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NUMERICAL COMPUTATIONS USING GABOR FRAMES FOR

PERIODIC SPACES

PETER LEMPEL SØNDERGAARD.

Resume

This thesis presents Gabor frames for L2(T) and for the space of periodic sequences CLp , asbeing analogue to Fourier series and the discrete Fourier transform. Results from the standardtheory of Gabor frames for L2(R) are transferred to the periodic setting. This includes resultsabout sampling Gabor systems and existence of Gabor frames. The interrelation of Gaborframes for the continuous space L2(T) and the discrete space CLp is afterwards used to constructa new method for numerical di�erentiation with properties close to the method of spectraldi�erentiation. Finally, the various results are illustrated by numerical examples.

Resumé

Rapporten præsenterer Gabor frames for L2(T) og for rummet af periodiske talfølger CLpsom erstatning for fourierrækker og den diskrete fouriertransformation. Resultater fra Gaborframe teori for L2(R) overføres til L2(T). Dette inkluderer resultater omhandlende samplingaf Gabor systemer og eksistens af Gabor frames. Sammenhængen mellem Gabor frames fordet kontinuerte rum L2(T) og det diskrete rum CLp bliver herefter brugt til at konstruere enny metode til numerisk di�erentiation, med egenskaber meget lig spectral di�erentiation. Tilslut i rapporten illustreres de forskellige resultater med numeriske eksempler.

Contents

Resume 1Resumé 1List of symbols 21. Introduction. 32. Periodic Spaces. 33. Fourier analysis. 43.1. The discrete Fourier transform. 43.2. Fourier series. 84. Gabor frames for CLp 104.1. Frames 104.2. Basic properties. 124.3. Algorithms. 195. Gabor frames for L2(T). 255.1. Basic properties. 255.2. Sampling. 305.3. Bounds. 415.4. Existence. 44

1

2 PETER LEMPEL SØNDERGAARD.

6. Interpolation and Numerical di�erentiation 476.1. Polynomial di�erentiation 486.2. Spectral di�erentiation 496.3. The Gabor derivative. 537. Numerical experiments. 577.1. Dual windows. 577.2. Di�erentiation. 648. Concluding remarks. 68References 69

List of symbols1

Symbol: Description:

a.e. �almost everywhere�. Statements holds true except on a set of measure zero.RLp ,CLp The spaces of real and complex periodic sequences of length L.T The circle parameterized by [0; 1[.L2(T) The space of square-integrable functions on the circle T.Cp(T) The space of p times di�erentiable functions with continuous p'th derivative on T.Rm,n,Cm,n The spaces of real and complex matrices of size m× n.{fn},{gn} Sequences.f ,g,h Vectors.A,B,C Matrices.A:,i The i'th column of a matrix.A∗ Conjugate transpose of the matrix A.A⊗B Kronecker product.f The DFT or Fourier series for a vector f .F The discrete Fourier transform or the Fourier series operator.f Complex conjugation of f .f (p) The p′th derivative of f .⌊ab

⌋The result of a divided by b rounded down.

O(·) Bounded from above, used only for computing-speed considerations.

ωL The �rst complex L-root of unity, ωL = e2πiL .

θk,L The k'th mode of length L, θk,L ={

1√Le2πi

knL

}n.

FL The Fourier-matrix of order L.I,IL The identity operator and the identity matrix of size L.δj Kronecker's delta sequence, j being an integer.C,(Cγ) The analysis operator (with window γ).D,(Dg) The pre-frame operator (with window g).S The frame operator.Tk The translation operator.

1The notation used in this thesis di�er somewhat from the typical notation used in Gabor analysis literature.All operators are written using the \mathcal font: D,M, T , matrices are written using bold font: A,B,C andGabor systems are written with the \mathfrak font G,H,K. All sequences are indexed by subscript gk insteadof g(k). This make Fourier series look a little bit out of the ordinary.

NUMERICAL COMPUTATIONS USING GABOR FRAMES FOR PERIODIC SPACES 3

Ml The modulation operator.G,H,J,K Gabor systems.g,γ Gabor windows.Dg,Dγ Gabor matrices with window sequences {gk} and {γk}.Gn The n'th correlation function.G,G The operator and matrix de�ned in theorem 72.PϕW The periodized Gaussian of width W .

1. Introduction.

When solving partial di�erential equations, one of the best methods is based on Fourierseries. This works by approximating a function via a truncated Fourier series expansion,and di�erentiating this series termwise. This gives a remarkably good approximation to thederivative of the function, the so-called spectral derivative.

This only works well if the function is smooth. If the function is not smooth, or perhaps noteven di�erentiable, then the spectral derivative has oscillations. Furthermore these oscillationsa�ect the whole domain, because the complex exponentials in the Fourier series have globalsupport. The complex exponentials have perfect frequency resolution but no time resolution.

Gabor frames have a good trade-o� of time- and frequency resolution, and also a great�exibility in the choice of window function and parameters.

The idea behind this thesis is to use Gabor frames that resembles Fourier series and thediscrete Fourier transform to construct a new method for numerical di�erentiation. Hopefullythis new method will inherit some of the strengths of the spectral di�erentiation, while limitingsome of the weaknesses.

Section 2 presents the relevant function spaces needed. Section 3 contains some necessaryresult from Fourier analysis. Section 4 presents some theory of Gabor frames for periodicsequences. These are the type of Gabor frames that will be used for the actual computationon a computer. The section therefore also contains some results to improve the computationale�ciency.

Section 5 presents the theory of Gabor frames for the square integrable function on thecircle, L2(T). These Gabor frames will be used for the error-analysis of the numerical dif-ferentiation. The results presented in this section have all been adapted from similar resultsfor L2(R), sometimes with big simpli�cations in the complexity of the proofs. The resultsconcern sampling of a Gabor frame, decay properties of the Gabor coe�cients and existenceresults. The similar existence results for L2(R) are very complicated to check, but for L2(T)they simplify to something that can be checked numerically.

Section 6 presents the various methods for numerical di�erentiation and derives some errorbounds, and section 7 presents numerical results for existence of Gabor frames and for theaccuracy of the numerical di�erentiation methods.

2. Periodic Spaces.

Numerical computations on a computer can only be done on �nite dimensional spaces. Thiscould be a sequence of complex numbers, CL, enumerated by the numbers 0..L− 1. However,when doing calculations values enumerated by −1 and L are frequently needed. The simplestsolution to this is to only consider periodic sequences. This means that the �nite sequence ofcomplex numbers will be enumerated by the ring ZL = ZmodL. The value enumerated by


−1 is the value enumerated by L− 1 and so on. This solves any problems associated with theends of the sequence, because there is no ends!

A complex, periodic sequence {fn} of length L will be denoted by CLp , p for periodic. The

space CLp will be equipped with the inner product

(1) 〈{fn} , {gn}〉 =L−1∑n=0

fngn.

This de�nition is very close to the the de�nition of the inner product for CL and the argumentsthat show that CL is a Hilbert space can also be used to show that CLp is a Hilbert space withinner product (1), and that (1) really is an inner product.

When there is no need to use the periodicity or the index of a sequence {fn}, it will bereferred to as a vector f .

The notation here presented, CLp , is my own contribution.

The continuous space corresponding to CLp is the circle, T, T for torus. In this thesis it willalways be parameterized by the interval [0; 1[. A function on the circle T can be associatedwith a periodic function of period one on the real line R. This identi�cation will be usedfrequently.

The function space L2(T) is a Hilbert space with the inner product:

〈f, g〉 =

∫ 1

0f(x)g(x)dx.

The space will sometimes be written as L2([0; 1]) if the mapping with periodic functions on Ris used. Other spaces of this kind will also be used, for example written as L2([0; 1

N ]). This

corresponds to the L2 space of periodic functions with period 1N on R.

Note that the relations between common function spaces are somewhat simpler than forthe real line:

Cp(T) ⊂ C(T) ⊂ L∞(T) ⊂ L2(T) ⊂ L1(T)

Perspectives. Working with periodic spaces is not always convenient - not all problems areperiodic of nature. In this thesis everything will be periodic for the sake of simplicity, and Iwill not make any considerations about non-periodic spaces. If results from this thesis shouldbe applied to non-periodic problems, a way to do it would be to periodize the problems insome way and adapt the problem to the periodic space.

3. Fourier analysis.

3.1. The discrete Fourier transform. This section is mostly inspired by [Briggs95] withthe normalization of the DFT �tted to match [Stroh98], and the notation θj,L being my ownaddition.

Fourier analysis on �nite, periodic sequences is done through the discrete Fourier transform:

De�nition 1. The discrete Fourier transform of {fn} ∈ CLp is a sequence{fk

}given by

F {fn} ={fk

}=

{1√L

L−1∑n=0

fne−2πink

L

}k∈Z

.

The discrete Fourier transform is abbreviated as the DFT.


De�nition 1 only tells us that{fk

}is a sequence. The following proposition will show that

it is also periodic.

Proposition 2. The DFT of a periodic sequence {fn} ∈ CLp is again a periodic sequence{fk

}∈ CLp .

Proof. Proving this amounts to proving that fk = fkmodL for any k. Since the complex

exponential function is periodic with period 2π then e−2πinkL = e−2πin

(kmodL)L . �

The DFT can be also be formulated as a matrix-vector product. For convenience de�ne

ωL = e2πiL . This is the �rst complex L-root of unity, i.e. (ωL)L = 1. Furthermore, de�ne the

modes:

De�nition 3. The k'th mode θk,L ∈ CLp of period L is de�ned by

θk,L =

{1√Le2πi

knL

}n∈Z

=

{1√LωknL

}n∈Z

,

for k ∈ Z and L ∈ N.

The modes are periodic sequences of period L: Since the exponential function is periodicwith period 2π, then the modes are periodic with at most period L.

Using ωL and the modes, the de�nition of the DFT can be written as{fk

}=

{1√L

L−1∑n=0

fnω−nkL

}k∈Z

(2)

= {〈f, θk,L〉}k∈Z .This can be reformulated as a matrix product:

f = F∗Lf,

where the j, k'th entry of the matrix FL is given by

(FL)j,k =1√LωjkL .

This matrix is called the Fourier matrix 2:

FL =1√L

1 1 1 · · · 1

1 ω1L ω2

L · · · ωL−1L

1 ω2L ω4

L · · · ω2(L−1)L

......

......

1 ω(L−1)L ω

2(L−1)L · · · ω

(L−1)(L−1)L

.(3)

The modes appears as the columns of the matrix.To gain more insight in the structure of the Fourier matrix, the following important lemma

about the modes is needed.

2The matrix de�ned this way is the transpose of the similar matrix de�ned in [Stroh98]. This makes thematrix methods for the DFT and IDFT resemble those for the Gabor frame methods more closely.


Lemma 4. Orthogonality relations. Let k, j ∈ Z and L ∈ N. Then

〈θj,L, θk,L〉 = δ|j−k| mod L,

where δj is Kronecker's delta function, de�ned by

δj =

{1 if j = 00 if j 6= 0

.

Proof. All terms of the form ωkL are roots of the polynomial zL − 1. This polynomial can befactored as

zL − 1 = (z − 1)(zL−1 + zL−2...+ z + 1)(4)

= (z − 1)

L−1∑n=0

zn.

Now the inner product can be expanded as:

〈θj,L, θk,L〉 =

L−1∑n=0

1√LωnjL

1√Lω−nkL

=1

L

L−1∑n=0

ωn(j−k)L =

1

L

L−1∑n=0

(ωj−kL

)n.(5)

If j = k modulo L then z = ωj−kL = 1 and so 〈θj,L, θk,L〉 = 1.

If j 6= k modulo L then z = ωj−kL 6= 1, and zL − 1 = 0 because ωj−kL is a complex root

of unity. Inserting this into (4) it is seen that∑L−1

n=0 zn must be zero. This is the last term

occurring in (5) so the conclusion is that 〈θj,L, θk,L〉 = 0.These two cases prove the lemma. �

If this lemma is applied to the Fourier matrix the following proposition comes as the result.

Proposition 5. The Fourier matrix FL is a unitary matrix.

Proof. This amounts to proving that 〈F:,j ,F:,k〉 = δ|j−k|, where F:,j denotes the j'th columnof F. Since the columns are the modes:

〈F:,j ,F:,k〉 = 〈θj,L, θk,L〉= δ|j−k|modL

the result follows from 4. The �modL� can be discarded since the indices only run in therange 0, ..., L− 1. �

The inverse matrix of a complex unitary matrix is the conjugate transpose of the matrix,so this proves the existence of the inverse discrete Fourier transform, the IDFT, and providesa matrix expression:

f = (F∗L)−1 f = FLf .

From this a summation expression for the IDFT can be obtained:


Corollary 6. The inverse discrete Fourier transform {fn} of a sequence{fk

}is a mapping

from CLp → CLp and is given by

F−1{fk

}= {fn} =

{1√L

L−1∑k=0

fke2πink

L

}n∈Z

.

Proof. The summation expression is read directly from the matrix expression, and the peri-odicity is proven in the same manner as for the DFT. The only structural di�erence in thede�nition of the DFT and the IDFT is a minus-sign in the exponential function. �

An important consideration for applications is how the DFT of a real signal {fn} ∈ RLpbehaves. In that case fn = f∗nand so

f∗k =

(1√L

L−1∑n=0

fne−2πink

L

)∗

=1√L

L−1∑n=0

fne−2πin(−k)

L

= f−k.(6)

This means that one does only need to compute and store half the complex DFT coe�cientsfor a real signal, as the other half can be found by complex conjugation.

The DFT (and the IDFT) can be computed numerically using the matrix methods

f = F∗Lf

f = FLf .

This method requiresO(L2)�oating point operations to compute either one of the transforms,

where O(L2) means that it is bounded from above by a constant times L2.A more e�cient way of computing the transforms is through the use of a fast Fourier

transform, an FFT or for the inverse transform, an IFFT. These methods require O(LlogL)�oating point operations.

Perspectives. So far I have chosen to enumerate �nite periodic sequences with the integersrunning from 0 to L− 1. One could just as well use the indices −L

2 + 1 to L2 . This gives the

alternate de�nition of the DFT:

De�nition 7. Using the indices k ∈ −L2 + 1, ..., L2 and n ∈ −L

2 + 1, ..., L2 the DFT is de�nedby {

fk

}=

1√L

L2∑

n=−L2+1

fne−2πink

L

k∈Z

.

The coe�cients obtained this way are exactly the same as for the original de�nition of theDFT, because everything work modulo L. This de�nition only works when L is even, but thisis not a problem. It can easily be formulated for L being odd. Similarly, the IDFT

(7) {fn} =

1√L

L2∑

k=−L2+1

fke2πink

L

n∈Z


At �rst these de�nitions look more troublesome because of the need to divide by two. Thereare however good reasons for using them, and these are the problems of aliasing.

The coe�cient fL−1 is calculated by the inner product of f and θL−1,L. The vector θL−1,Lis formed from L samples of the complex function e−2πi(L−1)x in the interval x ∈ [0; 1]. Ifthese samples are plotted one might expect to see some points forming L − 1 waves on theinterval [0; 1]. This is done on �gure 1.

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

θ11,12

Figure 1. Plot of the mode θ11,12, with lines tracing out e2πi(11) and e2πi(−1).Only the real part is displayed.

The �gure shows the 12 points of the mode θ11,12 tracing out the wave e2πi(−1) instead

of the wave e2πi(11). This illustrates that the high-frequency mode θ11,12 is aliased to thelow-frequency mode θ−1,12.

When using an already programmed DFT routine on a computer, one should check whichde�nition of the DFT it uses, de�nition 1, de�nition 7 or a third de�nition. The softwarepackage Matlab uses de�nition 1 and so does the software package Octave, in which thenumerical examples for this thesis are written.

The fast Fourier transform (FFT) methods provide a fast way of computing the DFT. Theyexist in many variants. Di�erent algorithms for the computation of the FFT are presented in[Briggs95, chapter 10].

3.2. Fourier series. The section contains statements necessary to develop the theory of Ga-bor frames for L2(T) and to derive error estimates for the numerical di�erentiation methodsthat will be presented later.

Theorem 8. Plancherel's theorem for the circle. Let f ∈ L2(T) and let

(Ff)k = fk =

∫ 1

0f(x)e−2πikxdx

be the k'th Fourier coe�cient. Then f can be expanded as a Fourier series

f(x) =∑k∈Z

fke2πikxdx a.e. x ∈ T.

and ∫ 1

0|f(x)|2 dx = ‖f‖2L2(T) =

∑k∈Z

∣∣∣fk∣∣∣2 =∥∥∥f∥∥∥2

l2(Z).

Proof. This is [Groch01, theorem 1.3.1] or [Rud66, theorem 9.13]. �


The next lemma establishes a Fourier expansion for a periodized function on T. The lemmawill be used to replace Poisson's summation formula for L2(R) when transferring proofs tothe periodic setting.

Lemma 9. Poisson summation for the circle. Let f ∈ C(T) and N ∈ N. ThenN−1∑n=0

f(x+n

N) = N

∑k∈Z

fkNe2πikNx, x ∈ [0;

1

N].

Proof. Let

ϕ(x) =N−1∑n=0

f(x+n

N), x ∈ [0;

1

N]

andψ(x) = ϕ(

x

N), x ∈ T.

Since ψ is periodic with period 1, because ϕ is periodic with period 1N , then ψ can be expressed

as a Fourier series with coe�cients

ψ(k) =

∫ 1

0ψ(x)e−2πikxdx

=

∫ 1

0ϕ(

x

N)e−2πikxdx

= N

∫ 1N

0ϕ(x)e−2πikNxdx

= N

∫ 1N

0

(N−1∑n=0

f(x+n

N)e−2πikNx

)dx

= N

N−1∑n=0

(∫ 1N

0f(x+

n

N)e−2πikN(x+ n

N)dx

)

= N

N−1∑n=0

(∫ n+1N

nN

f(x)e−2πikNxdx

)

= N

∫ 1

0f(x)e−2πikNxdx

= NfkN .

In line �ve it is used that since the complex exponential function is periodic, then e−2πikNx =e−2πikN(x+ n

N), and then the order of summation and integration is interchanged. This works

well since both are �nite. Now the summation and integration form a disjoint partition ofthe interval [0; 1] (the overlap has measure zero) in line six, and so they are replaced by onecombined integration.

With this, ψ can be expressed by its Fourier series

ψ(x) =∑k∈Z

ψke2πikx

= N∑k∈Z

fkNe2πikx


for x ∈ T and this gives a similar expansion of ϕ:

ϕ(x) = ψ(Nx)

= N∑k∈Z

fkNe2πikN ,

for x ∈ [0; 1N ]. �

Remark 10. The lemma can be extended by density ([Groch01, A.1]) to cover all f ∈ L2(T),but then only with convergence almost everywhere x ∈ [0; 1

N ].

Theorem 11. Let f ∈ Cp−1(T). Assume that f (p) is bounded and piecewise monotone. Thenthe Fourier series coe�cients of f satisfy

|ck| ≤C

|k|p+1 , ∀k ∈ Z,

where C is a constant independent of k.

Proof. This is [Briggs95, theorem 6.2]. �

The Fourier series coe�cients {ck} can be approximated by the DFT-coe�cients{

1√Lfk

}.The

error introduced by doing this is outlined in the following theorem.

Theorem 12. Let f ∈ Cp−1(T). Assume that f (p) is bounded and piecewise monotone. Then

the error of the DFT coe�cients f of order L as an approximation to the Fourier seriescoe�cients ck satisfy ∣∣∣∣fk − 1√

Lck

∣∣∣∣ ≤ C

Lp+1, k = −L

2+ 1, ...,

L

2,

where C is a constant independent of k and L.

Proof. [Briggs95, theorem 6.3]. �

Remark 13. In [Briggs95] the theorem is extended to also cover the situation where f is noteven continuous , but only bounded and piecewise monotone. In this case

(8)

∣∣∣∣fk − 1√Lck

∣∣∣∣ ≤ C

L

as could be expected from theorem 12.

4. Gabor frames for CLpThis section is a transfer of some of the results I showed in [Soen02] and the last part is

based upon [Stroh98] but with more details.

4.1. Frames. In the �rst part of this project [Soen02] I showed some general theory of framesfor Hilbert spaces. Since CLp is a Hilbert space all these results are applicable. I will brie�yrepeat some important results and de�nitions here.

De�nition 14. A family of elements {ej}j∈J is a frame for a separable Hilbert space H if

constants 0 < A ≤ B <∞ exists such that

(9) A ‖f‖2H ≤∑j∈J

∣∣〈f, ej〉H∣∣2 ≤ B ‖f‖2H , ∀f ∈ H.The constants A and B are denoted the lower and upper frame bound.


A sequence that only satis�es the upper frame bound is called a Bessel sequence:

De�nition 15. A family of elements {ej}j∈J , ej ∈ H is a Bessel sequence if∑j∈J

∣∣〈f, ej〉H∣∣2 ≤ B ‖f‖2H , ∀f ∈ Hfor some constant 0 < B <∞. The constant B is called the Bessel bound.

De�nition 16. Let {ej}j∈J be a frame for a Hilbert space H and let f ∈ H. Then the

analysis operator is de�ned by

Cf ={〈f, ej〉H

}j∈J .

The analysis operator returns the inner products of the vector f with the frame elements{ej}j∈J .

Proposition 17. Let {ej}j∈J be a frame for a Hilbert space H. Then the analysis operator

C : H → l2(J) is a bounded operator C : H → l2(J).

Proof. [Soen02] or [Groch01, prop. 5.1.1]. �

De�nition 18. Let {ej}j∈J be a frame for a Hilbert space H. Then the pre-frame operator

D : l2(J) → H is de�ned by

D{cj} =∑j∈J

cjej .

The pre-frame operator assembles a vector f from its decomposition {cj} in the frame{ej}j∈J .

Proposition 19. Let {ej}j∈J be a frame for a Hilbert space H. Then the pre-frame operator

D is the adjoint operator of the analysis operator C.

Theorem 20. Let {ej}j∈J be a frame for a Hilbert space H. Then the frame operator de�nedas

S : H → H, Sf = DCf =∑j∈J〈f, ej〉H ej

is bounded, self-adjoint and invertible.

Proposition 21. Let {ej}j∈J be a frame for a Hilbert space H with frame operator S. The

family{S−1ej

}j∈J is also a frame, the so-called dual frame.

Theorem 22. Let {ej}j∈J be a frame for a Hilbert space H. Then any f ∈ H can be writtenas

f =∑j∈J

⟨f,S−1ej

⟩H ej .

The proofs of the preceding theorems and propositions can all be found in [Soen02, section3.1] or in [Chr02, Groch01].


4.2. Basic properties. The de�nition of a Gabor frame for CLp requires the de�nition of thetranslation and modulation operators.

De�nition 23. The translation Tn and the modulationMm operators working on a sequence{fk} ∈ CLp are de�ned by

Tn {fk} = {fk+n}k∈ZMm {fk} =

{e−2πi

mkL fk

}k∈Z

={ω−mkL fk

}k∈Z

for all m,n ∈ Z.

The translation of a periodic sequence is again a periodic sequence, since all the values havejust been shifted by the same distance. Similarly for the modulation operator, the modulationof a periodic sequence is again a periodic sequence, since the modulation consist of pointwisemultiplying the sequence with a periodic sequence, and all the periods are equal, namely L.

The operators are also periodic in their parameters:

Tn {fk} = TnmodL {fk}Mm {fk} = MmmodL {fk} .

The periodicity in m ofMm follows from the periodicity of the exponential function.The inverse operators are simply the opposite action of the operators:

(Tn)−1 {fk} = T−n {fk}(Mm)−1 {fk} = M−m {fk} .

Composing the operators is straightforward:

TnTl {fk} = Tn+l {fk}MmMl {fk} = Mm+l {fk}

The two operators do not commute:

MmTn {fk} ={e−2πi

mkL fk+n

}(10)

={e2πi

mnL e−2πi

mL·(k+n)fk+n

}(11)

= e2πimnL TnMm {fk} .(12)

It is now possible to de�ne a Gabor system for CLp .

De�nition 24. A Gabor system in CLp is a family of sequences de�ned by:

G = G({gk} , a,b

L) = {MmbTnag}m=0,...,M−1,n=0,...,N−1 ,

with {gk} ∈ CLp and where a, b ∈ N and Mb = Na = L. This means that L must be divisibleby a and b.

Each element in the system is called a Gabor atom:{(Gm,n)k

}= {MmbTnag}k =

{e−2πi

mbL·kgk+na

}k∈Z

.

A Gabor system that is also a frame is called a Gabor frame.


De�nition 25. The density of a Gabor system for CLp as de�ned in 24 is given by

(13)L

MN=ab

L=

a

M=

b

N.

The density is the ratio of the length of the vectors, and the number of vectors in the Gaborsystem. The density divides Gabor systems for CLp in three categories:LMN > 1: The system cannot be a frame or a basis, as there is to few vectors to span the

space CLpLMN = 1: If the system is a frame, then it is also a basis, since L vectors spanning an L-

dimensional space constitutes a basis.LMN < 1: The system cannot be a basis, but it can be a frame.

This categorization correspond to the similar case for Gabor frames for L2(R), [Soen02, corol-lary 3.14 and 3.15].

The next proposition will present the main result for Gabor frames for CLp .

Proposition 26. Let G({gk} , a, bL), {gk} ∈ CLp , a, b, L ∈ N be a Gabor frame for CLp . Then

there exists a sequence γ0 ∈ CLp such that H({γ0k}, a, bL) is a dual frame of G.

Proof. The method is to show that both the translation and modulation operator commuteswith the frame operator.

The frame operator can be written

(14) Sf =N−1∑n=0

M−1∑m=0

〈f,MmbTnag〉MmbTnag.

This is the de�nition of the frame operator from theorem 20 with the Gabor atoms inserted.Consider the following expression

(MrbTsa)−1 S (MrbTsa) f(15)

=N−1∑n=0

M−1∑m=0

〈MrbTsaf,MmbTnag〉 (MrbTsa)−1MmbTnag.(16)

for r, s ∈ Z. This was just a matter of inserting the de�nition of the frame operator (14) andmoving the inverted translation and modulation inside the summation. The is perfectly legalsince the summation is �nite. The inner product can be rewritten as

〈MrbTsaf,MmbTnag〉 =⟨f, T−saM(m−r)bTnag

⟩=

⟨f, e2πiab(m−r)sM(m−r)bT−saTnag

⟩= e−2πiab(m−r)s

⟨f,M(m−r)bT(n−s)ag

⟩,

using the inverse operators of translation and modulation and using the commutation relation(12). The phase factor changes sign when pulled outside the inner product, because the innerproduct is conjugate linear in the second slot.

The last part of the expression can be rewritten in a similar manner

(MrbTsa)−1MmbTnag = T−saM(m−r)bTnag

= e2πiab(m−r)sM(m−r)bT−saTnag

= e2πiab(m−r)sM(m−r)bT(n−s)ag.


Inserting these two expressions in (16) shows that the phase factor cancel

N−1∑n=0

M−1∑m=0

〈MrbTsaf,MmbTnag〉 (MrbTsa)−1MmbTnag

=

N−1∑n=0

M−1∑m=0

⟨f,M(m−r)bT(n−s)ag

⟩M(m−r)bT(n−s)ag(17)

=

N−1∑n=0

M−1∑m=0

〈f,MmbTnag〉MmbTnag

= Sf.The second line shows something that looks like the Gabor frame operator, except that allthe summation indices are shifted. However, since both the translation and the modulationoperator are L-periodic in their parameter, the second and the third line contain exactly thesame terms, just summed in a di�erent order.

To sum up, it has been shown that

(MrbTsa)−1 S (MrbTsa) = S.This can be transferred to the inverse operator:

S−1f =(

(MrbTsa)−1 S (MrbTsa))−1

f

= (MrbTsa)−1 S−1 (MrbTsa) f,so the same relation holds true for the inverse operator. A simple application of this commu-tation relation is

S−1MrbTsag = MrbTsaS−1g= MrbTsaγ0,

whereγ0 = S−1g.

Applying the inverse frame operator to a Gabor atom is the same as applying the translationand modulation directly to a new window function. This new window function γ0 = S−1g iscalled the canonical dual sequence of the Gabor frame. �

The pre-frame (or synthesis) operator for a Gabor frame for CLp with window {gn} is givenby

(18) Dg{cm,n} =N−1∑n=0

M−1∑m=0

cm,nMmbTnag,

from de�nition 18. The pre-frame operator for a Gabor frame for CLp will be called the inversediscrete Gabor transform, the IDGT.

The operator can be formulated using a matrix

(19) Dg{cm,n} = Dgc

if an ordering of the double indices m and n is chosen.The ordering used in this thesis is that cm,n is placed in a vector c such that cm+nM = cm,n,

that is c0,0 is the �rst element followed by c1,0 and the M + 1'th element is c0,1 followed by


c1,1 and so on. This forces an ordering such that the Gabor atomM0T0g is the �rst columnin the matrix DG followed byM1bT0g as the second column andM0T1ag as the (M + 1)'thcolumn.

This structure serves as the de�nition of a Gabor matrix:

De�nition 27. AGabor matrix with window g is a matrixDg ∈ CL,MN with elements (Dg)m,ngiven by:

(Dg)m,n =(Mbb nM cTa(nmodM)g

)m

= e−2πibbnM cmL gm+a(nmodM),(20)

where⌊nM

⌋denotes the result of the division n

M rounded towards zero.

The operator norm of the pre-frame operator can be easily calculated.

Proposition 28. Let {cm,n} ∈ CM,N and let {gk} ∈ CLp , M,N, a, b, L ∈ N with Ma = Nb =L. Then then pre-frame operator has operator norm∥∥D{g} {cm,n}∥∥op ≤ √Lmax

j|gj |

Proof. De�ne f ∈ CLp by

{fk} =

N−1∑n=0

M−1∑m=0

cm,nMmbTna {gk} .

Then

‖{fk}‖22 =

L−1∑k=0

∣∣∣∣∣N−1∑n=0

M−1∑m=0

cm,ne−2πimb k

L gk+na

∣∣∣∣∣2

≤L−1∑k=0

(N−1∑n=0

M−1∑m=0

∣∣∣cm,ne−2πimb kL gk+na∣∣∣)2

≤ L

(maxj|gj |)2(N−1∑n=0

M−1∑m=0

1 · |cm,n|

)2

= L

(maxj|gj |)2√

MN ‖{cm,n}‖22 .

In line three the reference to {gk} is bounded by it maximum value. This can then be pulledoutside the summations and the summation over k can be removed. Then Cauchy-Schwartzinequality can be used on the two sequences {1}m,n and {cm,n}, and this gives the result. �

Remark 29. The previous proposition does not require that {gk} generates a frame, so thestatement can be used even if the pre-frame operator is not associated to a frame.

The analysis operator for a Gabor frame for CLp with window sequence {γn} is given by

(21) Cγf = {〈f,MmbTnaγ〉}m=0,..,M−1, n=0,...,N−1 ,

from de�nition 16. This operator can be formulated using a matrix

(22) Cγf = D∗γf,


where Dγ is a Gabor matrix with window γ. The analysis operator for a Gabor frame for CLpwill be called the discrete Gabor transform, the DGT.

The next expression for S is the so-called Walnut's representation [Groch01, theorem 6.3.2]in the discrete, periodic case.

Proposition 30. Walnut's representation. The Gabor frame operator S can be expressedas a matrix S ∈ CL,L with

(23) Sm,n =

{L∑N−1

k=0 (Tkag)m (Tkag)n if (n−m) modM = 00 otherwise

.

When {gk} ∈ RLp then S ∈ RL,L and consequently the canonical dual is real:{(γ0)k

}∈ RLp .

Proof. As the frame operator by de�nition (as de�ned in theorem 20) is the composition ofthe pre-frame operator and the analysis operator, then it follows from (19) and (22) that

(24) Sf = DgCgf = DgD∗gf = Sf.

Fully written out, the components of S are:

Sm,n =

N−1∑k=0

M−1∑l=0

(MlbTkag)m (MlbTkag)n.

This expression can be simpli�ed:

Sm,n =N−1∑k=0

M−1∑l=0

e−2πilbLmgm+kae

2πi lbLngn+ka

=

N−1∑k=0

gm+kagn+ka

M−1∑l=0

e−2πilbLme2πi

lbLn

=N−1∑k=0

gm+kagn+ka

M−1∑l=0

e2πiln−mM .

The �nal line contains the term∑M−1

l=0 e2πiln−mM . This is the inner product⟨√

Mθn−m,M ,√Mθ0,M

⟩= Mδn−mmodM ,

because√Mθ0,M = {1}l and

√Mθn−m,M =

{e2πil

n−mM

}l. The result follows from lemma 4.

Using this, the expression for S becomes:

Sm,n =

{M∑N−1

k=0 (Tkag)m (Tkag)n if (n−m) modM = 00 otherwise

It shows, that since the matrix elements of S are composed only of translations of {g} and nomodulations are involved, then if {g} is real then S is real as well. This means that also S−1

is a real matrix and so the statement about the canonical dual follows. �

Theorem 31. Janssens representation. Let G({gk} , a, bL), {gk} ∈ CLp , be a Gabor frame

for CLp .

S {fk} =N−1∑n=0

N−1∑m=0

〈{fk} ,MmbTna {gk}〉MmbTnag


can be written as

(25) S {fj} =N

b

b−1∑l=0

a−1∑r=0

〈{gj} ,MrNTlM {gj}〉MrNTlM {fj} .

Proof. The matrix form of Walnut's expression for the frame operator can be written bysummations:

(S {fl})j =L−1∑l=0

Sj,lfl

= M

b−1∑l=0

(N−1∑k=0

gj+kagj+lM+ka

)fj+lM .(26)

This last expression comes from only including the contributing terms from (23).The sequences Gl, l = 0, ..., b− 1

{Gl}j =

{N−1∑k=0

gj+kagj+lM+ka

}j

are periodic with period a and so they can be written using a discrete Fourier transform oflength a:

{Gl}j =1√a

a−1∑r=0

(Gl

)re2πir

ja , l = 0, ..., a− 1,

with (Gl

)r

=1√a

a−1∑j=0

(Gl)j e−2πij r

a

=1√a

a−1∑j=0

N−1∑k=0

gj+kagj+lM+kae−2πij rN

L

=1√a

L−1∑j=0

gjgj+lMe−2πij rN

L

=1√a〈{gj} ,MrNTlM {gj}〉CLp

In line three the summations over k and j are combined into one and j + ka is replaced byj. Because the exponential function is periodic, then the term containing the exponentialfunction does not change.

This gives an expression for Gl:

{Gl}j =1

a

a−1∑r=0

〈{gj} ,MrNTlM {gj}〉 e2πirjNL .


Inserting this into (26):

S {fl} =

{M

a

b−1∑l=0

(a−1∑r=0

〈{gj} ,MrNTlM {gj}〉 e2πirjNL

)fj+lM

}j

=N

b

b−1∑l=0

a−1∑r=0

〈{gj} ,MrNTlM {gj}〉MrNTlM {fj} .

The last line follows from Ma = N

b . �

Proposition 32. Let G(g, a, bL), g ∈ CLp , a, b, L ∈ N be a Gabor frame for CLp . Let γ ∈ CLp bea dual window of g. Then

(27) 〈MmbTnag,MrbTsaγ〉 =

{1 if r = mmoduloM andn = smoduloN0 otherwise

for all m,n, r, s ∈ Z.

Proof. When γ is a dual window of g then f = DgCγf for all f ∈ CLk . Using the Gabor matrixnotation this is

f = DgD∗γf, ∀f ∈ CLk ,

which by standard linear algebra gives

DgD∗γ = IL,

where IL is the identity matrix of size L. Looking at the structure of the matrices shows theresult for m, r ∈ 0, ...,M − 1 and n, s ∈ 0, ..., N − 1. The extension to m,n, r, s ∈ Z followsfrom the periodicity of the translation and modulation operators. �

Remark 33. A more careful statement of the previous result would yield the Wexler-Raz bi-orthogonality relations for CLp (see [Groch01, theorem 7.1] for the L2(Rd) relations). Therelations characterize all possible dual window functions by requiring them to satisfy (27).

Considerations. The requirements for the numbers a, b,M,N,L for a Gabor system for CLp asde�ned in de�nition 24, that Mb = Na = L and a, b,M,N,L ∈ N put some restrictions onthe freedom to choose these numbers. A simple example: If one wishes to keep the densityLMN and the ratio between the number of time shifts and frequency shifts M

N �xed, then it isnot possible to double L.

Another example is the very common practice of choosing L to be a power of 2. This willforce M and N to also be powers of 2, and then the density can only be 1

2p for some p ∈ N,so one could only �nd a Gabor basis with redundancy 1, or a rather redundant Gabor framewith redundancy 1

2 or 14 . It would not be possible to �nd frames with an intermediate density.

Good choices of a, b,M,N tends to be numbers divisible by di�erent small prime factors 2,3 and 5. An example would be

L = 360

M = N = 20

a = b = 18,

giving a square system with a redundancy of 360400 = 0.9.

The proof of proposition 26 that shows how to �nd the canonical dual sequence resemblesthe corresponding result for Gabor frames for L2(R) [Soen02, prop. 3.11] very much. Similarly


in [Soen02] there is one de�nition of the translation- and modulation operators and of a Gaborframe, and in this thesis there is another as well.

It would be tempting to search for a common theory covering both the continuous and thediscrete case of Gabor frames. Such a theory is described in [Groech98].

The combined theory for the discrete and the continuous theory abstracts the space (CLp or

L2(R)) with a locally compact Abelian group. The group composition is then used to de�nethe translation operator. The modulation operator is de�ned by the use of the characters,which for the two spaces considered are

χy(x) = e2πiyx, y ∈ R, x ∈ R

{(χm)k} = e2πimkL , m ∈ Z, k ∈ Z.

The modulation operator is then de�ned by point wise multiplication by a character.In this manner translation and modulation and Gabor frames can be de�ned in a generic

way.I will not pursue the theory of locally compact Abelian groups in this thesis, instead I have

just repeated the proof of proposition 26 adapted to CLp . I does however seem quite fruitful tohave one combined theory instead of two or more separate theories, but de�ning the theoryin all details would be more work than simply repeating the few proofs needed for this thesis.

4.3. Algorithms. This section is written solely from [Stroh98], but the proofs for proposition36, theorem 37 and theorem 38 are my own additions as there are only hinted at in the article.Most notably is the use of

⌊ab

⌋not used in the article.

This section will present some faster ways of calculating c = D∗gf , f = Dgc and γ = S−1g.To do so, some linear algebra tools and de�nitions are needed.

De�nition 34. The Kronecker product A ⊗ B where A is a j × k matrix and B a l × mmatrix is a jl × km matrix de�ned block wise by

A⊗B =

a0,0B · · · a0,k−1B...

...aj−1,0B · · · aj−1,k−1B

.For Kronecker products it holds that

(28) (A⊗B)∗ = A∗ ⊗B∗,

see [vL92].Another simple rule for Kronecker products that will be needed is

(29) (IN ⊗A) (IN ⊗B) = IN ⊗AB,

where IN denotes the identity matrix of size N , IN ∈ CN,N and A ∈ Cb,M ,B ∈ CM,b.The matrix IN ⊗ A is block-diagonal with the matrix A along the diagonal. By a little

inspection of the matrix products it shows that multiplying matrices of this kind only multipliescorresponding blocks on the diagonal, and so the result follows.


De�nition 35. The mod p perfect shu�e permutation matrix Pp,L ∈ CL,L where p, q, L ∈ Nand pq = L is de�ned by

(Pp,L)j,k =

{1 if kp+

⌊kq

⌋modL = j

0 otherwise

= δ∣∣∣kp+⌊ kq ⌋−j∣∣∣modL.(30)

or equivalently

(Pp,L)j,k =

{1 if jq +

⌊jp

⌋modL = k

0 otherwise

= δ∣∣∣kp+⌊ jp⌋−k∣∣∣modL.(31)

The de�nition (30) can be reformulated using k = rq + s, where r = 0, ..., p − 1 ands = 0, ..., q − 1. Then

(Pp,L)j,rq+s = δ∣∣∣(rq+s)p+⌊ rq+sq ⌋−j∣∣∣modL

= δ|sp+r−j|modL

= δsp+r−j.

The last line follows from the fact that sp + r < L for all s, r. From this it follows that amatrix element is one if and only if j = sp+ r, so the ones are placed in the positions

(Pp,L)sp+r,rq+s .

If j is replaced by sp+ r in (31) the result is the same, and so the two de�nitions are equal.Since Pp,L is a permutation matrix it follows from basic linear algebra that it is also a

unitary matrix, i.e. P∗p,L = P−1p,L.The de�nition implies that Pp,L has exactly one element per column with the value 1

instead of 0. For the �rst Lp columns this element is placed when kpmodL = j. For the next

Lp columns this pattern is repeated, except that all the columns are shifted by 1. The following

picture may illuminate this

p rows

Pp,L =

1 0 · · · 0 0 · · · 01

01

......

1

0 1

︸︷︷︸q columns

.


Some de�nitions for diagonal matrices are also needed. The notion diag(W0,W1, ...,WM−1)will mean a block-diagonal matrix of size Mm×Mn if each Wk are of size m× n:

diag(W0,W1, ...,WM−1) =

W0 0 · · · 00 W1 0...

. . .

0 0 WM−1

,where 0 is the zero-matrix of size m× n.

Proposition 36. Let Dg be a Gabor matrix with window g. Then

(Dg (IN ⊗ FM ))m,n =

{ √Mgm+ab nM c if |m− n| modM = 0

0 otherwise,

where IN denotes the identity matrix of size N .

Proof. The matrix IN ⊗ FM is a block-diagonal matrix with FM as each block. MultiplyingDg from the right with this matrix amounts to inner products between panels of the rows ofDg and the columns of FM , i.e. the modes of length M . The panels of the rows are thoseelements of the rows that corresponds to the same translation of the window function g, whichare the elements 0, ..,M − 1, M, ..2M − 1 and so on.

The j'th panel of M elements of the m'th row of the Gabor matrix Dg consists of[Tjagm, (M1bTjag)m , (M2bTjag)m , ...,

(M(M−1)bTjag

)m

]= gm+ja

[ω0L, ω

−mbL , ω−2mbL , ..., ω

−(M−1)mbL

]= gm+ja

√Lθ−m,M .(32)

Line three comes from ω−jmbL = e−2πijmbL = e−2πi

jmM = ω−jmM , since L

b = M .With (32) and knowledge of the columns of the Fourier matrix, the term (Dg (IN ⊗ FM ))m,n

can be calculated:

(Dg (IN ⊗ FM ))m,n =

M−1∑k=0

√Lgm+ab nM c (θ−m,M )k (θnmodM,M )k

=√Lgm+ab nM c 〈θ−m,M , θ−n,M 〉

=√Lgm+ab nM cδ|m−n| mod M .

The panel index j in (32) has been replaced by⌊nM

⌋, as in (20). The last line follows from

the orthogonality relations, lemma 4. �

Rearranging the terms in Dg (IN ⊗ FM ) leads to the following proposition.

Theorem 37. Let Dg ∈ CL,MN be a Gabor matrix with window g. Then

Ddiag = P∗M,LDg (IN ⊗ FM )P∗N,MN

where Ddiag ∈ CL,MN is a block-diagonal matrix

Ddiag = diag(W0, ...,WM−1)


with Wk ∈ Cb,N :(Wk)m,n =

√Mgk+mM+na

for m = 0, ..., b− 1, n = 0, .., N − 1 and k = 0, ..,M − 1.

Proof. Proposition 36 proves the sparse structure of Dg (IN ⊗ FM ).De�nition 35 gives the structure of the permutation matrices(

P∗M,L

)m,n

= δ|mM+bmb c−n|modL(P∗N,MN

)m,n

= δ|nN+b nN c−m|modMN .

The �rst line is derived from (30) and the second from (31).Using this and proposition 36(

Dg (IN ⊗ FM )P∗N,MN

)m,n

=MN−1∑k=0

√Lgm+ab kM cδ|m−k| mod Mδ|nM+b nN c−k|modMN

=√Lg

m+a

⌊nM+b nN cmodMN

M

⌋δ|m−(nM+b nN cmodMN)| mod M

=√Lgm+anδ|m−b nN c| mod M ,

for m = 0, ..., L − 1 and n = 0, ...,MN − 1. The summation in line two is removed becauseP∗N,MN is a permutation matrix, and so there is exactly one element per column that is non-

zero. Therefore k can be replaced by nM +⌊nN

⌋modMN . The term a

⌊nM+b nN cmodMN

M

⌋reduces to an since n < MN and therefore

⌊b nN cmodMN

M

⌋= 0 always.

The full product can now explicitly be written out.

Ddiag =(P∗M,LDg (IN ⊗ FM )P∗N,MN

)m,n

=L−1∑k=0

δ|mM+bmb c−k|modL

√Lgk+anδ|k−b nN c| mod M

=√Lg(mM+bmb cmodL)+anδ|(mM+bmb cmodL)−b nN c| mod M

=√LgmM+bmb c+anδ|bmb c−b nN c| mod M

=√LgmM+bmb c+anδ|bmb c−b nN c|,

for m = 0, ..., L − 1 and n = 0, ...,MN − 1. This follows the same line of reasoning as theprevious formulas, k is replaced by mM +

⌊mb

⌋modL. Since g is periodic with period L , the

�modL� in the �rst part of line three is unnecessary. The second �modL� can be removed,since L

M = b, so M always divides L. The �nal �modM � can be removed since m < L = bM

and N < MN so∣∣⌊m

b

⌋−⌊nN

⌋∣∣ is always less than M .

The last line shows that the result is only non-zero if⌊mb

⌋=⌊nN

⌋. This is only the

case for elements in the b × N blocks on the diagonal, so it proves the structure Ddiag =

diag(W0, ...,WN−1) where Wk ∈ Cb,N .


Setting k =⌊mb

⌋=⌊nN

⌋shows that

(Wk)m,n =√Lg(m+kb)M+k+a(n+kN)

=√LgmM+k+an,

for m = 0, .., b − 1 and n = 0, ..., N − 1. Going from indices m,n running over the wholematrix to indices running in the sub-matrices, it is necessary to add the terms kb and kN ,but they can be removed because of the periodicity of g. �

The factorization from theorem 37 can be used to factor S into a block-diagonal matrix.

Theorem 38. Let S be the matrix of the frame operator S for a Gabor frame G(g, a, b) forCLp . Then

Sdiag = P∗M,LSPM,L

where Sdiag is a block-diagonal matrix

Sdiag = diag (B0, ...,BM−1)

with

Bk = WkW∗k

for m = 0, ..., b− 1, n = 0, ..., b− 1 and k = 0, ...,M − 1 and Wk de�ned as in theorem 37.

Proof. A rearrangement of the result from theorem 37 shows that

Dg = PM,LDdiagPN,MN (IN ⊗ FM )∗ .

Inserting this into the matrix expression for the frame operator (24)

S = DgD∗g

= PM,LDdiagPN,MN (IN ⊗ FM )∗ (PM,LDdiagPN,MN (IN ⊗ FM )∗)∗

= PM,LDdiagPN,MN (IN ⊗ F∗M ) (IN ⊗ FM )P∗N,MND∗diagP

∗M,L

= PM,LDdiagD∗diagP

∗M,L.

In line three (28) is used to put the transposition inside the Kronecker product IN ⊗ F∗M .Then (29), as well as the fact that FM is a unitary matrix, are used to remove the Fouriermatrices. Similarly the permutation matrices are also unitary and the pair PN,MNP

∗N,MN can

be removed.From this and (29) the result follows:

P∗M,LSPM,L = DdiagD∗diag = Sdiag

�

The factorization from theorem 37 provides a faster way of numerically computing theIDGT than explicitly calculating

f = Dgc.

Using theorem 37 this can be expanded to

f = PM,LDdiagPN,MN (IN ⊗ F∗M ) c.


This is more e�cient to calculate as the matrices can be applied in turn to the vector c:

c1 = (IN ⊗ F∗M ) c(33)

c2 = PN,MNc1(34)

c3 = Ddiagc2(35)

f = PM,Lc3(36)

Calculating c1 amounts to N small FFTs of length M of panels of c, c2 is just a reorderingof c1, c3 can be calculated by multiplying panels of c2 by the matrices Wk, and �nally f is areordering of c3. The total amount of �oating point operations needed is

O (NM log(M)) +O (MbN) = O (NM log(M)) +O (LN) .

The reordering from the permutation matrices have not been included, as the operationsneeded for this are memory moves and not �oating point operations. They amount to a costof O(MN).

A completely similar thing can be done for the DGT:

c = D∗gf.

Again using theorem 37, this expands to

c = (IN ⊗ FM )P∗N,MND∗γ,diagP

∗M,Lf.

This corresponds to everything done in the reverse order as it was done for the IDGT, andthe computational cost is exactly the same.

The computation of the canonical dual window can similar be done more e�ciently than

γ0 = S−1g.

Using theorem 38 and inserting

γ0 =(PM,LSdiagP

∗M,L

)−1g

= PM,L (Sdiag)−1P∗M,Lg.

Again it was used that permutation matrices are unitary matrices. Inverting a block-diagonalmatrix is done by inverting the blocks, and so γ0 can be calculated by

c1 = P∗M,Lg

c2 = diag(B−10 , ...,B−1M )c1

γ0 = PM,Lc2.

Inverting and applying the blocks can be done in various ways. The total cost is

O(Mb3) = O(Lb2).

Perspectives. The algorithms presented in this section speed up the computation of the DGTand the IDGT from O(L2) to O (NM log(M)) + O (LN). It depends on M , the number ofmodulations, which of the terms O (NM log(M)) and O (LN) will dominate. If the Gaborframe has many modulations then the �rst term O (NM log(M)) will dominate. If the densityis kept close to one, then this corresponds to a situation with many modulations and fewtranslations. For other choices of M and N the second term seems more likely to dominate.

Another aspect of these algorithms is that they require much less memory. The originalalgorithms c = D∗γΓ∗f , f = Dgc and γ = S−1g required that the matrices Dg, Dγ and Swere formed explicitly. This requires the storage of an L × L or L ×MN complex matrix.


In an easy-to-use programming language like Matlab, setting up the matrices Dg and S willactually take far longer time than computing the matrix product, so one could be tempted toclaim, that one of the main reason for developing the improved algorithms was to avoid thebig matrices.

The improved algorithms only require the storage of an b × N or an N × N real matrix.The matrices are real (if the window sequence is real) because the matrix Wk only consistsof elements of g.

The article [Stroh98] presents some improvements of these algorithms, especially the algo-rithm for �nding the canonical dual window. I choose to stop here, the presented algorithmsare fast enough for my use. The purpose for which I will use the algorithms, solving partialdi�erential equations, only requires a �xed Gabor frame. This means that the canonical dualwindow can be calculated once before the real computation starts, and then the DGT andthe IDGT are computed numerous times. This is the reason why I have not focused more ondeveloping algorithms for inverting S.

There exists other kind of algorithms for these problems than matrix factorization algo-rithms. Most notable are iterative methods and methods using the Zak-transform. Since Ifound the algorithms in [Stroh98] adequate for my needs, I have not pursued any of the otherkinds of algorithms.

The methods based on the Zak-transform avoids the use of the dual window altogether,and calculates the DGT directly. However, they only work for Gabor frames with a densityof 1 (which is then a Gabor basis), and they can be extended to Gabor frames with integeroversampling, that is a Gabor frame with density 1

p , p ∈ N. An example of the Zak-transform

method is in [Orr93]. An example of an conjugate-gradient and matrix factorization methodis in [Qiu98].

5. Gabor frames for L2(T).

5.1. Basic properties. To de�ne a Gabor frame for L2(T), the translation and modulationoperators will have to be de�ned once again.

De�nition 39. The translation Ty and the modulationMm operators for L2(T) are de�nedby

(Tyf) (x) = f(x+ y)

(Mmf) (x) = e−2πimxf(x)

for all m ∈ Z and y ∈ T.

The symbols for the translation and modulation operators are identical to those of de�nition23, but it should always be clear from the context which operator a symbol refers to.

The translation and modulation operators for L2(T) obeys similar relations as those for CLp .The modulation operator is not periodic in its parameter, only the translation operator is:

(Tyf) (x) = (Tymod 1f) (x) .

The inverse and adjoint operators are the opposite action of the operators:

(Ty)−1 f = (Ty)∗ f = T−yf(Mm)−1 f = (Mm)∗ f =M−mf.(37)

This also means that the translation and modulation operators are unitary operators on L2(T).


The commutation relation (similar to (12)) is slightly di�erent:

(MmTyf) (x) = e−2πimxf(x+ y)(38)

= e2πimye−2πim(x+y)f(x+ y)(39)

= e2πimy (TyMmf) (x) .(40)

The commutation relation can be used to commute pairs of the operators:

(MaTbMcTdf) (x) = e−2πibc (MaMcTbTdf) (x)

= e−2πibc (Ma+cTb+df) (x)

= e−2πibc (McMaTdTbf) (x)

= e2πi(ad−bc) (McTdMaTbf) (x)(41)

These operators are related to the corresponding operators for CLp as de�ned in de�nition23 by the following relation:

(42)(MmbT n

Ng)( k

L

)= (MmbTna {gk})k ,

with {gk}k=0,...,L−1 = g( kL) and Na = L.

Gabor frames for L2(T) can now be de�ned.

De�nition 40. A Gabor system in L2(T) with window g ∈ L2(T) is a family of functionsde�ned by:

G = G(g,1

N, b) =

{MmbT n

Ng}m∈Z,n=0,...,N−1

,

where b,N ∈ N.Each element in the system is called a Gabor atom:

Gm,n(x) =(MmbT n

Ng)

(x) = e−2πimbxg(x+n

N), x ∈ T.

For Gabor frames for L2(T) a canonical dual window function exists, just as the canonical,dual sequence for a Gabor frame for CLp :

Proposition 41. Let G(g, 1N , b), g ∈ L2(T), M, b ∈ N be a Gabor frame for L2(T). Then

there exists a function γ0 ∈ L2(T) such that H(γ0, 1N , b) is a dual frame of G.

Proof. The proof follows the exact same steps as the proof for proposition 26. The method isto show that both the translation and modulation operator commute with the frame operator.The notable di�erences are thatMmb is no longer periodic in m and that the sum over m inthe de�nition of the frame operator, corresponding to (14), is an in�nite sum.

The step corresponding to (17) is(MrbT s

N

)−1S(MrbT s

N

)f

=

N−1∑n=0

∑m∈Z

⟨f,M(m−r)bTn−s

Ng⟩M(m−r)bTn−s

Nag

=N−1∑n=0

∑m′∈Z

⟨f,Mm′bT n

Ng⟩Mm′bT n

Ng

= Sf.


A substitution of variables m′ = m− r and the fact that Tn−sN

is periodic with period N are

used when going from line two to line three.The rest of the proof follows exactly the proof for prop. 26. �

The following de�nitions and theorems are adaptations to L2(T) of similar statements in[Groch01].

The �rst statements establish that the pre-frame, analysis and frame operators are boundedwhen g ∈ L∞(T). They have been transferred from similar statements for L2(T) presented in[Groch01]. For the L2(T) theory the Wiener space W is used. A function is in the Wignerspace if it is locally in L∞ and globally in L1. Since decay properties are not meaningful forfunctions de�ned on T, the circle-equivalent of W is the space L∞(T).

Proposition 42. XXX Let g ∈ L∞(T) and let N, b ∈ N. Then

(43) D {cm,n} =N−1∑n=0

∑m∈Z

cm,nMmbT nNg

is a bounded operator from l2(Z× {0, ..., N − 1}) into L2(T) with operator norm

‖D‖op ≤ ‖g‖∞ .

Proof. Consider {cm,n} ∈ l2(Z × {0, ..., N − 1}) with �nite support, i.e. only �nitely manycm,n are non-zero. Then it is safe to set

f =N−1∑n=0

∑k∈Z

cm,nMmbT nNg,

and the norm of f can be easily bounded:

‖f‖22 =

∥∥∥∥∥N−1∑n=0

∑m∈Z

cm,nMmbT nNg

∥∥∥∥∥2

2

=

∫ 1

0

∣∣∣∣∣N−1∑n=0

∑m∈Z

cm,ne2πimbxg(x+

n

N)

∣∣∣∣∣2

dx

≤N−1∑n=0

∑m∈Z

∫ 1

0

∣∣∣cm,ne2πimbxg(x+n

N)∣∣∣2 dx

≤ ‖g‖2∞N−1∑n=0

∑m∈Z|cm,n|2 .

Since only �nitely many cm,n are non-zero, the summation overm is �nite, and it can be pulledoutside the integral. Since g ∈ L∞(T), it is bounded by ‖g‖∞ and similarly the exponentialfunction is bounded by one. This gives the result.

By the density principle [Groch01, A.1] the result extends to all {cm,n} ∈ l2(Z×{0, ..., N − 1})and the results is proven. �

Corollary 43. Let g ∈ L∞(T) and let N, b ∈ N. Then

Cf ={⟨f,MmbT n

Ng⟩}

m,n


is a bounded operator from L2(T) into l2(Z× {0, ..., N − 1}) with operator norm

‖C‖op ≤ ‖g‖∞ ,and

(44) Sf =

N−1∑n=0

∑m∈Z

⟨f,MmbT n

Ng⟩MmbN n

Ng

is a bounded operator from L2(T) into L2(T) with operator norm

‖S‖op ≤ ‖g‖∞ .

Proof. The operators D, C and S are the pre-frame-, analysis- and frame- operators for aGabor system G(g, 1

N , b). In [Groch01, prop. 5.1.1] it is shown that C is the adjoint operatorof D and so ‖D‖op = ‖C‖op = ‖S‖op and from this the result follows. �

De�nition 44. Let g ∈ L2(T) and let b,N ∈ N. Then the correlation functions of g arede�ned as

Gn(x) =N−1∑k=0

g(x+n

b+k

N)g(x+

k

N), x ∈ T.

for n = 0, ..., b− 1.

Lemma 45. Let g ∈ L2(T), b,N ∈ N and let Gn, n = 0, ..., b− 1 be the correlation functionsas de�ned in de�nition 44. Then Gn has the Fourier series

Gn(x) = N∑k∈Z

⟨g,M−kNTn

bg⟩e2πikNx, a.e. x ∈ [0;

1

N].

Proof. Let

hn(x) = g(x+n

b)g(x) =

(Tnbgg)

(x), x ∈ T.

The correlation function Gn are a periodization of hn with period 1N and hn ∈ L2([0; 1

N ]) since

g ∈ L2(T). Using lemma 9:

Gn(x) =N−1∑k=0

g(x+n

b+k

N)g(x+

k

N)

= N∑k∈Z

(hn

)kN

e2πikNx, a.e. x ∈ [0;1

N](45)

The kN 'th Fourier coe�cients of hn are:(hn

)kN

=

∫ 1

0g(x+

n

b)g(x)e−2πikNxdx

=⟨g,M−kNTn

bg⟩.

Inserting this in (45):

Gn(x) = N∑k∈Z

⟨g,M−kNTn

bg⟩e2πikNx, a.e. x ∈ [0;

1

N].

�


Theorem 46. Walnut's representation. Let g ∈ L∞(T) and let b,N ∈ N. Then theoperator S : L2(T)→ L2(T)

Sf =N−1∑n=0

∑m∈Z

⟨f,MmbT n

Ng⟩MmbT n

Ng

can be written using de�nition 44:

Sf =1

b

b−1∑k=0

(GkT k

bf).(46)

Proof. The fact that S is bounded on L2(T) follows from corollary 43.First an important identity for frequency-time shifts is needed:

⟨f,M−mbT n

Ng⟩

=

∫ 1

0f(x)g(x+

n

N)e−2πimbxdx

=

∫ 1

0

(f(x)g(x+

n

N))e−2πimbxdx

=(F(fT n

Ng))

mb.

Line two is recognized as a Fourier coe�cient, and from this the result follows.Consider Cgf , where

Cgf ={⟨f,M−mbT n

Ng⟩}

m∈Z,n=0,...,N−1.

Then{⟨f,M−mbT n

Ng⟩}

m∈Z,n=0,...,N−1∈ l2(Z× {0, ..., N − 1}) from corollary 43. Therefore,

the Fourier series mn(x)

(47) mn(x) =∑m∈Z

⟨f,M−mbT n

Ng⟩e2πimbx =

∑m∈Z

(F(fT n

Ng))

mbe2πimbx, a.e. x ∈ T.

are in L2(T) by Plancherel's theorem, theorem 8. Also mn(x) is periodic with period 1b since

e2πimbx is periodic with period 1b .

The Fourier series mn(x) can now be rewritten using lemma 9:

mn(x) =1

b

(b∑m∈Z

(F(fT n

Ng))

mbe2πimbx

)

=1

b

b−1∑k=0

(fT n

Ng)

(x+k

b)

=∑m∈Z

⟨f,M−mbT n

Ng⟩e2πimbx, a.e. x ∈ [0;

1

b].


The last line is (47) again. Substituting this into the de�nition S

Sf(x) =

N−1∑n=0

∑m∈Z

⟨f,MmbT n

Ng⟩MmbT n

Ng

=N−1∑n=0

(∑m∈Z

⟨f,MmbT n

Ng⟩e−2πimbx

)g(x+

n

N)

=N−1∑n=0

(1

b

b−1∑k=0

f(x+k

b)g(x+

k

b+n

N)

)g(x+

n

N), a.e. x ∈ T.

Since both sums are �nite they can be interchanged

Sf(x) =b−1∑k=0

(1

b

N−1∑n=0

g(x+k

b+n

N)g(x+

n

N)

)f(x+

k

b), a.e. x ∈ T,

or in short notation using de�nition 44 with γ = g:

Sf(x) =1

b

b−1∑k=0

Gk(x)f(x+k

b), a.e. x ∈ T.

�

Corollary 47. Let G(g, 1N , b), b,N ∈ N be a Gabor frame for L2(T) with a real window

function g ∈ L∞(T). Then the canonical dual window γ0 ∈ L2(T) is also a real function.

Proof. This follows from Walnut's representation. When g is real, the correlation functionsare also real, and so it follows from Walnut's representation that S is a real operator. Theinverse operator of a real operator is also real, and because

γ0 = S−1g

then γ0 is real. �

Remark 48. Theorem 46 and corollary 47 are the L2(T) equivalent of proposition 30.

5.2. Sampling. The following subsection is a conversion of results in [Jans97] to a periodicsetting. It will present two main results: That a Gabor frame from CLp can be obtained from

a Gabor frame for L2(T) by sampling, and that the canonical dual windows for these twoframes are closely related.

The following de�nitions will ensure that the process of sampling is well de�ned.

De�nition 49. Let f be a measurable function on T. Then x0 is a Lebesgue point of f if

limε→0

1

ε

∫ 12ε

− 12ε|f(x+ x0)− f(x0)| dx = 0.

With a little more care Lebesgue points can be de�ned for any measurable space.

Lemma 50. Let f ∈ L1(T). If f is continuous at a point x0, then x0 is a Lebesgue point off .


Proof. Since f is continuous at x0 then for all ε > 0 there exist a δ such that

|f(x+ x0)− f(x0)| < ε, ∀x ∈ [−1

2δ,

1

2δ].

Now

1

δ

∫ 12δ

− 12δ|f(x+ x0)− f(x0)| dx ≤ sup

x∈[− 12δ, 1

2δ]

|f(x+ x0)− f(x0)| < ε.

Here the integral is bounded by the maximum value. This shows that when ε → 0 then1δ

∫ 12δ

− 12δ|f(x+ x0)− f(x0)| dx→ 0 and this is exactly the de�nition of a Lebesgue point. �

Lemma 51. Let f ∈ L1(T). If x0 is a Lebesgue point of f then

f(x0) = limε→0

1

ε

∫ 12ε

− 12εf(x+ x0)dx.

Proof. Consider for ε > 0∣∣∣∣∣f(x0)−1

ε

∫ 12ε

− 12εf(x+ x0)dx

∣∣∣∣∣ =

∣∣∣∣∣1ε∫ 1

2ε

− 12εf(x0)dx−

1

ε

∫ 12ε

− 12εf(x+ x0)dx

∣∣∣∣∣≤ 1

ε

∫ 12ε

− 12ε|f(x0)− f(x+ x0)| dx.

Since x0 is a Lebesgue point then 1ε

∫ 12ε

− 12ε|f(x0)− f(x+ x0)| dx→ 0 as ε→ 0, and therefore

limε→0

∣∣∣∣∣f(x0)−1

ε

∫ 12ε

− 12εf(x+ x0)dx

∣∣∣∣∣ = 0,

and this proves the lemma. �

Proposition 52. Let f, g ∈ L2(T) such that f = g in L2-sense. If x0 is a Lebesgue point off and g then f(x0) = g(x0).

Proof. Since f = g in L2-sense, then f(x) = g(x) except on a set of measure zero, and so∫E f(x)dx =

∫E g(x)dx for any set E. Using lemma 51 then

f(x0) = limε→0

1

ε

∫ 12ε

− 12εf(x+ x0)dx = lim

ε→0

1

ε

∫ 12ε

− 12εg(x+ x0)dx = g(x0).

�

De�nition 53. Let f be a measurable function on T and let L ∈ N. Then f satis�es conditionR if:

(48) limε→0

L−1∑k=0

1

ε

∫ 12ε

− 12ε

∣∣∣∣f(x+k

L)− f(

k

L)

∣∣∣∣ dx = 0.

Remark 54. It shows that f satis�es condition R if and only if all the points x = kL , k =

0, ..., L − 1 are Lebesgue points of f . From lemma 50 it then follows, that a continuousfunction satis�es condition R, since all points of a continuous function are Lebesgue points.


With a condition to ensure that sampling is well de�ned, the �rst main theorem of thissection can be stated.

Theorem 55. Let M,N, a, b, L ∈ N such that Mb = Na = L. Let

G

(g,

1

N, b

), g ∈ L∞(T)

be a Gabor frame for L2(T) as de�ned in de�nition 40 with frame bounds 0 < A ≤ B < ∞,and let g satisfy condition R (de�nition 53). Let

H

({gk} , a,

b

L

), {gk} ∈ CLp

be a Gabor system in CLp as de�ned in de�nition 24 such that {gk} = g( kL).

Then H is a frame for CLp with frame bounds A and B.

Proof. Let 0 < ε < 1L and let

δε(x) =1

εχ]− 1

2ε; 1

2ε[(x)

δεk(x) =1

εχ]− 1

2ε; 1

2ε[(x+

k

L)

for k ∈ 0, ..., L− 1. Let {ck} , {dk} ∈ CLp and set

f ε =L−1∑r=0

crδεr

hε =L−1∑s=0

dsδεs.

Then f ε ∈ L2(T) since the sum over r is �nite and

‖f ε‖22 =

∫ 1

0

∣∣∣∣∣L−1∑r=0

crδεr

∣∣∣∣∣2

dx

=

L−1∑k=0

∫ 1

0|crδεr |

2 dx

=1

ε

L−1∑k=0

|cr|2 dx =1

ε‖c‖22 .(49)

Line two follows from the fact that the δεr have disjoint support. De�ne

Υm,n(ε) = ε∑l∈Z

⟨f ε,M−lL+mbT n

Ng⟩⟨M−lL+mbT n

Ng, hε

⟩,

with m = 0, ...,M − 1, n = 0, ..., N − 1 and ε ∈ [− 12L ; 1

2L ]. Since G is a frame then{⟨f ε,M−lL+mbT n

Ng⟩}

l∈ l2(Z) by the frame inequality. Then the series de�ning Υ is in

l1(Z) from Cauchy-Schwartz inequality. Therefore Υm,n(ε) is well-de�ned.


It follows by rearranging the summations that

(50)N−1∑n=0

M−1∑m=0

Υm,n(ε) = ε

N−1∑n=0

∑m∈Z

⟨f ε,MmbT n

Ng⟩⟨MmbT n

Ng, hε

⟩.

The function Υm,n can be written as

Υm,n(ε) = ε∑l∈Z

⟨L−1∑r=0

crδεr ,M−lL+mbT nN g

⟩⟨M−lL+mbT n

Ng,

L−1∑s=0

dsδεs

⟩

=

L−1∑r=0

L−1∑s=0

crdsε∑l∈Z

⟨δεr ,M−lL+mbT nN g

⟩⟨M−lL+mbT n

Ng, δεs

⟩.(51)

The summations over r and s can be pulled outside the inner product as they are both �nite.Consider the 1

L -periodic functions

αt(u) =L−1∑j=0

δεt (u−j

L)MmbT n

Ng(u− j

L), t = 0, ..., L− 1, n = 0, ..., N − 1.

Using lemma 9 then αt has the expansion

αt(u) = L∑k∈Z

(F(δεtMmbT n

Ng))

kLe2πikLx, a.e. x ∈ [0;

1

L],

with (F(δεtMmbT n

Ng))

kL=

∫ 1

0δεt (x)MmbT n

Ng(x)e−2πikLxdx

=⟨δεt ,M−kL+mbT nN g

⟩.

Since the complex exponentials Le2πikLx, k ∈ Z form an orthonormal basis for L2([− 12L ; 1

2L ])XXX then Parseval's equation can be used on the expansion of αt:

ε∑l∈Z


⟩⟨M−lL+mbT n

Ng, δεs

⟩= ε

∫ 12L

− 12L

αr(x)αs(x)dx

= ε

∫ 12L

− 12L

(∑k∈Z

δεr(x−k

L)MmbT n

Ng(x− k

L)

)(∑l∈Z

δεs(x−l

L)MmbT n

Ng(x− l

L)

)dx

=1

ε

∫ 12ε

− 12εMmbT n

Ng(x+

r

L)MmbT n

Ng(x+

s

L)dx

The summations over k and l have been removed because

δεr(u+k

L) =

{1ε if − k

L + rL ∈ [−1

ε ; 1ε ]

0 otherwise.

and since 0 < ε < 1L the terms in the sums are only non-zero when k = r and l = s. Similarly,

the range over which to integrate is restricted to the support of the indicator functions, whichthen can be removed.


Since rL and s

L are Lebesgue points of MmbT nNg and because g ∈ L∞(T) is essentially

bounded then

limε→0

1

ε

∫ 12ε

− 12εMmbT n

Ng(x+

r

L)MmbT n

Ng(x+

s

L)dx

= MmbT nNg(r

L)MmbT n

Ng(s

L).

This can be inserted into (51):

limε→0

Υm,n(ε) = limε→0

L−1∑r=0

L−1∑s=0

crdsε∑l∈Z


⟩⟨M−lL+mbT n

Ng, δεs

⟩

=

L−1∑r=0

L−1∑s=0

crdsMmbT nNg(r

L)MmbT n

Ng(s

L)

From this

limε→0

ε

N−1∑n=0

∑m∈Z

⟨f ε,MmbT n

Ng⟩L2(T)

⟨MmbT n

Ng, hε

⟩L2(T)

= limε→0

N−1∑n=0

M−1∑m=0

Υm,n(ε)

=N−1∑n=0

M−1∑m=0

limε→0

Υm,n(ε)(52)

=N−1∑n=0

M−1∑m=0

L−1∑r=0

L−1∑s=0

crdsMmbT nNg(r

L)MmbT n

Ng(s

L)

=

N−1∑n=0

M−1∑m=0

〈{ck} ,MmbTna {gk}〉CLp 〈MmbTna {gk} , {dk}〉CLp .(53)

Second line comes from (50). The translation and modulation operators in the last line are

the discrete versions. Since ‖f ε‖2 = 1ε ‖c‖

2 from (49) and

εA ‖f ε‖2 = A ‖c‖2 ≤ ε∑N−1

n=0

∑m∈Z

∣∣∣⟨f ε,MmbT nNg⟩∣∣∣2 ≤ εB ‖f ε‖2 = B ‖c‖2

from the frame condition (9), then

A ‖c‖2 ≤∑N−1

n=0

∑M−1m=0 |〈{ck} ,MmbTna {gk}〉|2 ≤ B ‖c‖2 ,

from (53). Since {ck} was arbitrarily chosen, the proof is complete. �

Remark 56. In (52) the limit is pulled inside the two summations. This is not possible in thesimilar proof for non-periodic frames, so the proof of [Jans97, prop. 2] deviates at this point.

To prove the relationship between the canonical duals of the frames G and H from the-orem 55 another representation of the Gabor frame operator is needed. This is Janssensrepresentation. To make sure it is convergent an additional condition is needed.


De�nition 57. Let g ∈ L2(T) and N, b ∈ N. The g satis�es condition A if:

(54)N−1∑n=0

∑m∈Z

∣∣∣⟨g,MmNTnbg⟩∣∣∣ <∞.

The corresponding condition A for L2(R) is delicate to satisfy [Groch01, p. 132], but forL2(T) the situation is simpler.

Proposition 58. Let g ∈ C1(T), N, b ∈ N. Then g satis�es condition A.

Proof. Consider

N−1∑n=0

∑m∈Z

∣∣∣⟨g,MmNTnbg⟩∣∣∣ =

N−1∑n=0

∑m∈Z

∣∣∣∣∫ 1

0g(x+

n

b)g(x+

n

b)e2πimNxdx

∣∣∣∣=

N−1∑n=0

∑m∈Z

∣∣∣∣(F (Tnb (gg)))−mN

∣∣∣∣The term in line two is bounded if

F(Tnb

(gg))∈ l1(Z), ∀n ∈ {0, ..., N − 1} .

From theorem 11 then this is satis�ed if Tnb

(gg) is continuous with a bounded and piecewise

monotone derivative. This is easily satis�ed if requiring g ∈ C1(T). �

The previous proposition is by no means meant to be exhaustive, it is only meant to providea simple, easy to check condition to be used for the construction of Gabor frames.

With this condition then Janssens representation can be de�ned.

Theorem 59. Janssen's representation. Let G(g, 1N , b), g ∈ L

2(T), N, b ∈ N be a Gabor

frame for L2(T) and let g satisfy condition A. Then the Gabor frame operator

Sf =

N−1∑n=0

∑m∈Z

⟨f,MmbT n

Ng⟩MmbT n

Ng

can be written as

(55) Sf =N

b

b−1∑s=0

∑r∈Z

⟨g,MrNT s

bg⟩MrNT s

bf

with unconditional convergence.

Proof. De�ne the operator S by

Sf =N

b

b−1∑s=0

∑r∈Z

⟨g,M−rNT s

bg⟩M−rNT s

bf.

The operator S is equal to (55), except for a trivial change in the sign of r.


It must be shown that S equals S. Since g satisfy condition A, the series de�ning S isabsolutely convergent and therefore independent of the order of summation.

(Sf)

(x) =1

b

b−1∑s=0

(N∑r∈Z

⟨g,M−rNT s

bg⟩e2πirNx

)(T sbf)

(x)

=1

b

b−1∑s=0

(GsT s

bf)

(x), a.e. x ∈ T.

The last line comes from the Fourier series expansion of the correlation functions, lemma 45.The term in the last line is Walnut's representation of the frame operator, theorem 46, andfrom this the result follows. �

Remark 60. The proof of the preceding theorem is similar to, but conceptually more di�cultthan the proof of theorem 25, because Janssens representation of the frame operator for aGabor frame for CLp does not require a condition A to converge.

Before the second main theorem can be proven, an additional lemma and proposition isneeded.

Lemma 61. Let f ∈ L2(T) satisfy condition R and assume that the Gabor system in L2(T):G(g, 1

N , b), g ∈ L∞(T), N, b ∈ N is a Bessel sequence with Bessel bound B. Let {ck,l} ∈l1(Z× 0, ..., N − 1.

Then

ϕ =N−1∑l=0

∑k∈Z

ck,lMkNT lbg

satis�es condition R and the Gabor system in L2(T): H(ϕ, 1N , b) is a Bessel sequence with

Bessel bound

B = BN−1∑l=0

∑k∈Z|ck,l|2 .

Proof. The function ϕ is well-de�ned since {ck,l} ∈ l1(Z× 0, ..., N − 1).


For f ∈ L2(T) there holds

N−1∑n=0

∑m∈Z

∣∣∣⟨f,MmbT nNϕ⟩∣∣∣2

=N−1∑n=0

∑m∈Z

∣∣∣∣∣⟨f,MmbT n

N

(N−1∑l=0

∑k∈Z

ck,lMkNT lbg

)⟩∣∣∣∣∣2

=

N−1∑n=0

∑m∈Z

∣∣∣∣∣N−1∑l=0

∑k∈Z

⟨f, ck,lMmbT n

NMkNT l

bg⟩∣∣∣∣∣

2

=

N−1∑n=0

∑m∈Z

∣∣∣∣∣N−1∑l=0

∑k∈Z

ck,le−2πi(ml−nk)e−2πikl

Nb e⟨M−kNT− l

bf,MmbT n

Ng⟩∣∣∣∣∣

2

≤

N−1∑l=0

∑k∈Z|ck,l|

(N−1∑n=0

∑m∈Z

∣∣∣⟨M−kNT− lbf,MmbT n

Ng⟩∣∣∣2)

12

2

≤

(N−1∑l=0

∑k∈Z|ck,l|

(B∥∥∥M−kNT− l

bf∥∥∥2) 1

2

)2

= B ‖f‖2(N−1∑l=0

∑k∈Z|ck,l|

)2

In the third line the summation can be pulled outside the inner product since ϕ is well-de�nedand the inner product is continuous. In the fourth line the commutation relations (41) and(40) and the adjoint operators of the translation and modulation operators (37) are used. Inthe �fth line the triangle inequality∥∥∥∥∥∥

∑k,l

cklψk,l

∥∥∥∥∥∥ ≤∑|ck,l| ‖ψk,l‖

for ψk,l ∈ l2(Z× 0, ..., N − 1) is used. In the last line the frame condition (9) is used and thefact that the translation and modulation operators are unitary operators on L2(T).

This calculation shows that H(ϕ, 1N , b) has a �nite upper frame bound B. To show that ϕ

satis�es condition R then consider:

N−1∑j=0

1

ε

∫ 12ε

12ε|ϕ(j + x)− ϕ(j)|2 dx

=

N−1∑j=0

1

ε

∫ 12ε

12ε

∣∣∣∣∣N−1∑l=0

∑k∈Z

ck,l

(MkNT l

bg(j + x)−MkNT l

bg(j)

)∣∣∣∣∣2

dx

≤

N−1∑l=0

∑k∈Z|ck,l|

N−1∑j=0

1

ε

∫ 12

12ε

∣∣∣MkNT lbg(j + x)−MkNT l

bg(j)

∣∣∣2 dx 1

2

2

.(56)


The last line follows from the triangle inequality for L2(0, ..., N − 1 × [−12ε;

12ε]). The part

inside the integral can be estimated∣∣∣MkNT lbg(j + x)−MkNT l

bg(j)

∣∣∣≤

∣∣∣∣e−2πixNg(j + x+l

b)− g(j +

l

b)

∣∣∣∣≤

∣∣∣∣(e−2πixN − 1)g(j + x+l

b)

∣∣∣∣+

∣∣∣∣g(j + x+l

b)− g(j +

l

b)

∣∣∣∣ .In the second line the term e−2πijN is pulled outside the absolute signs and bounded by one.

From this

N−1∑j=0

1

ε

∫ 12ε

12ε

∣∣∣MkNT lbg(j + x)−MkNT l

bg(j)

∣∣∣2 dx≤

∫ 12ε

12ε

∣∣e−2πixN − 1∣∣2

ε

N−1∑j=0

∣∣∣∣g(j + x+l

b)

∣∣∣∣2 dx+

N−1∑j=0

1

ε

∫ 12ε

12ε

∣∣∣∣g(j + x+l

b)− g(j +

l

b)

∣∣∣∣2 dx.(57)

Since|e−2πixN−1|2

ε → 0 as ε→ 0 and since∑N−1

j=0

∣∣g(j + x+ lb)∣∣2 ≤ N ‖g‖∞ it shows that the

term in line two goes to zero as ε→ 0. The term in line three goes to zero because g satis�escondition R. Both of the terms are bounded in k and l since g is essentially bounded. Since{ck,l} ∈ l1(Z× 0, ..., N − 1 then it follows from (56) that ϕ satis�es condition R. �

Proposition 62. Let G(g, 1N , b) and H(h, 1

N , b), g ∈ L∞(T), h ∈ L2(T), N, b ∈ N be Gabor

frames for L2(T). Assume that g satisfy condition R and condition A, de�nition 57. LetL = Ma = Nb with L,M, a ∈ N. Let Sg be the frame operator corresponding to G and let

S{g} be the frame operator corresponding to the Gabor frame for CLp : J({g( kL)

}k, a, bL).

Then Sh satis�es condition R, and the Gabor system in L2(T): K(Sgh, a, bL) is a Besselsequence, and

(Sgh)

(k

L

)=

1

LS{g}

{h(k

L)

}k

, ∀k ∈ {0, ..., L− 1} .

Proof. The claim that J is a Gabor frame for CLp follows from theorem 55.Since g satis�es condition A then

N−1∑n=0

∑m∈Z

∣∣∣⟨g,MmNTnbg⟩∣∣∣ <∞

which means that {cm,n} ={⟨g,MmNTn

bg⟩}

m,n∈ l1(Z × {0, ..., N − 1}. Using lemma 61

on h with the sequence {cm,n} shows that Sh satis�es condition R and that K is a Besselsequence.


Since g satis�es condition A, then S can be expressed using Janssens representation, andsince Sh satis�es condition R, then it can be sampled:

Sh(k

L) =

N

b

b−1∑n=0

∑m∈Z

⟨g,MmNTn

bg⟩MmNTn

bh(k

L)

=N

b

b−1∑n=0

a−1∑m=0

∑l∈Z

⟨g,MmN+lLTn

bg⟩MmN+lLTn

bh(k

L)

=N

b

b−1∑n=0

a−1∑m=0

∑l∈Z

⟨g,MmN+lLTn

bg⟩MmNTn

bh(k

L).(58)

In line two the summation over m is replaced by summations over m and l. The simpli�cationin the modulation operator in line three is possible because of the sampling.

Next consider the 1L periodic function ψ de�ned by

ψ(x) =L−1∑k=0

g(k

L+ x)g(

k

L+ x+

n

b)e−2πimN( k

L+x).

for m = 0, ...,M − 1 and n = 0, ..., N − 1 Then

1

ε

∫ 12ε

− 12ε|ψ(x)− ψ(0)| dx

≤ 1

ε

L−1∑k=0

∫ 12ε

− 12ε

∣∣∣∣g(k

L+ x)g(

k

L+ x+

n

b)e−2πimNx − g(

k

L)g(

k

L+n

b)

∣∣∣∣ dx≤ 1

ε

L−1∑k=0

∫ 12ε

− 12ε

∣∣∣∣g(k

L+ x)g(

k

L+ x+

n

b)

∣∣∣∣ ∣∣e−2πimNx − 1∣∣ dx

+1

ε

L−1∑k=0

∫ 12ε

− 12ε

∣∣∣∣g(k

L+ x)g(

k

L+ x+

n

b)− g(

k

L)g(

k

L+n

b)

∣∣∣∣ dx.The last two lines go to zero as ε → 0 because of the exact same reasoning as for (57). Thisshows that

limε→0

1

ε

∫ 12ε

− 12ε|ψ(x)− ψ(0)| dx = 0

which means that 0 is a Lebesgue point of ψ. A Fourier expansion for ψ can be found fromlemma 9:

ψ(x) = L∑l∈Z

(F(g(x)g(x+

n

b)e−2πimNx

))lLe2πilLx, a.e. x ∈ [0;

1

L],

where (F(g(x)g(x+

n

b)e−2πimNx

))lL

=

∫ 1

0g(x)g(x+

n

b)e−2πimNxe−2πilLxdx

=⟨g,M−mN−lLTn

bg⟩,


so

(59) ψ(x) = L∑l∈Z

⟨g,M−mN−lLTn

bg⟩e2πikLx, a.e. x ∈ [0;

1

L].

The right hand side of (59) is a continuous function, since{⟨g,M−mN−lLTn

bg⟩l

}∈ l1(Z),

[BuNe71, prop. 4.1.5]. Since the right hand side is continuous, it coincides with ψ at allLebesgue points. This follows from lemma 50 and proposition 52. Therefore it also coincidesfor x = 0. From this

ψ(0) = L∑l∈Z

⟨g,M−mN−sLTn

bg⟩

=L−1∑k=0

g(k

L)g(

k

L+n

b)e−2πimN

kL

=⟨{gk} ,M−mNTn

b{gk}

⟩.(60)

This can be inserted in (58):

Sh(k

L) =

N

b

b−1∑n=0

a−1∑m=0

∑l∈Z

⟨g,MmN+lLTn

bg⟩MmNTn

bh(k

L)

=1

L

(N

b

b−1∑n=0

a−1∑m=0

〈{gk} ,MmNTnM {gk}〉MmNTnM {hk}

)k

=1

L

(S{g} {hk}

)k.

Line two comes from inserting (60), and using (42) to express the term using the translationand modulation operators for CLp . The second line is Janssens representation of the frameoperator for J (25), and so the result follows. �

Remark 63. The factor 1L between Sh and

(S{g} {hk}

)kdoes not appear in the similar state-

ment for non-periodic spaces L2(R) and l2(Z) presented in [Jans97, proposition 3]. It is aresult of the need to normalize the sampled functions. If a function f ∈ L2(T) has ‖f‖ = 1then it does not hold that ‖{fk}‖ = 1. The same normalization constant appears in thede�nitions of the DFT and the IDFT, de�nition 1 and corollary 6 and in proposition 28.

To prove the main theorem and additional result is needed, which I will state as a conjecture.

Conjecture 64. Let G(g, 1N , b), g ∈ L

2(T), N, b ∈ N be a Gabor frame for L2(T) and assume

that g satis�es condition A. Then the canonical dual window γ0 also satis�es condition A.

For Gabor frames for G(g, α, β) for L2(Rd) with rational oversampling, that is αβ ∈ Q,the corresponding result is proven in [Jans95a]. The proof utilizes the Zak-transform and theso-called Zibulski-Zeevi representation of the Gabor frame operator. I have chosen not toinclude this theory in the thesis, but based on my experience of translating the other resultsin this thesis, I see no reason why it should not be possible. The Zak transform can be de�nedfor general locally compact Abelian groups, and therefore also for T. In the following theoremit is made an assumption that the canonical dual window should satisfy condition A, but it ismost likely a super�uous assumption.

Theorem 65. Let G, H, g be as de�ned in theorem 55. Furthermore assume that g satis�escondition A (de�nition 57). (Assume that canonical dual window for G: γ0 also satis�escondition A).


Then γ0 satis�es condition R, and{Lγ0k

}={Lγ0( kL)

}k=0,...,L−1 is the canonical dual se-

quence for H.

Proof. Let Sg be the frame operator of G and let S{g} be the frame operator of H.

Since γ0 is the canonical dual of g then γ0 = S−1g g. Since S−1g = Sγ0 is the frame operator

for the dual frame J(γ0, 1N , b) and since γ0 satis�es condition A, then S−1g can be written as

Janssen's representation (theorem 61). This gives the expression

γ0 = S−1g g = Sγf =N

b

b−1∑s=0

∑r∈Z

⟨γ0,MrNT s

bγ0⟩MrNT s

bg.

By lemma 61 then γ0 satis�es condition R. From g = Sgγ0 and from proposition 62 then{1

Lgk

}= S{g}

{γ0k},

which is the same as {Lγ0k

}=(S{g}

)−1 {gk} .This shows that

{Lγ0k

}is the canonical dual sequence of H. �

5.3. Bounds. This section is greatly inspired by the corresponding proofs for Fourier seriespresented in [Briggs95].

Theorem 66. XXX Let f ∈ Cp−1(T), p ≥ 1. Assume that f (p) is bounded and piecewisemonotone and let G

(g, 1

N , b), N, b ∈ N be a Gabor frame for L2(T) with window function

g ∈ L2(T) and canonical dual window function γ ∈ C∞(T). Then the Gabor coe�cients of f ,

{cm,n} = Cγfsatisfy

|cm,n| ≤C

|mb|p+1 , ∀m ∈ Z,∀n ∈ 0, ..., N − 1

where C is a constant independent of m and n.

Proof. The coe�cient cm,n can be integrated by parts

cm,n =⟨f, e−2πimbxγ

(x+

n

N

)⟩=

∫ 1

0f(x)e2πimbxγ

(x+

n

N

)dx

=

[e2πimbx

2πimbf(x)γ(x+

n

N)

]10

− 1

2πimb

∫ 1

0e2πimbx

(f(x)γ

(x+

n

N

))′dx

= − 1

2πimb

∫ 1

0e2πimbx

(f(x)γ

(x+

n

N

))′dx.

The �rst term in the third line goes away because of the periodicity of f , γ and the complexexponential. Repeating this process a total of p times gives

cm,n =

(− 1

2πimb

)p ∫ 1

0e2πimbx

(f(x)γ

(x+

n

N

))(p)dx.


According to [Briggs95, theorem 6.2], the integral can be bounded by∫ 1

0e2πimbx

(f(x)γ

(x+

n

N

))(p)dx ≤ Mn

mp

where M is a constant independent of m. This leads to the bound

|cm,n| ≤Mn

|2πimb|p+1 =Cn

|mb|p+1 ,

where C is a constant independent of m but dependent on n. Choosing the constant C asC = maxn=0,...,N−1Cn gives the �nal result. �

Proposition 67. Let f ∈ C(T) and assume that f (1) is piecewise monotone. Let G, H, g andγ0 be as de�ned in theorem 65, and assume that γ ∈ C∞(T). Let

{cm,n} ={⟨f,MmbT n

Nγ⟩}

m∈Z,n∈0,...,N−1

and

{cm,n} = {〈{fk} ,MmbNna {Lγk}〉}m=0,...,M−1,n=0,...,N−1 ,

where fk = f( kL) and γk = γ( kL) for all k = 0, ..., L− 1. Then

cr,s =1

L

∑j∈Z

cr+jM,s,

for all r = 0, ...,M − 1 and s = 0, ..., N − 1.

Proof. The expansion of f in G is

(61) f(x) =N−1∑n=0

∑m∈Z

cm,ne−2πimbxg(x+

n

N), x ∈ T,

with {cm,n} ∈ l1(Z× {0, ..., N − 1}, by theorem 66. This means that the series (61) is domi-nated by an absolutely convergent series

‖g‖∞N−1∑n=0

∑m∈Z|cm,n| ,

and is therefore uniformly and pointwise convergent to f(x). The expansion of f thereforeholds pointwise in the samples xk = k

L , k = 0, ..., L− 1:

fk = f(k

L) =

N−1∑n=0

∑m∈Z

cm,ne−2πimb k

L g(k

L+n

N)

=

N−1∑n=0

∑m∈Z

cm,ne−2πimb k

L gk+na(62)

for k = 0, .., L− 1. Notice that g( kL + nN ) = gk+na since L = Na.


Since the requirements of theorem 65 are satis�ed, then {Lγk} is the canonical dual windowof H. Analyzing {fk} in H gives:

cr,s =⟨{fk} ,

{e−2πirb

kLLγk+sa

}k

⟩= L

L−1∑k=0

fke2πirb k

L γk+sa

= L

L−1∑k=0

(N−1∑n=0

∑m∈Z

cm,ne−2πimb k

L gk+na

)e2πirb

kL γk+sa.

In the �nal line (62) was inserted. Since the sum over k is �nite, then it can be moved insidethe sum over m:

cr,s = LN−1∑n=0

∑m∈Z

cm,n

L−1∑k=0

e−2πimbkL gk+nae

2πirb kL γk+sa

=N−1∑n=0

∑m∈Z

cm,n 〈MmbTna {gk} ,MrbTsa {Lγk}〉 .

Together with the bi-orthogonality relations for Gabor frames for CLp , proposition 32, thisshows that the inner product in the �nal line is equal to one if r = mmodM and s = nmodNand zero otherwise. Using this, the result follows:

cr,s =∑j∈Z

cr+jM,s

�

Remark 68. It should be possible to loosen the assumptions in the previous theorem. Therequirements of γ0 ∈ L∞(T) and f ∈ C(T) with f (1) being piecewise monotone, are onlyneeded to ensure that the Gabor expansion of f is pointwise convergent. Hopefully this canbe ensured in a more clever way.

The Gabor coe�cients {cm,n} obtained from expanding f Gabor frame for L2(T) can beapproximated by expanding {fk} in a Gabor frame from CLp . The error introduced by doingthis is outlined in the following theorem.

Theorem 69. Let f ∈ Cp−1(T), p ≥ 1. Assume that f (p) is bounded and piecewise monotone.Let G, H, g, γ0 be as de�ned in theorem 65, and assume that γ0 ∈ L∞(T). Let

{cm,n}m∈Z,n=0,...,N−1 = Cγ0f

and

{cm,n}m=−M2+1,...,M

2, n=0,...,N−1 = C{Lγ0k} {fk} .

Then the error of using {cm,n} as an approximation of {cm,n} is

|cm,n − cm,n| ≤C

Lp+1, ∀m = −L

2+ 1, ...,

L

2, ∀n = 0, ..., N − 1

where C is a constant independent of k and L.


Proof. Using proposition 67:

|cm,n − cm,n| =

∣∣∣∣∣∣∑

j∈Z\{0}

cm+jM,n

∣∣∣∣∣∣ .The decay of the Gabor coe�cients of f expanded in G are known from theorem 66, and sothe sum can be bounded as∣∣∣∣∣∣

∑j∈Z\{0}

cm+jM,n

∣∣∣∣∣∣ =

∣∣∣∣∣∣∑j∈N

cm−jM,m + cm+jM,n

∣∣∣∣∣∣≤

∑j∈N

C1

|(m− jM) b|p+1 +C1

|(m+ jM) b|p+1

=C1

Lp

∑j∈N

1∣∣mM − j

∣∣p+1 +1∣∣m

M + j∣∣p+1

≤ C2

Lp+1,

for all m = −L2 +1, ..., L2 and n = 0, ..., N −1 and p ≥ 1. The constant C1 is the one occurring

in theorem 66 and C2 is independent of L. �

Lemma 70. Let g ∈ C(T) and N, b ∈ N. Assume that {cm,n} ∈ l1(Z). Then the summation

N−1∑n=0

∑m∈Z

cm,nMmbT nNg

is uniformly convergent.

Proof. This is proven by dominated convergence of an absolute convergent series. For all x ∈ Tit holds ∣∣∣cm,ne−2πimbxg(x+

n

N)∣∣∣ ≤ |cm,n| ∣∣∣∣sup

x∈Tg(x)

∣∣∣∣ .Since {cm,n} ∈ l1(Z) then it holds that

N−1∑n=0

∑m∈Z|cm,n|

∣∣∣∣supx∈T

g(x)

∣∣∣∣ =

∣∣∣∣supx∈T

g(x)

∣∣∣∣N−1∑n=0

∑m∈Z|cm,n| <∞.

Now the result follows from [Jens92, p. 176]. �

5.4. Existence. All previous statements have characterized Gabor frames, but so far nothinghas been mentioned about the existence of Gabor frames. Theorem 55 speci�es under whichconditions a Gabor frame for CLp can be obtained by sampling a Gabor frame for L2(T).

However, this still leaves the question about when a Gabor system in L2(T) is a frame.This subsection will show some results of when a Gabor system is a frame.For the following results let

Gj,l(x) =

N−1∑k=0

g(x+l

b+k

N)g(x+

j

b+k

N),


for j, l = 0, ..., b−1. These Gj,l are identical to the correlation functions (de�nition 44, denotedby only one index on G: Gn when j = 0: Gn(x) = G0,n(x). Observe that

(63) Gn(x+j

b) = Gj,j+n(x).

The following corollary of Walnut's representation, theorem 46, is needed to prove the nexttheorem.

Corollary 71. Let g ∈ L∞(T) and let b,N ∈ N. Then the operator S : L2(T)→ L2(T)

Sf =N−1∑n=0

∑m∈Z

⟨f,MmbT n

Ng⟩MmbT n

Ng

can be written as

(64) 〈Sf, h〉 =b−1∑l=0

b−1∑k=0

1

b

∫ 1b

0

(Gl,k(x)T k

bf(x)

)T lbh(x)dx

for all f, h ∈ L2(T).

Proof. The fact that S is bounded on L2(T) follows from corollary 43. Substituting Walnut'srepresentation (46) for Sf :

〈Sf, h〉 =1

b

∫ 1

0

(b−1∑k=0

Gk(x)f(x+k

b)

)h(x)dx

=1

b

∫ 1b

0

(b−1∑l=0

b−1∑k=0

Gk(x+l

b)f(x+

l + k

b)

)h(x+

l

b)dx

=b−1∑l=0

b−1∑k=0

1

b

∫ 1b

0

(Gl,l+k(x)T l+k

bf(x)

)T lbh(x)dx

=

b−1∑l=0

b−1∑k=0

1

b

∫ 1b

0

(Gl,k(x)T k

bf(x)

)T lbh(x)dx.

Line two comes from splitting the interval [0; 1] in b pieces. In line three the summations arerearranged, but this is �ne since they are �nite. The correlation function Gk is replaced as in(63). In line four k + l is replaced by k. This is valid since Gl,l+k and T l+k

bare both periodic

in l+ k with period b, and since the summation k runs over 0, .., b− 1. After the replacement,the terms are just added in di�erent order. �

Theorem 72. Let g ∈ L2(T) and let N, b ∈ N. De�ne for each x ∈ T the operator G(x) :Cb → Cb by

(65){G(x) {cl}l=0,...,b−1

}j=0,...,b−1

=

b−1∑l=0

Gj,l(x)cl = G(x)c,

where G(x) ∈ Cb,b is the matrix de�ned by

(66) G(x)j,l = Gj,l(x).


Then S as de�ned in theorem 46 is invertible on L2(T) if and only if

(67) G(x) ≥ aICb , a.e. x ∈ [0;1

N],

for some constant a > 0.

Proof. Since all Gj,l are periodic with period 1N , then G(x) = G(x+ k

N ), k = 0, ..., N + 1. Thismeans that (67) is equivalent to G(x) ≥ aICb , a.e. x ∈ T, because of the periodicity of theG(x).

Assume that

(68) G(x) ≥ aICb , a.e. x ∈ T.Then 〈G(x) {ck} , {ck}〉 ≥ a ‖{ck}‖ holds for any sequence {ck} ∈ Cb and so it also holds for

the sequence {cj} ={T jbf(x)

}jfor every f ∈ L∞(T). From this

b−1∑j=0

b−1∑l=0

Gj,l(x)T lbf(x)T j

bf(x) ≥ a

b−1∑j=0

∣∣∣T jbf(x)

∣∣∣2 .Integrating this over the interval [0; 1

b ], the left hand side is recognized as part of (64). Fromthis it follows

〈Sf, f〉 ≥ a

b

∫ 1b

0

b−1∑j=0

∣∣∣T jbf(x)

∣∣∣2 dx =a

b‖f‖2 .

This shows that S is a positive operator. From the spectral theorem it follows that S isinvertible.

To prove the �only if� part, assume (68) as before and furthermore assume that S is notinvertible. Then there exists a function f 6= 0 such that Sf = 0 and this function canbe approximated by a sequence of bounded functions. From this sequence a subsequencefn ∈ L∞(T) can be chosen such that

〈Sfn, fn〉 <1

n‖fn‖22 .

Using (64) then

0 <1

n‖fn‖22 − 〈Sfn, fn〉

=1

n

∫ 1

0|fn(x)|2 dx−

b−1∑l=0

b−1∑k=0

1

b

∫ 1b

0

(Gl,k(x)T k

bfn(x)

)T lbfn(x)dx

=

∫ 1b

0

1

n

b−1∑j=0

∣∣∣∣fn(x+j

b)

∣∣∣∣2 − 1

b

b−1∑l=0

b−1∑k=0

(Gl,k(x)T k

bfn(x)

)T lbfn(x)

dx.

The third line comes from writing the �rst integration in line two as an integration over [0; 1b ]

of a periodization of fn, and then collecting everything in one integration.Therefore there exists a set En ⊆ [0; 1

b ] with positive measure such that

1

n

b−1∑j=0

∣∣∣∣fn(x+j

b)

∣∣∣∣2 − 1

b

b−1∑l=0

b−1∑k=0

(Gl,k(x)T k

bfn(x)

)T lbfn(x) > 0, ∀x ∈ En.


Writing T jbf(x) as cj then it follows that

〈Gc, c〉 < b

n

b−1∑j=0

|cj |22 ,∀x ∈ En,

or di�erently phrased

G(x) <b

nI, ∀x ∈ En.

This contradicts (68) because En has positive measure, and therefore S must be invertible. �

Remark 73. If S, g, N, b are as de�ned in theorem 46 and S is a bounded and invertibleoperator then G(g, 1

N , b) is a frame for L2(T). This follows from standard frame theory, see[Chr02].

The operator G(x) is a positive, semi-de�nite operator:

Proposition 74. The operator G(x) as de�ned in (65) is a positive semi-de�nite operator inthe sense that

〈G(x)c, c〉 ≥ 0, ∀c ∈ Cb.

Proof. By direct calculation

〈G(x)c, c〉 =

⟨{b−1∑l=0

Gj,l(x)cl

}j

, {cj}

⟩

=b−1∑j=0

b−1∑l=0

Gj,l(x)clcj

=N−1∑k=0

b−1∑j=0

cjg(x+j

b+k

N)

(b−1∑l=0

clg(x+l

b+k

N)

)

=

N−1∑k=0

∣∣∣∣∣∣b−1∑j=0

cjg(x+j

b+k

N)

∣∣∣∣∣∣2

≥ 0.

In line three the expression (65) is inserted and the summations are rearranged. Since thesummations over j and l are independent, the expressions in the parenthesis in line three is anumber and its complex conjugate, and from this the result follows. �

Theorem 72 provides a way to test if a set g, b,N with g ∈ L∞(T), b,N ∈ N will generatea Gabor frame. How to use the test in practice will be discussed in section 7.

6. Interpolation and Numerical differentiation

This is my own collection of results from standard books on analysis, and for the secondpart, also from [Briggs95, Tref].

On a computer it is impossible to represent a general, continuous function f , as it wouldrequire in�nite storage. A common practice is instead to store a �nite number of samples off : {fn} at grid points {xn}, and use the samples to represent the function f . As there existin�nitely many continuous functions interpolating a given set of discrete points, then one must


choose a method to interpolate the points, and then hope that this interpolating function fthrough the samples {fn} closely match the original sampled function f .

The process of numerical di�erentiation consists of di�erentiating the interpolating functionf , and using this derivative f ′ as an approximation to the derivative f ′ of the original functionf .

All considered functions will belong to L2(T), and all grids will be equidistantly spaced onT denoted by the interval [0; 1]. For a grid {xn}, n = 0, ..., L− 1:

xk = kh =k

L, ∀k ∈ 0, ..., L− 1,

so h is

h =1

L.

The next section de�nes two ways of using polynomials to do numerical di�erentiation, andderives an error bound for one of them.

6.1. Polynomial di�erentiation. Polynomial di�erentiation could be done by interpolatingthe samples {fn} of the function f at the points {xn} by a polynomial, di�erentiating thispolynomial, and sampling the derivative. However, the method described here is even simpler,because no global interpolating function is constructed. Instead, for each fk = f(xk), a localinterpolating polynomial is found, and then this polynomial is di�erentiated and sampled atthe point xk.

De�nition 75. Let f be a continuous function f ∈ C(T) with hL = 1 and let {fk} = f(xk)with xk = k

L . The second-order centered3 �nite di�erence is given by{

f ′k

}=

{fk+1 − fk−1

2h

}k

.

Similarly the fourth-order centered �nite di�erence can be de�ned:

De�nition 76. Let f be a continuous, periodic function f ∈ C0(T) with period hL = 1 andlet {fk} = f(xk) with xk = hk. The fourth-order centered �nite di�erence is given by:{

f ′k

}=

{−fk+2 + 8fk+1 − 8fk−1 + fk−2

12h

}k

.

An error bound for the use of the second-order di�erence is:

Proposition 77. Let f ∈ C3(T) with hL = 1. Then the relative , global error of using thesecond-order centered �nite di�erence as de�ned in de�nition 75 to approximate the sampledderivative {f ′k} = f ′(xk) of f is: ∥∥∥{f ′n} − {f ′n}∥∥∥

‖{f ′n}‖≤ Ch2,

where C is a constant independent of h.

3�Centered� refers to the approximation of the derivative being calculated in the same place as one of the

grid points.


Proof. The Taylor expansion for fk+1 developed around the point xk is

fk+1 = fk + hdf

dx(xk) +

h2

2

d2f

dx2(xk) +

h3

6

d3f

dx3(ξ1),

where ξ1 ∈ [xk;xk+1]. Similarly for fk−1:

fk−1 = fk − hdf

dx(xk) +

h2

2

d2f

dx2(xk)−

h3

6

d3f

dx3(ξ2),

where ξ2 ∈ [xk−1;xk]. Constructing the second-order centered �nite di�erence from thesegives

fk+1 − fk−12h

=df

dx(xk) +

h2

12

(d3f

dx3(ξ1) +

d3f

dx3(ξ2)

).(69)

The term d3fdx3

(ξ1) + d3fdx3

(ξ2) can always be bounded from above by a constant

(70) C = 2

(supy∈[0;1]

∣∣∣∣d3fdx3 (y)

∣∣∣∣),

since f is assumed to be both continuous and periodic.

This shows, that the error∣∣∣f ′(xi)− f ′i∣∣∣ decreases as h2 if the function f is sampled on a

�ner grid.

The global, relative error‖{f ′n}−{f ′n}‖‖{f ′n}‖

also decreases as h2 since∥∥∥{f ′n} − {f ′n}∥∥∥‖{f ′n}‖

≤∥∥{h2C}

n

∥∥‖{C1C}n‖

=1

C1h2 = C2h

2

where C is as in (70) and C1 = 1C infy∈[0;1]

∣∣∣ dfdx(y)∣∣∣. Both C and C1 are independent of h. �

A similar analysis requiring f to be �ve times di�erentiable would show that the error offourth-order centered �nite di�erence decreases as h4.

The assumption of proposition 77 is that f is di�erentiable. This assumption excludes manyinteresting situations, where f might only be piece-wise continuous. I will not analyze thissituation just make the following remark:

Remark 78. The computation of the second order �nite di�erence is very localized. To computean approximation to the derivative at xk it only uses the values fk+1 and fk−1, so if just f isdi�erentiable in a region around xk−1, xk and xk+1, then it does not in�uence the computationthat f might have discontinuities elsewhere. Same argument goes for the fourth order �nitedi�erence, except that the region around xk where f must be di�erentiable is larger.

6.2. Spectral di�erentiation. Spectral di�erentiation is done by interpolating the samples{fk} of a function f with the interpolant

p(x) =1√L

L2∑

k=−L2+1

fke2πikx, x ∈ [0; 1].


This is the IDFT from (7) extended to a continuous function on T. It can easily be di�eren-tiated and sampled

{p′n}

=

2π√L

L2∑

k=−L2+1

ikfke2πi kn

L

n

.

Here the di�erent terms in the summation has been straightforward di�erentiated. This is notquite the way do it - the problem is the modes with highest and lowest wavenumber: θL

2,L and

θ−L2,L. These modes are equal, and could both equally well be included in the summation.

Sampling θL2,L = θ−L

2,L on the grid gives

e±2πin2 =

{1 if n is even−1 if n is odd,

a real sequence, as shown on �gure 2. Therefore the derivative of the function represented by

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

θ6,12

Figure 2. The mode√Lθ6,12 and it's interpolant cos(2πi x12).

this sequence should be real and zero at the grid points, and not a complex exponential. Thislead to the de�nition:

De�nition 79. Let f ∈ C(T). The spectral derivative of f is de�ned by

(71){f ′n

}=

2π√L

L2−1∑

k=−L2+1

ikfke2πi kn

L

n

.

The L2 -term has been removed, because it should not contribute.

To analyze the validity and accuracy of de�nition 79, f and f ′ will be expanded in Fourierseries.

Now the accuracy of the spectral derivative can be stated.

Theorem 80. Let f ∈ Cp−1(T). Assume that f (p) is bounded and piecewise monotone. Then

the global error of using the spectral derivative{f ′k

}as an approximation to the sampled

derivative {f ′k} is ∥∥∥{f ′k}− {f ′k}∥∥∥ ≤ C 1

Lp−1,

where L is the number of samples and C is a constant independent of L.


Proof. The Fourier expansions of f are:

f(x) =∑k∈Z

cke2πikx,

where

ck =

∫ 1

0f(x)e−2πixdx, ∀k ∈ Z.

Similarly then f ′ can be expanded as

(72) f ′(x) =∑k∈Z

dke2πikx,

where

dk =

∫ 1

0f ′(x)e−2πixdx, ∀k ∈ Z.

Using partial integration on dk:

dk =[f(x)e−2πikx

]10

+ 2πik

∫ 1

0f(x)e−2πikxdx

= 2πikck,(73)

for all k ∈ Z. This shows that the Fourier series for f ′: {dk} can be found from the Fourier

series for f , {ck}. Theorem 12 and (73) shows that an approximation{dk

}to the Fourier

series {dk} for the derivative of f can by calculated from the DFT of f :

dk = 2πik1√Lfk.

Inserting this in (72) and truncating the summation and sampling at the grid point gives theexpression

{f ′n

}=

1√L

L2−1∑

k=−L2+1

2πikfke2πik n

L

n∈Z

.

This is the spectral derivative as de�ned in de�nition 79.The error in using the spectral derivative as an approximation to the sampled derivative is

the residual sequence {rn} ={f ′n − f ′n

}. This can be analyzed using (72) and sampled on


the grid:

{rn} ={f ′n − f ′n

}=

{∑k∈Z

dke2πik n

L

}n

−

1√L

L2−1∑

k=−L2+1

2πikfke2πik n

L

n

=

L2−1∑

k=−L2+1

(dk −1√L

2πikfk)e2πik n

L

n

+

∑

k∈Z\{−L2 +1,...,L2−1}

dke2πik n

L

n

=

{∑k∈Z

rke2πik n

L

}n

=

∑j∈Z

L−1∑k=0

rk+jLe2πi(k+jL) n

L

n

=

∑j∈Z

L−1∑k=0

rk+jLe2πik n

L

n

,

where

rk =

{dk − 2πik 1√

Lf = 2πik(ck − 1√

Lfk) if k ∈

{−L

2 + 1, ..., L2 − 1}

dk otherwise, ∀k ∈ Z.

Since the exponential function is periodic, the last term can be simpli�ed.The terms |rk| can be estimated using theorem 11 and theorem 12:

|rk| =

{|k| C

Lp+1 if k ∈{−L

2 + 1, ..., L2 − 1}

C|k|p+1 otherwise , ∀k ∈ Z,

where the constant C is the largest of the constants in the two theorems and the factor 2π isincluded. Notice that theorem 66 is not directly applicable to dk − 2πik 1√

Lfk but instead to

2πik(ck − 1√Lfk), and this makes the term |k| enter the bound.


Since the modes form an orthonormal basis for CLp then Parseval's equation and the estimatefor |rk| can be used to calculate ‖{rk}‖2:

‖{rk}‖22 = L

∥∥∥∥∥∥∑j∈Z

L−1∑k=0

rk+jL1√Le2πik

nL

∥∥∥∥∥∥2

2

≤ L∑j∈Z

L−1∑k=0

|rk+jL|2

= L∑k∈Z|rk|2

= CL

∣∣∣∣ 1

Lp+1

∣∣∣∣2 ∑k∈{−L2 +1,...,L

2−1}|k|2 +

∑k∈Z\{−L2 +1,...,L

2−1}

∣∣∣∣ 1

|k|p+1

∣∣∣∣2

≤ C1L

(1

L2p+2L3 +

1

L2p+2

)≤ C2

1

L2p−2 .

The constants C1and C2 are independent of L. This gives the result:

‖{rk}‖ ≤ C21

Lp−1.

�

From this it can be seen that the error of the spectral di�erentiation decays faster than anypolynomial when f is smooth. When f is not smooth the rate of convergence is limited byhow many continuous derivatives f have. Further work is done in [Tref, theorem 4], where theresult is improved for analytical and band-limited functions.

To compare with remark 78 it is clear that computing the spectral derivative at a point xkinvolves all samples of a function f . Therefore if f has a discontinuity, then this discontinuitya�ects the computation of all f ′k, and not just those f ′k close to the discontinuity.

6.3. The Gabor derivative. Gabor di�erentiation is done in the same manner as the spectraldi�erentiation: By interpolating the samples {fk} of a function f with an interpolant

f(x) =

N−1∑n=0

M2−1∑

m=−M2+1

cm,ne−2πimbxg(x+

n

N), x ∈ T,

which is the IDGT from (18) extended to a continuous function on T and cm,n are the corre-sponding Gabor coe�cients. Di�erentiating this yields:

(74) f ′(x) =

N−1∑n=0

M2−1∑

m=−M2+1

−2πimbcm,ne−2πimbxg(x+

n

N) + cm,ne

−2πimbxg′(x+n

N)

Sampling this leads to the de�nition:


De�nition 81. Let f ∈ C(T) and H({gk} , a, b) be a Gabor system for CLp with canonical

dual sequence{γ0k}. The Gabor derivative of f is de�ned by

{f ′k

}=

N−1∑n=0

M2−1∑

m=−M2+1

(−2πimbcm,ne

−2πimMkgk+na + cm,ne

−2πimMkg′k+na

)k

,

where

cm,n =⟨{f ′k}k,MmbTna

{γ0k}⟩

are the coe�cients of the expansion of {f ′k} in H and fk = f( kL), k = 0, ..., L− 1.

Lemma 82. Let f ∈ C1(T) and assume that f (2) is piecewise monotone. let G(g, 1N , b),

g ∈ C1(T). Assume that the canonical dual window of G: γ0 ∈ L∞(T). Then

(75) f ′(x) =

N−1∑n=0

∑m∈Z

(−2πimbcm,ne

−2πimbxg(x+n

N) + cm,ne

−2πimbxg′(x+n

N)),

where

cm,n =⟨f,MmbT n

Nγ0⟩.

Proof. Since G is a Gabor frame then f has the expansion

(76) f(x) =

N−1∑n=0

∑m∈Z

cm,ne−2πimbxg(x+

n

N).

A standard result from analysis [Jens92, prop. 23.3] is that if (76) is pointwise convergent andif the right hand side of (75) is uniformly convergent, the right hand side of (75) converges to

f ′(x) for all x ∈ T. Since f ∈ C1(T) and f (2) is piecewise monotone then from theorem 66 thedecay of cm,n is |cm,n| < C

|mb|3 , and so {cm,n} ∈ l1(Z× {0, ..., N − 1}). Lemma 70 then shows

that (76) is uniformly convergent, and therefore also pointwise convergent.The right hand side of (75) can be written as

(77)N−1∑n=0

∑m∈Z−2πimbcm,ne

−2πimbxg(x+n

N) +

N−1∑n=0

∑m∈Z

cm,ne−2πimbxg′(x+

n

N).

The sequence {−2πimbcm,n} ∈ l1(Z × {0, ..., N − 1}) since |−2πimbcm,n| < C|mb|2 and since

g′ ∈ C(T), then lemma 70 is satis�ed for both summations in (77) and so the right hand sideof (75) is uniformly convergent and converges to f ′(x). �

Theorem 83. Let f ∈ Cp−1(T), p ≥ 2. Assume that f (p) is piecewise monotone. Let G, H,g, γ0 be as de�ned in theorem 65. Assume that g ∈ C1(T), and that γ0 ∈ C∞(T). Then the

global error in using the Gabor derivative{f ′k

}as de�ned in de�nition 81 as an approximation

to {f ′k} is

(78)∥∥∥{f ′k}− {f ′k}∥∥∥ ≤ C 1√

L

MN

Lp−1,

where C is a constant independent of L,M and N .


Proof. Consider the expansion of f in G:

f(x) =N−1∑n=0

∑m∈Z

cm,ne−2πimbxg(x+

n

N),

where

cm,n =⟨f,MmbT n

Nγ0⟩,

and

cm,n =⟨{fk} ,MmbT n

N

{Lγ0k

}⟩Using lemma 82 then f ′(x) can be written as:

f ′(x) =N−1∑n=0

∑m∈Z

(−2πimbcm,ne

−2πimbxg(x+n

N) + cm,ne

−2πimbxg′(x+n

N)).

The residual sequence {rk} ={f ′( kL)

}−{f ′k

}where

{f ′k

}is the Gabor derivative from

de�nition 81) can now be calculated

rk = f ′(k

L)− f ′k

=

N−1∑n=0

∑m∈Z

(−2πimbcm,ne

−2πimb kL g(

k

L+n

N) + cm,ne

−2πimb kL g′(

k

L+n

N)

)

−N−1∑n=0

M2−1∑

m=−M2+1

(−2πimbcm,ne

−2πimb kL gk+na + cm,ne

−2πimb kL g′k+na

)

=

N−1∑n=0

∑j∈Z

M−1∑m=0

−2πimbsm+jM,ne−2πi(m+jM)b k

L gk+na +

N−1∑n=0

∑j∈Z

M−1∑m=0

sm+jM,ne−2πi(m+jM)b k

L g′k+na

=∑j∈ZD{g} {−2πi(m+ jM)bsm+jM,n}m=0,...,M−1,n +

∑j∈ZD{g′} {sm+jM,n}m=0,...,M−1,n ,

where

sm,n =

{cm,n − cm,n if m ∈

{−M

2 + 1, ..., M2 − 1}

cm,n otherwise, m ∈ Z, n = 0, ..., N − 1,

and {sm,n} ∈ l2(Z × {0, ..., N − 1}) because {cm,n} ∈ l2(Z × {0, ..., N − 1}). In line four thesummations are rearranged, and because the exponential functions are periodic, then line �vefollows. The operator D{g} is the pre-frame operator of the frame H and D{g′} is an operatoras de�ned in proposition 28.

From proposition 28, the operator norm of the two pre-frame operators are bounded byC1 =

√LMN maxk |gk| and C2 =

√LMN maxk |g′k| respectively. Using this


‖{rk}‖22 ≤∑j∈Z

∥∥D{g} {−2πi(m+ jM)bsm+jM,n}∥∥22

+∑j∈Z

∥∥D{g′} {sm+jM,n}∥∥

≤ 4π2C21LMN

∑j∈Z

N−1∑n=0

M−1∑m=0

|(m+ jM)bsm+jM,n|2 + C21LMN

∑j∈Z

N−1∑n=0

M−1∑m=0

|sm+jM,n|2

= LMNN−1∑n=0

∑m∈Z

(4π2Bm2b2 + C2) |sm,n|2

≤ LMN2

M2−1∑

m=−M2+1

(4π2Bm2b2 + C2)C1

L2p+2+

∑m∈Z\{−M2 +1,...,M

2−1}

C2(4π2Bm2b2 + C2)

|mb|2p+2

≤ LMN2

(4π2

C3

L2p+2ML2 +

C4

L2p

)≤ C

M2N2

L

1

L2p−2

In line three the summations over m and j are collapsed to one summation. In line �ve sm,n isexpanded and replaced by the bounds from theorem 69 and theorem 66, and the summationover n is removed because the bounds are independent of n. The �rst term in line �ve comesfrom b2

∑mm

2 ≤ CM3b2 = cML2. All constants are independent of M ,N and L. �

Perspectives. In the discussion so far it has been assumed that a known function f is sampled,and that the samples then are used to construct an approximation to the derivative of f . Thereis little point in doing this if f is already known analytically. Then it would probably be mucheasier and more accurate to just di�erentiate f analytically.

On the other hand, if f is unknown, and only samples of f are known, there exists in�nitelymany functions interpolating the samples, so asking for the accuracy of the method makelittle sense: It might just as well be a cubic spline interpolating the samples as it might bea superposition of sine and cosine functions. In the �rst case, a fourth order �nite di�erencewould give the exact result, and in the second case a spectral method would give the exactresult.

When solving partial di�erential equations the situation is a mixture: The function f mightbe an analytically known initial condition to a partial di�erential equation

Lu = g

where

u(x, 0) = f(x),

but as time evolves then u(x, t) for t �xed is no longer known analytically. In this case theerror bounds make perfect sense. They allow for an estimate of the error

‖{u(xk, t)}k − {uk(t)}k‖

where uk(t) is the numerical approximation to u(xk, t).


7. Numerical experiments.

7.1. Dual windows. This section will show graphs and results for Gabor frames for L2(T).All these results have been numerically computed by sampling a Gabor frame for L2(T) toobtain a Gabor frame for CLp . This process works because of theorem 55 and theorem 65 aresatis�ed, so the assumptions of these theorems must also be satis�ed.

To construct Gabor frames for L2(T) the following class of window functions will be used

PϕW (x) =∑n∈Z

ϕW (x+ n) = 214

∑n∈Z

e−Wπ(x+n)2 , x ∈ T

These are the periodized Gaussian functions. Since the Gaussians are in the Schwartz classof smooth, faster than polynomial decaying functions, the Gaussians are also in L1(R), andtherefore their periodizations are well de�ned.

Lemma 84. The periodized Gaussians are smooth, PϕW ∈ C∞(T).

Proof. The summation de�ning the periodized Gaussian can be bounded by∑n∈Z

ϕW (x+ n) =∑

n∈N∪{0}

ϕW (x+ n) +∑n∈N

ϕW (x− n)

≤∑n∈N

ϕW (n) +∑n∈N

ϕW (−n+ 1).

In line two the summation is split in two, and each interval is bounded by its largest element.This shows that the summation can be bounded by an absolutely convergent series, and

the series is therefore uniformly convergent. Di�erentiating each term in the summation willmultiply each term by a factor −2Wπ(x+ n), but this is just a polynomial, so the new seriescan be bounded by an absolutely convergent series as well:∑

n∈Z−2πW (x+ n)ϕW (x+ n) ≤

∑n∈N

2πWnϕW (n) +∑n∈N

2πWnϕW (−n+ 1).

This shows that the series is also absolutely convergent, and therefore converges to the deriv-ative of PϕW .

This argument was enough to prove that PϕW is di�erentiable. Repeating it on the deriva-tives of PϕW shows that PϕW is smooth. �

Since the periodized Gaussians are continuous, they satisfy condition R (remark 54) andcondition A (proposition 58). This means that the periodized Gaussians satis�es all therequirements of theorem 65, except from the requirement that the canonical dual windowhave to satisfy condition A. I choose to believe in conjecture 64, since I cannot verify thecondition.4

The parameter W controls the �width� of the function. If W is close to zero the function isalmost constant, and if W is large the function has one large spike. Examples of periodizedGaussians of di�erent width can be seen on �gure 3.

To compute the function numerically, it is only necessary to do enough iterations to makesure that e−Wπ(x+n) has decayed to the machine precision.

Some examples of periodized Gaussians and their dual window can be seen on �gure 3. The

4Looking at the calculated canonical dual windows many of them look di�erentiable, so by proposition 58they would satisfy condition A. The problem is that they cannot be calculated without using theorem 65, sothis is not the way to solve the problem.


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600

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Figure 3. Comparison of canonical dual windows for three di�erent windowfunctions. The windows are on the right hand side and their correspondingcanonical dual windows are on the left hand side. The Gabor frames has theparameters N = 12 and b = 10. The three windows are PϕW with values ofW of 4, 10 and 30.

�gure shows three sets of windows and dual windows. The parameters N and b have �xedvalues, and the windows and dual windows are shown for three di�erent values of W .

The �rst row shows a very wide window with its dual, which has big, regular oscillationsspread out across the whole interval. The dual window in the middle is smooth, quite narrowand oscillates a little bit, but the oscillations die out quickly. The last row shows a narrowwindow. The dual window is narrow as well, and have no oscillations, but instead it has twosharp spikes. From the graph it can not be seen if the window is at all di�erentiable.

It has not yet be veri�ed that the windows and parameters shown on �gure 3 actuallygenerate Gabor frames. Since the periodized Gaussians are bounded, the operator S from (44)is bounded by corollary 43. To show that S is also invertible and prove that G(PϕW , 1

N , b) is

a Gabor frame for L2(T), the test from theorem 72 can be done with the parameters for eachplot.

The test from theorem 72 requires that the minimum eigenvalue of G(x) is found over allx ∈ [0; 1

N ]. As a start this can be done by doing a �ne grained sampling of the interval [0; 1N ],

and plotting the result. This is done on �gure 4. Three of the four plot corresponds to thewindows and dual windows on �gure 3. The �gure shows that the minimum eigenvalue isnon-zero for all four plots.


0.001

0.01

0.1

1

10

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 4

0.001

0.01

0.1

1

10

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 4

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 10

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 10

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 11

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 11

0.0001

0.001

0.01

0.1

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 30

0.0001

0.001

0.01

0.1

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

N = 12 b = 10 W = 30

Figure 4. The �gure shows all the eigenvalues of the matrices G(x) over therange x ∈ [0; 1

N ]. The window functions used to generate G(x) are periodizedGaussians of width W .

Since the eigenvalues have only been calculated for certain samples of x ∈ [0; 1N ], then it

could be possible that G(x) for some x not among the samples had a zero eigenvalue. Judgingfrom the shapes of the curves of the eigenvalues it seems highly unlikely. The curves are �at(to within the margin of the machine accuracy) or have regular oscillations. WhenW is small,corresponding to a wide window, the eigenvalues are constant over all x, and as the width ofthe pulse decreases the eigenvalues start to oscillate.

Looking at �gure 4 it seems like the eigenvalues have a periodic behavior over x ∈ [0; 1N ]. I

have not tried to prove this mathematically, but all the plots I have looked at show the samestructure. By counting the periods over a range of parameters N, b and window functions g,I have found that there are always b

gcd(b,N) periods on one interval [0; 1N ], where gcd(b,N)

denotes the greatest common divisor of b and N . Another way of saying this is that there islcm(b,N) periods on the interval [0; 1], where lcm(b,N) is the least common multiple of Nand b.

The �sampling� approach for �nding the minimum eigenvalue is not adequate, because thereis no guarantee that the minimum eigenvalue has actually been found. Two possible solutionsare the following:

• Use a numerical method to �nd the minimum eigenvalue. Since a matrix G(x) can beconstructed for any x, it is a simple matter of using a numerical method to �nd a localminimum. If the behavior of the eigenvalues are indeed periodic, then considering an


interval x ∈ [0; gcd(N,b)Nb ] would su�ce, and a local minimum on this interval would

most likely be a global minimum.• Assume that g must satisfy certain nice properties: g should be continuous, symmetricaround x = 0 and monotone on the interval [0; 1

2 ]. Combined with the hypothesis thatthe eigenvalues are periodic, it would probably be enough to show, that the minimum

eigenvalue for all G(x), x ∈ [0; 1N ] is found for G(x) with x = 0 or x = 1

2gcd(N,b)Nb .

This is illustrated on �gure 4 by the crosses on the lines. They are all placed in points

corresponding to x = 0 or x = 12gcd(N,b)Nb .

In this thesis I have just calculated the minimum eigenvalue of G(x) over a �ne grained sam-pling of the interval [0; 1

N ], and made sure that the points include the points just mentioned.This should produce the correct result.

Figure 4 show that the minimum eigenvalue is highest for the plots with W = 10 andW = 11 and considerably lower for the other two. To study this e�ect �gure 5 shows theminimum eigenvalue of a range of W .

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

1 10 100

W

Minimum eigenvalue.

N=12 b=10N=24 b=20N=12 b=5

N=24 b=10N=8 b=6

Figure 5. Plot of the minimum eigenvalue of the matrix G for di�erent sets ofparameters N and b over a range of window functions. The window functionsare all periodized Gaussians PϕW and so the eigenvalues are plotted againstthe value W . Note that the graphs have been calculated over all displayedvalues of W , but only the non-zero, non-negative results have been displayed.

Figure 5 shows results from �ve tests. The parameters N and b are kept �xed, and thewidth of the window W is varied. For each set N, b,W the minimum eigenvalue of G(x) iscomputed and plotted. The two �rst curves represent frames with little redundancy b

N = 0.83

and the next two represent frames with the double redundancy bN = 0.42.

All �ve curves have been calculated for values of W in the interval [0.3; 120], but some ofthe results turn out negative and close to the machine precision, so the graphs do not extendover the whole axis. However, from proposition 74 G cannot have negative eigenvalues, so the


negative results indicates numerical errors, in this case an eigenvalue that is zero within themachine precision.

I cannot tell whether the matrix G(x) really has a zero eigenvalue, or whether the curveswould continue toward zero without hitting it if the computations could be done in in�niteprecision. Numerically however, it does not make a di�erence. If G(x) has an eigenvalue thatis numerically zero, then S cannot be numerically inverted and the dual window cannot becalculated.

The regions where the graphs are not displayed correspond to intervals of W where theperiodized Gaussian does not generate a frame for that particular set N, b. The regions arevery unstable. The slightest deviation from the shape of the periodized Gaussian, like failingto compute the periodization to a su�ciently high precision, will result in the minimumeigenvalue growing several orders of magnitude. Since I am not acquainted with perturbationtheory for Gabor frames, I will not pursue the interesting question about what happens inthese regions.

All �ve graphs on �gure 5 rise and fall in the same manner. The three windows on �gure3 correspond to points on the rise, straight in the middle and on the fall of the graphs. Thethree dual windows are quite representative for the di�erent varieties of dual windows for theperiodized Gaussians. Compare with the two top dual windows in the right column of �gure6. The top one represents a point on the fall of a graph, and the middle one represent a point

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Window function.

-5

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20

25

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N = 6 b = 5

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N = 12 b = 10

-50-40-30-20-10

0102030405060

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

N = 24 b = 20

-0.50

0.51

1.52

2.53

3.54

4.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

N = 12 b = 5

-10123456789

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

N = 24 b = 10

Figure 6. The �gure shows a window function and the corresponding canoni-cal dual window function for di�erent values of N and b. The window functionis the periodized Gaussian PϕW with W = 10.


on the rise of a graph. The have di�erent width and decay properties, but the overall shapeis the same.

The rise seems to be dictated by the parameter b and the fall by the parameter N . I haveno explanation why the rise seems to be related to b, but the fall of the graphs are quite easyto explain: When W is big compared to N , the translations of g do not seem to cover thewhole axis, see �gure 7.

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Figure 7. The �gure shows three translations of the windows and dual win-dows on �gure 3.

The �gure shows how the translations cover the axis. When the window is wide, there ismuch overlap as seen in the �rst row. When the window is small, as seen in the last row, thereseems to be almost no overlap, and the high peaks of the dual window occurs exactly in thearea where the translations of the window functions do not seem to overlap.

Since the Gaussians decay exponentially, they never decay fully to zero. Therefore thewindow functions are supported on all of T. Therefore the translations overlap no matterhow small the width of the function is, and the dual window grows as the size of the overlapshrinks.

Two of the graphs on �gure 5 correspond to highly redundant frames with densities bN =

0.43. Two dual windows corresponding to points on these graphs can be seen in the last rowon �gure 6. They seem much softer than the other dual windows: They have no oscillationsor no sharp peaks. This could be a bene�t for applications.

The graphs on �gure 5 seem identical for identical densities bN of the frames, just shifted.

This also holds true when looking at the dual windows. This e�ect is displayed on �gure 8 .


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Dual window, N = 12 b = 10

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70

80

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Dual window, N = 24 b = 20

Figure 8. The �gure shows the window and canonical dual window for twoGabor frames, where N , b and the width of the window are scaled in the samemanner. Notice that the second dual window seems identical to the �rst exceptthat it is half as narrow and twice as high. The window function are periodizedGaussians PϕW with W = 10 and W = 20.

The �gure shows that if W and b,N are scaled the same way, the dual windows look likedilated versions of each other. Since the functions live on the circle, γ ∈ L2(T), then truedilations are not possible, but the e�ect seems real because the region where the functionsdi�er consist of values close to zero.

This e�ect shows that when working with two Gabor frames G and H as de�ned in theorem65, there is two main ways of increasing the sampling rate L:

• Keeping b, N and the window width W �xed while increasing L. This will retain thewindow g and dual window γ of G, and these windows will be better resolved as theybecome sampled more �nely. Since L = bM this corresponds to increasing the numberof modulations in the system H.• Letting b, N and the window width W scale with L. This will narrow the windowg and the dual window γ, but the number of samples across the main part of thewindows remains constant. Because L = bM this corresponds to keeping the numberof modulations in the system H constant and increasing the number of translations.

Since theorem 83 shows that the precision of the Gabor derivative increases with the numberof modulations, the �rst method of scaling a Gabor system will be used for testing. Thesecond method of scaling is well suited for very long signals, like musical signals, where alloscillations have a relatively short duration compared to the length of the signal.

The �gure 9 shows some completely di�erent windows and their corresponding dual win-dows. The windows have been constructed from indicator functions of di�erent width. Noneof the widths divides any of the numbers a, b,M,N or L. The graphs have been computedwith very many points (14400 points along the x-axis), so the structures of the dual windows


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Figure 9. The �gure shows three windows constructed from the indicatorfunction, and their corresponding canonical dual windows. The windows havewidth 0.88, 0.42 and 0.11, and N = 12 and b = 10.

are well resolved, and not a result of insu�cient computations. The �gure just serves an illus-trative purpose. I will not use these windows for any computations, or make any statementsof whether they really do generate Gabor frames.

7.2. Di�erentiation. Theorem 83 provides an order of accuracy for the Gabor derivative asa function of the smoothness of f , if f ∈ C1(T) and f (2) is piecewise monotone and if thecanonical dual window is smooth. From looking at the dual windows on �gure 3 and �gure6some of them look smooth. Though I have not found a mathematical proof to justify thatthis holds true. Some results of smoothness of the canonical dual window for Gabor framesfor L2(Rd) are presented in [Groch01, chapter 13.] and in [Jans95b], but I have chosen not totransfer these results to the L2(T) setting.

A comparison of the accuracy of various numerical methods for di�erentiation can be seenon �gure 10. It compares second order polynomial di�erentiation, fourth order polynomialdi�erentiation, spectral di�erentiation and di�erentiation using Gabor frames. The smoothtest function is f(x) = esin(2πx). It is shown on �gure 11 along with its analytical derivative.The accuracy of the second and fourth order methods are clearly visible. The spectral methodusing a DFT converges very quickly in the beginning, and then grows slowly. The reason forthe slow growth of the spectral method is an accumulation of numerical round-o� errors. TheGabor derivative starts out terrible but still converges with faster than polynomial growth,until it is stopped by numerical round-o� errors at the same level as the spectral method. Thefaster than polynomial growth is predicted by theorem 83, so this serves as an illustration ofthat.

Figure 12 shows how the accuracy of the Gabor derivative change when the width of theperiodized Gaussian is varied. For small values of W the method is very accurate in thebeginning, but fails to reach the full precision, and starts to diverge. For larger values the


1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

10 100 1000

rela

tive

err

or

L

exp(sin) loglog plot

DFTDGT2nd4th

Figure 10. Comparison of the accuracy of numerically di�erentiating thefunction f(x) = esin(2πx), x ∈ [0; 1]. The Gabor frames used has the parame-ters N = 8, b = 6 and the window is a periodized Gaussian with W = 4. Thesampling rate L is placed on the x-axis.

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exp(sin)

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analytical derivative

Figure 11. The �gure shows the function f(x) = esin(2πx), x ∈ [0; 1] and itsanalytical derivative.

method starts out much worse, but reaches the full precision at some point. The canonicaldual windows corresponding to the �rst four lines can be seen on �gure 13. Perhaps a littlesurprising the parameters N, b,W of the best working method does not correspond to a max-imum point on the graph of �gure 5, but instead to a point a little to the left. This is alsoshown on �gure 13: The dual window corresponding to the best working method (numberthree on the plot) has oscillations. This certainly makes the dual Gabor atoms resemble thecomplex exponentials as much as possible, and since the spectral method is always the mostaccurate for smooth functions, it probably explains why this particular dual window seems towork best.


1e-14

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100

10 100 1000

Ab

so

lute

err

or

L

exp(sin) loglog plot

W=2W=3W=4W=6W=8

Figure 12. The �gure shows the accuracy of the Gabor derivative for di�erentchoices of window functions. The window functions are periodized Gaussiansof width W , as shown in the plot. The testing function is f(x) = esin(2πx), x ∈[0; 1].

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Figure 13. The �gure shows the canonical dual windows corresponding tothe �rst four lines on �gure 12. The Gabor frames have parameters N = 8,b = 6 and W equal to 2, 3, 4 and 6.

The only test function considered so far has been smooth. Some non-smooth test functionsare displayed on �gure 14.

The �rst test function is discontinuous. The second is not di�erentiable, and the third isdi�erentiable, but not two times di�erentiable. The third function are two parabolas combined.The rate of convergence is strongly dependent on the smoothness of the function for all threefunctions, and the Gabor derivative is no better or worse than the other methods. Since thetwo �rst functions are not even di�erentiable, it does not make much sense to calculate theglobal error, because it will be dominated by what happens in the discontinuity, which is notproperly de�ned anyway.


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Figure 14. The left column shows three non-smooth test functions, and theright column shows the relative, global error of using the di�erent numericalmethods on that particular test function. The Gabor frames used are the sameas in �gure 10.

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Figure 15. The �gure shows the pointwise error of the numerical di�erenti-ation over the interval [0; 1]. The test functions in each row are the same ason �gure 14. In the left column a Gabor frame with parameters N = 8, b = 6and W = 4 are used, and for the right column W = 10 instead. The samplingrate is L = 504.


Figure 15 shows the pointwise error over the interval [0; 1]. At the discontinuities bothmethods make exactly the same error. The interesting thing is what happens away from thediscontinuities. Away from the discontinuity the spectral method has slowly decaying errors.This corresponds to oscillations around the correct solution - an e�ect of the so-called Gibbsphenomena. The Gabor method does not do much better on the plots in the left column.The error decays much faster, but returns again to the level of the spectral method. This isprobably due to the wide support of the dual window, see �gure 16. Since the dual windownever decay to zero, then all translations of the dual window overlap the discontinuity.

On the plots in the right hand column, the window of the Gabor frame have much morenarrow support, and this changes the situation. For the �rst two examples the Gabor methodis signi�cantly better at reducing the oscillations away from the discontinuities. For the thirdexample the spectral method starts to catch up.

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Figure 16. Two pairs of windows and dual windows corresponding to the oneused in �gure 15. The �rst pair has parameters N = 8 b = 6, and W = 4. Thesecond has parameters N = 8, b = 6 and W = 10.

This example shows that for the same parameters N, b, the optimal window seems to bedependent on the test function. It illustrates that the accuracy of the Gabor derivative isdependent on N , b, the window function and on the test function.

This was just a short introduction to the Gabor derivative. A deeper understanding of thedependency of the accuracy on N , b, the window function and on the test function requires amuch more detailed study.

8. Concluding remarks.

This project started out dealing only with di�erentiation, and ended up dealing also withGabor frames for periodic spaces.

The spaces L2(T) and CLp seems to be a very good setting for Gabor frame theory. The

structure of Gabor frames for CLp allows for e�cient computations, and Janssens sampling

theorem shows that they can be used for approximation of Gabor frames for L2(T). Thespace L2(T) is in�nite dimensional, but compared to L2(R) it is much simpler. It is easy to�nd windows that satisfy condition A, and it is also relatively easy to verify that a Gaborsystem is a frame. These aspects are much more delicate in L2(R). I �nd the method ofconsidering eigenvalues to show that a Gabor system is a frame, especially interesting. If itcan be shown for a class of nice functions that the minimum eigenvalue can be found for aparticular and explicitly known x, the method would be both e�cient and accurate.


Furthermore the two types of Gabor frames resemble Fourier series and the discrete Fouriertransform. In this thesis this relationship was used to replace the DFT in the computation ofthe spectral derivative, and to there by get a new method of di�erentiating. There might beother areas where the same substitution could be done.

The Gabor derivative has the same convergence properties as the spectral derivative, butit generally requires a higher sampling rate to converge to the same margin of error. I cannotsay whether it can be useful for some applications. It is never more accurate than the spectralderivative. It does handle discontinuities better, but the error at the discontinuity is the exactsame. It is far more �exible, because there are so many parameters to vary, but this alsomeans that without some detailed study one might chose some very suboptimal parameters.

However, a new numerical method for di�erentiation with speed and accuracy close to thatof spectral di�erentiation is here by founded.

Peter Lempel Søndergaard.

References

[BuNe71] Butzer, P.L, Nessel, R.J. Fourier Analysis and Approximation, vol 1. Birkhäuser Verlag Basel,Stuttgart 1971.

[Briggs95] Briggs, William L. The DFT, An Owner's Manual for the Discrete Fourier Transform. SIAMPhiladelphia 1995.

[Chr02] Christensen, Ole. An Introduction to Frames and Riesz Bases. to appear.[Groech98] Gröchenig, Karlheinz. Aspects of Gabor Analysis on Locally Compact Abelian Groups. In Gabor

Analysis and Algorithms, page 211 - 231. Birkhaüser 1998.[Groch01] Gröchenig, Karlheinz. Foundations of Time-Frequency Analysis. Birkhäuser 2001.[Orr93] Orr, Richard s. The Order of Computation or Finite Discrete Gabor Transforms. IEEE Transactions

on Signal Processing. vol 41. no 1. 1993[Jans95a] Janssen, A.J.E.M. On Rationally Oversampled Weyl-Heisenberg Frames. Signal Processing. vol. 47,

page 239 - 245. 1995[Jans95b] Janssen, A.J.E.M. Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl.

1(4), page 403-436. 1995[Jans97] Janssen, A.J.E.M. From Continuous to Discrete Weyl-Heisenberg Frames Through Sampling. The

Journal of Fourier Analysis and Applications, volume 3 number 5, 1997.[Jens92] Jensen, Helge Elbrønd Matematisk Analyse, bind 4. DTH 1992[Qiu98] Qiu, Sigang. Discrete Gabor Transforms: The Gabor-Gram Matrix Approach. The Journal of Fourier

Analysis and Applications. vol 4, issue 1, 1998[Rud66] Rudin, Walter. Real and complex analysis. McGraw-Hill 1966.[Soen02] Søndergaard, Peter Gabor Frames and Pseudo-Di�erential Operators. Forprojekt, MAT, DTU 2002.[Stroh98] Strohmer, Thomas. Numerical Algorithms for Discrete Gabor Expansions. in Gabor Analysis and

Algorithms, page 267 - 294. Birkhaüser 1998.[Tref] Trefethen, Lloyd N. Spectral Methods in Matlab. SIAM 1998.[vL92] van Loan, Charles. Computational Frameworks for The Fast Fourier Transform. SIAM, Philadelphia

1992.

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