the frame conditions are satisfied if and only if
either K≥N+ 1, or K=Nand PN
k=1(jk, lk)6= (N2/2 + N n, N 2/2 + Nm), for
all n, m ∈Z.
If N=Kis odd, then N2/2 + nN =N k + 1/2, for some k∈Z. Therefore, the
condition (ii) is automatically satisfied as PN
k=1(jk, lk)∈N2.
If N=Kis even, then for every k∈Zthere exist n∈Zsuch that N2/2+ nN =
Nk. Therefore, we get a frame if and only if PN
k=1(jk, lk)/∈NN2.
This result can be reformulated in the following sense: the discrete Gabor system
generated by the sampled, periodized Gaussian is in general linear position if Nis
odd (and almost in general linear position if Nis even). If any Npoints from the
set {(j, k) : j, k ∈0, ..., N }yield a finite Gabor frame for CNwith window g, then g
is said to be in general linear position, see for example [20]. In [18] it is shown that
almost every vector in CNis in general linear position for every N∈N. However,
explicitly known examples of vectors in general linear position are not localized and
thus of no use for practical purposes of finite Gabor analysis. Our result on the
other hand allows to use localized windows (for appropriate choices of λ) for the
prize that one needs to use one additional sampling point if the number of points
N∈Nis even and enjoys a particular arithmetic structure.
Theorem 14 and Theorem 3 can also be seen as ’Nyquist-type’ necessary and
sufficient results: with less than Nsamples the system is never a frame, but by
increasing the number of samples above N, one is assured to have a frame with
higher redundancy. Such a property comes in handy for situations where one is
given a signal representation sampled on a grid with more than Npoints (allowing
for perfect reconstruction by our result) and wishes to increase the grid resolution
for some numerical purpose. Then sampling again at a higher density is possible,
and it still leads to perfect reconstruction. This can be done until all the possible
N2points of the grid are used. By resorting to the extension to the torus and to
Theorem 14 one can further use off-grid points and increase the resolution to ar-
bitrary levels. The analogue necessary and sufficient result for infinite-dimensional
Gabor frames with Gaussian window is stated in terms of Beurling density and was
proved by Seip and Wallst´en [22, 21] and independently by Lyubarskii [17].
References
[1] L. Ahlfors. Complex Analysis: An Introduction to the Theory of Analytic Functions of one
Complex Variable. McGraw-Hill, 1979.
[2] L. Auslander and Y. Meyer. A generalized Poisson summation formula. Appl. Comp. Harmon.
Anal., 3(4):372–376, 1996.
[3] J. J. Benedetto, C. Heil, and D. F. Walnut. Gabor systems and the Balian-Low theorem.
In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and
Applications, pages 85–122. Birkh¨auser Boston, 1998.
[4] M. Combescure and D. Robert. Coherent States and Applications in Mathematical Physics.
Springer, 2012.
[5] J. Cooley, P. Lewis, and P. Welch. Application of the fast Fourier transform to computa-
tion of Fourier integrals, Fourier series, and convolution integrals. IEEE Trans. Audio and
Electroacoustics, 15(2):79–84, 1967.
[6] H. G. Feichtinger. On a new Segal algebra. Monatsh. Math., 92:269–289, 1981.
[7] H. G. Feichtinger. Modulation spaces: Looking back and ahead. Samp. Theory Signal Image
Process., 5:109–140, 2006.
[8] H. G. Feichtinger and K. Gr¨ochenig. Banach spaces related to integrable group representations
and their atomic decompositions, I. J. Funct. Anal., 86(2):307–340, 1989.
TIME-FREQUENCY ANALYSIS ON FLAT TORI 19
[9] T. J. Gardner and M. O. Magnasco. Sparse time-frequency representations. Proceedings of
the National Academy of Sciences, 103(16):6094–6099, 2006.
[10] J.-P. Gazeau. Coherent States in Quantum Physics. Wiley, 2009.
[11] K. Gr¨ochenig. Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal.
Birkh¨auser Boston, 2001.
[12] M. S. Jakobsen. On a (no longer) new Segal algebra. J. Fourier Anal. Appl., 24(6):1579–1660,
2018.
[13] M. S. Jakobsen and F. Luef. Sampling and periodization of generators of Heisenberg modules.
Int. J. Math., 30(10):1950051, 2019.
[14] A. J. E. M. Janssen. From continous to discrete Weyl-Heisenberg frames through sampling.
J. Fourier Anal. Appl., 3(5):583–596, 1997.
[15] N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J.
Fourier Anal. Appl., 11:25–42, 02 2005.
[16] P. Leboeuf and A. Voros. Chaos-revealing multiplicative representation of quantum eigen-
states. J. Phys. A: Mathematical and General, 23(10):1765, 1990.
[17] Y. I. Lyubarskii. Frames in the Bargmann space of entire functions. In Entire and Subhar-
monic Functions, volume 11 of Adv. Sov. Math., pages 167–180. American Mathematical
Society (AMS), 1992.
[18] R. D. Malikiosis. A note on Gabor frames in finite dimensions. Appl. Comp. Harmon. Anal.,
38(2):318–330, 2015.
[19] F.W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. NIST Handbook of Mathematical
Functions. Cambridge University Press, 1st edition, 2010.
[20] G. E. Pfander. Gabor frames in finite dimensions. In P. G. Casazza and G. Kutyniok, editors,
Finite Frames: Theory and Applications, pages 193–239. Birkh¨auser Boston, Boston, 2013.
[21] K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space. I. J.
Reine Angew. Math., 429:91–106, 1992.
[22] K. Seip and R. Wallst´en. Density theorems for sampling and interpolation in the Bargmann-
Fock space. II. J. Reine Angew. Math., 429:107–113, 1992.
[23] P. Søndergaard. Finite discrete Gabor analysis. PhD thesis, Institut for Matematik, DTU,
2007.
[24] P. Søndergaard. Gabor frames by sampling and periodization. Adv. Comput. Math., 27:355–
373, 10 2007.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090
Vienna, Austria
Email address:abreuluisdaniel@gmail.com
Email address:michael.speckbacher@univie.ac.at
Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14,
1040 Vienna, Austria
Email address:peter.balazs@oeaw.ac.at
Email address:nicki.holighaus@oeaw.ac.at
Department of Mathematics, NTNU Trondheim, 7041 Trondheim, Norway
Email address:franz.luef@ntnu.no
