SIAM J. NUMER. ANAL c

SIAM J. NUMER. ANAL. c© 2006 Society for Industrial and Applied MathematicsVol. 44, No. 1, pp. 412–448

DISCRETE INGHAM INEQUALITIES AND APPLICATIONS∗

MIHAELA NEGREANU† AND ENRIQUE ZUAZUA‡

Abstract. In this paper we prove a discrete version of the classical Ingham inequality fornonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced bydiscrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer, the limit ofthe discrete Ingham inequality is the classical continuous one. This analysis is partially motivatedby control-theoretical issues. As an application we analyze the control/observation properties ofnumerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality providesobservability and controllability results which are uniform with respect to the mesh-size in suitableclasses of numerical solutions in which the high frequency components have been filtered. We alsodiscuss the optimality of these results in connection with the dispersion diagrams of the numericalschemes.

Key words. wave equation, numerical approximation schemes, nonharmonic analysis, discreteFourier transform, Ingham inequalities, observability, controllability, dispersion, group velocity

AMS subject classifications. 42C99, 65T50, 65M06, 65N06

DOI. 10.1137/050630015

1. Introduction. Families of “nonharmonic” exponentials{eiλkt

}appear in var-

ious fields of mathematics and signal processing. One of the central problems arisingin all of these applications is the question of the Riesz basis property.

The following inequality for nonharmonic Fourier series due to Ingham is wellknown (see [9] and [26, p. 162]): Assume that the strictly increasing sequence {λk}k∈Z

of real numbers satisfies the “gap” condition

λk+1 − λk ≥ γ for all k ∈ Z,(1.1)

for some γ > 0. Then, for all T > 2π/γ there exist two positive constants C1, C2

depending only on γ and T such that

C1(T, γ)

∞∑k=−∞

|ak|2 ≤∫ T

0

∣∣∣∣∣∞∑

k=−∞ake

itλk

∣∣∣∣∣2

dt ≤ C2(T, γ)

∞∑k=−∞

|ak|2 ,(1.2)

for every complex sequence (ak)k∈Z ∈ �2, where

C1(T, γ) =2T

π

(1 − 4π2

T 2γ2

)> 0,(1.3)

C2(T, γ) =8T

π

(1 +

4π2

T 2γ2

)> 0,(1.4)

∗Received by the editors April 26, 2005; accepted for publication (in revised form) September 26,2005; published electronically March 7, 2006. This research was supported by grant BFM2002-03345of the Spanish MCYT and the network “New materials, adaptive systems and their nonlinearities:modelling, control and numerical simulation” of the EU.

http://www.siam.org/journals/sinum/44-1/63001.html†Departamento de Algebra, Facultad de Matematica, Universidad Complutense de Madrid, 28040

Madrid, Spain (mihaela [email protected]).‡Departamento de Matematicas, Facultad de Ciencias, Universidad Autonoma de Madrid, 28049

Madrid, Spain ([email protected]).

412

DISCRETE INGHAM INEQUALITIES AND APPLICATIONS 413

and �2 is the Hilbert space of square summable sequences,

�2 =

{{ak} : ‖ak‖2

�2 =∑k∈N

|ak|2 < ∞}.(1.5)

This result shows that the sequence of exponentials{eiλkt

}forms a Riesz basis

of its span for T > 2π/γ (see [26, Chapter 3, p. 112]).As we have mentioned above, one of the main applications of Ingham’s inequality

and its variants is the control of wave-like equations and other closely related problemslike observability or inverse problems. The problem of observability for wave equationsconsists of analyzing whether the energy of the waves propagating in a domain withsuitable boundary conditions can be estimated in terms of the energy concentratedon a given subregion of the domain (or its boundary) where propagation occurs in agiven time interval. On the other hand, the goal in controllability problems is to drivethe solutions of a given dynamical system (continuous or discrete) to a given state ata given final time by means of a control acting on the system on that subregion (orits boundary). It is well known that the two problems are equivalent provided onechooses an appropriate functional setting, which depends on the equation (see, forinstance, [17]).

In the context of partial differential equations, using the Fourier representationof the solutions, the problem of observability can be reduced to an application ofIngham’s inequality in which the sequence {λk} is constituted by the spectrum ofthe generator of the underlying semigroup. However, the gap condition (1.1) thatis required to apply Ingham’s inequality often limits the range of applicability ofthis technique to 1-d problems like strings and beams. This has led to a significantnumber of controllability results (see [15]) and also to far reaching generalizationsof the Ingham theorem under weakened gap conditions (see [2], [4], [5], [8], [12],[13]). The most complete result in this direction has been obtained independently byBaiocchi, Komornik, and Loreti in [2], [3], [4], and Avdonin and Moran in [1].

In the numerical analysis of those observability inequalities and for studying thecontrollability properties of numerical schemes the need of a discrete version of thisinequality arises naturally (see [7], [19], [20], [21]). This paper is devoted to provinga discrete version of that Ingham inequality.

The inequality we prove is uniform with respect to the mesh-size Δt in the time-discretization and, in the limit as Δt → 0, yields the classical Ingham inequalityabove.

The discrete Ingham inequality we prove is the natural tool to prove observabil-ity/controllability properties for fully discrete schemes for the approximation of the1-d wave equation and other closely related models (vibrating beams, Schrodingerequation, etc.) and to show that the controls of the limiting continuous model arethe limit of the controls of the full discrete schemes. However, it is important torecall that, as it is by now well known [28], numerical approximation schemes oftenintroduce spurious high frequency solutions that may be an obstacle for uniform (withrespect to the mesh-size) observability/controllability results. Thus, one often needsto filter or cut-off those spurious numerical solutions. Our generalization of Ingham’sinequality to the discrete context explains how this filtering has to be done in orderto guarantee uniform results.

As an example of application of our discrete Ingham inequality we perform theanalysis of the observability/controllability properties of the most standard centeredfully discrete schemes for the wave equation.

414 MIHAELA NEGREANU AND ENRIQUE ZUAZUA

The main reason for the lack of uniform observability/controllability of the nu-merical high frequency spurious solutions, is that they generate high frequency wavepackets for which the group velocity is of the order of the mesh-size ([28]). Thus, asthe mesh-size tends to zero, since the velocity becomes smaller and smaller, the timefor observability/controllability increasing in a divergent way. This fact is relatedto the dispersion diagram associated to the numerical approximation scheme, since,roughly, the slope of the dispersion diagram is the group velocity of propagation ofwave packets and also coincides with the spectral gap. Part of this article is devoted toexplaining the connections of these notions and to show how combining the qualitativeinformation that the dispersion diagram provides with the discrete Ingham inequality,one can get precise information on how the filtering should be implemented, if needed.

As proved in the original article by Ingham (see [9, p. 368]), an L1-version ofinequality (1.2) also holds. More precisely, for every increasing sequence {λk}k∈Z ofreal numbers satisfying the “gap” condition (1.1) we also have

C1(T, γ) |ak| ≤∫ T

0

∣∣∣∣∣∞∑

k=−∞ake

itλk

∣∣∣∣∣ dt ≤ C2(T, γ) |ak| for all k ∈ Z(1.6)

for all T > 2π/γ.In this paper we also prove a discrete version of this inequality.Our proofs are strongly inspired in that by Ingham (see also [26]), which is based

on the use of a suitable cut-off, nonnegative function, with compact support on thetime interval (0, T ) and whose Fourier transform is “concentrated” around τ = 0.We use the same function in the physical space, but its Fourier transform has to bereplaced by the discrete one. One of the key points in the proof is a careful comparisonbetween the continuous and discrete transforms of this weight function. This is doneby using a key result by N. Trefethen [23].

This paper is organized as follows: in section 2 we state our discrete Inghaminequality (see Theorem 2.1), we analyze the necessity of its hypotheses and compareboth the continuous and discrete inequalities. We also formulate a discrete version ofthe L1 analogue (1.6) (see Theorem 2.2). In section 3 we discuss the application of thisresult to the study of the properties of the solutions of fully discrete approximationsof the wave equation. In section 4 the controllability problem for the discrete systemis addressed and the main results of existence, characterization, and convergence ofthe discrete controls are presented and proved. In section 5 we discuss these results inconnection with the dispersion diagrams of the discrete equations under consideration.Finally, section 6 is devoted to proving the discrete Ingham inequality and its discreteL1 version.

The discrete Ingham inequality we present in this paper has been announced in[21].

2. Main results. The main result of this paper is as follows.Theorem 2.1 (discrete Ingham inequality). Let {λk}k∈Z be an increasing se-

quence of real numbers satisfying for some γ > 0 the “gap” condition

λk+1 − λk ≥ γ > 0 for all k ∈ Z.(2.1)

Let T > 0 and 0 < Δt ≤ 1. Assume that {λk}|k|≤N satisfies the additional condition

|λk − λl| ≤2π − (Δt)p

Δtfor all |k| ≤ N, |l| ≤ N, for some 0 ≤ p < 1/2,(2.2)


where 2N ≤ M and M = [T/Δt− 1]. Then, there exists a positive number ε(Δt)such that, for all T > T0(Δt) := 2π/γ + ε(Δt), there exist two positive constantsCj(Δt, T, γ) > 0, j = 1, 2, such that

C1(Δt, T, γ)

N∑k=−N

|ak|2 ≤ Δt

M∑n=0

∣∣∣∣∣N∑

k=−N

akeinΔtλk

∣∣∣∣∣2

≤ C2(Δt, T, γ)

N∑k=−N

|ak|2 ,

(2.3)

for every complex sequence (ak)k∈Z ∈ �2.Moreover, if γ and p in (2.1) and (2.2) are kept fixed, then ε(Δt) = o(Δt)1−2p

and the constants in (2.3) satisfy

Cj(Δt, T, γ) = Cj(T, γ) + δj(Δt), j = 1, 2, with δ1(Δt) ≤ 0 and δ2(Δt) ≥ 0,(2.4)

where Cj(T, γ), j = 1, 2, are the Ingham constants in (1.3) and (1.4) and limΔt→0

δj(Δt) =

0, j = 1, 2.Concerning the L1-version of Ingham inequality in (1.6), the following theorem

holds.Theorem 2.2. Under the hypotheses of Theorem 2.1 we also have the following

discrete version of (1.6):

C1(Δt, T, γ) |ak| ≤ Δt

M∑n=0

∣∣∣∣∣N∑

k=−N

akeinΔtλk

∣∣∣∣∣ ≤ C2(Δt, T, γ) |ak| for all |k| ≤ N.

(2.5)

As in Theorem 2.1, the time T and the constants in this inequality remain uniformas Δt → 0 and converge to those of the continuous Ingham inequality (1.6).

Remark 2.3. Condition T > 2π/γ is optimal for the classical Ingham inequality(see [26, p. 163]). In this sense, the condition T > 2π/γ + ε(Δt) in Theorem 2.1 isasymptotically optimal since ε(Δt) → 0 as Δt → 0.

It is important to emphasize that the time T and the constants Cj , j = 1, 2 in(2.3) are uniform in Δt. This is essential for the applications in numerical analysisin which Δt → 0. The uniformity may be guaranteed because of the assumptions(2.1)–(2.2) on the sequence {λk}k.

More precisely, when comparing the continuous and discrete inequalities, thefollowing can be said:

• In both continuous and discrete cases, the sequence {λk}k is required tosatisfy the so-called gap condition (2.1).

• The restriction (2.2) imposed on {λk}k in Theorem 2.1 is not needed in theclassical continuous Ingham inequality (1.2).

• It is easy to see that, for every N ∈ N fixed, if we pass to the limit Δt → 0in (2.3), we get the classical Ingham inequality (1.2). Indeed, for (1.2) to betrue for all sequences (ak)k∈Z ∈ �2 it is sufficient, by density, to prove it forsequences with only a finite number of nonzero components.In that case (1.2) is the limit of (2.3) because of the convergence of theminimal time T and the constants Cj , j = 1, 2, in (2.3) to those of (1.2).We also have a discrete Ingham inequality (2.3) for every sequence (λk)kverifying conditions (2.1) and (2.2), with 0 ≤ p ≤ 1. But, if p ≥ 1/2,


ε(Δt) = o(Δt)1−2p → ∞, so T0(Δt) → ∞, and this makes it of little use inpractice because we are looking for a uniform (with respect to Δt) time T .

On the other hand, the restriction 2N ≤ M , with M = [T/Δt− 1], is sharp.Indeed, when 2N > M one can find nontrivial values of the coefficients {ak}k suchthat

N∑k=−N

akeinΔtλk = 0, 0 ≤ n ≤ M(2.6)

and

N∑k=−N

|ak|2 �= 0.(2.7)

Observe that (2.6) is a system of M + 1 homogeneous linear equations with 2N + 1unknown quantities ak. If 2N > M , this system necessarily has nontrivial solutions.This is in agreement with common sense. Indeed, in view of the fact that we onlymake M + 1 measurements for n = 0, . . . ,M one cannot expect to recover more thanM + 1 coefficients of the solution.

When 2N ≤ M , (2.6)–(2.7) do not hold. However if λk−λl ∈ 2πZ/Δt for certainvalues of k and l with k �= l the sequence ak = −al = 1, an = 0, n �= k, and l satisfies(2.6). Then, an inequality of type (2.3) is impossible. So, it is natural to impose onthe sequence {λk}k the condition λk − λl �∈ 2πZ/Δt for a discrete Ingham inequality(2.3) to hold.

In fact, to avoid aliasing one has to restrict the increasing sequence of real numbers{λk}k to be such that λk−λl ∈ [2πm/Δt, 2π(m + 1)/Δt], for some m ∈ Z. Therefore,it is natural to impose the condition

|λk − λl| <2π

Δt.

In our theorem this latter condition is implied by the stronger one, (2.2), whichis needed for the uniform estimates in (2.3) to hold. More precisely, the restriction0 ≤ p < 1/2 in (2.2) is needed to guarantee the asymptotically optimal time T >2π/γ + ε(Δt), with ε(Δt) → 0 as Δt → 0 since ε(Δt) = o(Δt)1−2p.

Remark 2.4. The condition T > T0(Δt) is necessary for the proof of the firstinequality in (2.3) and in (1.6) (to have C1(Δt, T, γ) > 0). The second inequality in(2.3) and (1.6), respectively, holds for all T > 0. In this respect the situation is thesame as for the continuous inequalities (1.2).

3. Application to the uniform observability of the full discretizationsof the 1-d wave equation.

3.1. The wave equation. This section is motivated by the classical problem ofcontrol of waves. More precisely, it is related with the controllability of the 1-d waveequation: given T > 0 and (u0, u1) ∈ L2(0, 1) × H−1(0, 1), the problem is to find acontrol function v ∈ L2(0, T ) such that the solution of the system⎧⎨

⎩utt − uxx = 0, 0 < x < 1, 0 < t < T,u(0, t) = 0, u(1, t) = v(t), 0 < t < T,u(x, 0) = u0(x), ut(x, 0) = u1(x), 0 < x < 1,

(3.1)


satisfies

u(T ) = ut(T ) = 0, 0 < x < 1.(3.2)

This property is well known to be true for T ≥ 2. This problem has been studiedand solved in a much more general setting and, in particular, for multidimensionalwave equations [17]. Several approaches to the problem have been developed. Inparticular, the Hilbert uniqueness method (HUM) introduced by Lions in [17] offersa general way of reducing the problem to the so-called observability problem for theadjoint (up to an inversion in time) wave equation in the absence of control:⎧⎨

⎩φtt − φxx = 0, 0 < x < 1, 0 < t < T,φ(0, t) = φ(1, t) = 0, 0 < t < T,φ(x, 0) = φ0(x), φt(x, 0) = φ1(x), 0 < x < 1.

(3.3)

It is well known that the energy

E(t) =1

2

∫ 1

0

(| φx(x, t) |2 + | φt(x, t) |2

)dx(3.4)

of the solutions of (3.3) satisfies

dE(t)

dt= E′(t) = 0 for all t ∈ [0, T ]

and therefore is conserved in time.The observability problem is as follows: To find T > 0 such that there exists a

constant C(T ) > 0 for which

E(0) ≤ C(T )

∫ T

0

| φx(1, t) |2 dt(3.5)

holds for every solution of (3.3).HUM allows showing that, once the observability inequality (3.5) is satisfied for

the adjoint system (3.3), system (3.1) is controllable in time T . Moreover, HUMprovides a systematic method to build the control v = φx(1, t) of minimal L2(0, T )-norm.

In the context of the 1-d wave equation (3.3), inequality (3.5) can be easily provedby several methods including Fourier series, D’Alembert Formula, multiplier tech-niques, and Ingham’s theorem (1.2), provided T ≥ 2.

In order to solve the problem (3.5) applying the classical Ingham inequality, oneuses Fourier series techniques. Indeed, the solution of (3.3) admits the Fourier devel-opment

φ(x, t) =∑

k∈Z\{0}ake

iλktϕk(x),(3.6)

with {λk}k, λk = kπ = −λ−k, k > 0, being the sequence of eigenvalues of thesystem, ϕk(x) = sin(kπx), the corresponding eigenfunctions and ak ∈ C the Fouriercoefficients, which can be computed explicitly in terms of the initial data in (3.3).

By definition (3.4) of the conserved energy of the solution φ of (3.3) given by(3.6), we have

Eφ =1

2

∑k∈Z\{0}

k2π2 |ak|2 .(3.7)


On the other hand, in view of the explicit form of φx(1, t), inequality (3.5) may bewritten as:

∑k∈Z\{0}

k2π2 |ak|2 ≤ C(T )

∫ T

0

∣∣∣∣∣∣∑

k∈Z\{0}(−1)kkπake

iλkt

∣∣∣∣∣∣2

dt.(3.8)

According to Ingham’s inequality (1.2), (3.8) holds for T > 2, since the gap of thesequence {λk}k is constant, γ = π, and, consequently, the minimal observability timeis 2π/γ = 2. In this particular case the inequality holds also for the minimal timeT = 2. This is due to the orthogonality properties of the trigonometric polynomials.But, in general, i.e., for a general sequence (λk)k∈Z satisfying the gap condition (1.1), itis well known that the Ingham inequality (1.2) may fail for the minimal time T = 2π/γ(see [26, p. 163]).

In order to obtain numerical approximations of the controls, it is natural toanalyze the controllability and observability properties of numerical approximationschemes. We first recall some well-known facts about the space semi-discretizationschemes to later address space-time discretizations.

3.2. Space semi-discretizations. First, we consider the semi-discrete versionof the observability problem (3.5): Take N ∈ N, set h = 1/(N + 1) and consider thefinite-difference space semi-discretization of (3.3):

⎧⎪⎨⎪⎩

φ′′j =

φj+1 − 2φj + φj−1

h2, t > 0, j = 1, . . . , N,

φ0 = φN+1 = 0, t > 0,φ(0) = φ0,j , φ′

j(0) = φ1,j , j = 1, . . . , N.

(3.9)

The energy of system (3.9) is given by

Eh(t) =h

2

N∑j=1

| φj(t) |2 +h

2

N∑j=0

| φj+1(t) − φj(t) |2h2

(3.10)

and it is also conserved in time.The semi-discrete version of (3.5) is

Eh(0) ≤ C

∫ T

0

∣∣∣∣φN (t)

h

∣∣∣∣2

dt.(3.11)

More precisely, one seeks for a positive constant C > 0 such that (3.11) holds.The corresponding eigenvalue problem is of the form{

−[ϕk+1 + ϕk−1 − 2ϕk]/h2 = λ2ϕk, k = 1, . . . , N,

ϕ0 = ϕN+1 = 0.(3.12)

The eigenvalues and eigenvectors of (3.12) may be computed explicitly (see [10,p. 456]); one then has⎧⎨

⎩λ2k(h) =

4

h2sin2

(πkh

2

), k = 1, . . . , N,

ϕk ≡ (ϕk,1, . . . , ϕk,N ); ϕk,j = sin(kπjh), j, k = 1, . . . , N.(3.13)


The solutions of (3.9) in Fourier series are

φ =

N∑k=−N,k =0

akeiλk(h)tϕk,(3.14)

where φ = (φ1, . . . , φN ).As pointed out in [11], (3.11) holds for all T > 0 and h > 0, but, the observability

constant in (3.11) may not remain uniformly bounded as h → 0, for any T > 0. Moreprecisely,

supφ∈Sh

[Eh(0)∫ T

0| φN (t)/h |2 dt

]→ ∞, as h → 0,(3.15)

where Sh is the set of all solutions of (3.9). This is due to the pathological behaviorof the high frequency numerical solutions.

In the light of Ingham’s inequality (1.2), the lack of uniform observability ash tends to zero may be explained because of the lack of gap between consecutiveeigenvalues (see [11], [28]). In particular, the gap between the largest eigenvaluesentering in the Fourier development of the solution of (3.9) may be bounded above asfollows:

λN (h) − λN−1(h) ≤ 3π2h

2→ 0, as h → 0.(3.16)

As it was proved in [11], a suitable cut-off or filtering of the spurious numerical highfrequencies may be a good cure for these pathologies. Given 0 < α < 1, we introducethe following classes of filtered solution of (3.9):

Cα(h) =

{φ sol. of (3.9) : φ =

∑|k|≤αN, k =0

akeiλktϕk

}.(3.17)

In the class Cα(h) the high frequencies corresponding to the indexes j > αN havebeen cut-off. This guarantees a uniform gap condition

λk+1(h) − λk(h) ≥ π cos(πα

2

), for k ≤ α/h.(3.18)

Consequently, applying Ingham’s inequality, we may deduce the uniform observ-ability in the class Cα(h) for

T > T (α) = 2/ cos(πα/2).(3.19)

Let us explain this in more detail.By definition (3.10) of the conserved energy and taking into account the orthog-

onality properties of the eigenvectors (see [11], [20]), we have

Eh =1

4

αN∑k=−αN,k =0

|ak|2(1 + λ2

k(h)).(3.20)

Then, inequality (3.11) in the class Cα(h) may be rewritten as

αN∑k=−αN,k =0

|ak|2(1 + λ2

k(h))≤ C(T )

∫ T

0

∣∣∣∣∣∣αN∑

k=−αN,k =0

sin(Nkπh)

hake

iλkt

∣∣∣∣∣∣2

dt.(3.21)


Applying now Ingham’s theorem (1.2) for the real sequence (λk(h))|k|≤αN , in viewof (3.18), it follows that if T > T (α) with T (α) as in (3.19), there exists a constantC > 0 such that

αN∑k=−αN,k =0

∣∣∣∣ak sin(Nkπh)

h

∣∣∣∣2

≤ C(T )

∫ T

0

∣∣∣∣∣∣αN∑

k=−αN,k =0

sin(Nkπh)

hake

iλkt

∣∣∣∣∣∣2

dt,(3.22)

holds for every solution of (3.9) in the class Cα(h). Finally, it is sufficient to observethat

αN∑k=−αN,k =0

∣∣∣∣ak sin(Nkπh)

h

∣∣∣∣2

∼ Eh,

to obtain a uniform observability inequality (3.11) in each class Cα(h) for all 0 < α < 1.Note, however, that the minimal time T (α) depends on the filtering parameter α and,in particular, T (α)→ 2 as α→ 0 and T (α)→∞ as α→ 1 (see [28] for a rigorous proof).

As a further step towards a complete theory of numerical approximations of con-trols it is natural to address the same issue for full space-time discretizations. Thisissue is addressed in the following section.

3.3. Fully discrete approximations. The main ingredient to derive the fullydiscrete analogue of (3.5) for a finite-difference full discretization of a homogeneous1-d wave equation (3.3) is the Fourier representation of solutions combined, this time,with our discrete Ingham inequality in Theorem 2.1.

Given M,N ∈ N we set Δx = 1/(N + 1) and Δt = T/(M + 1) and introduce thenets

0 = x0 < x1 = Δx < · · · < xN = NΔx < xN+1 = 1,

0 = t0 < t1 = Δt < · · · < tM = MΔt < tM+1 = T

with xj = jΔx and tn = nΔt, j = 0, 1, . . . , N + 1, n = 0, 1, . . . ,M + 1.We consider the following finite-difference discretization of (3.1):

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

un+1j − 2un

j + un−1j

(Δt)2=

unj+1 − 2un

j + unj−1

(Δx)2, j = 1, 2, . . . , N ; n = 1, 2, . . . ,M,

un0 = 0, un

N+1 = vnΔx, n = 1, 2, . . . ,M,

u0j = u0j , u1

j = Δtu1j + u0j , j = 1, 2, . . . , N.

(3.23)

We shall denote by un = (un1 , . . . , u

nN ) the solution at the time step n. As in the

context of the continuous wave equation above, we consider the uncontrolled system

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

φn+1j − 2φn

j + φn−1j

(Δt)2=

φnj+1 − 2φn

j + φnj−1

(Δx)2, j = 1, 2, . . . , N ; n = 1, 2, . . . ,M,

φn0 = φn

N+1 = 0, n = 1, 2, . . . ,M,

φ0j = φ0j , φ1

j = φ0j + Δtφ1j , j = 1, 2, . . . , N,

(3.24)


a central finite difference discretization of (3.3).Under the stability condition μ = Δt/Δx ≤ 1 (μ is the Courant number), the

scheme (3.24) is convergent of order 2.However, as observed in [14], the resulting discrete sequence of controls vnΔx =

−φnN/Δx obtained with a discrete HUM method may have an unstable behavior as

(Δt,Δx) → (0, 0). More precisely, it is possible to exhibit initial conditions such thatthe discrete controls vnΔx do not converge towards the control v for (3.1) (see [28]).Once more, filtering of high frequencies is an efficient cure for these instabilities andour discrete Ingham inequality is the tool to analyze how it behaves.

The energy of (3.24) is

En =Δx

2

N∑j=0

⎡⎣(φn+1

j − φnj

Δt

)2

+

(φn+1j+1 − φn+1

j

Δx

)(φnj+1 − φn

j

Δx

)⎤⎦ ≥ 0,(3.25)

which is a discretization of the continuous energy E in (3.4), and it is conserved inall the time steps En = E0, n = 1, . . . ,M , for the solutions of (3.24) (see [20]).

Solutions of (3.24) admit the Fourier development (see [20])

φn =

N∑k=−N,k =0

akeiλknΔtϕ|k|,(3.26)

with ak ∈ C, ϕk = (ϕk,1, . . . , ϕk,N ) = (sin(kπΔx), . . . , sin(NkπΔx)) and

λk = sgn(k)2

Δtarcsin

(Δt

Δxsin

kπΔx

2

).(3.27)

Our goal is to analyze the discrete version of the observability inequality (3.5)

E0 ≤ C

[Δt

M∑n=0

∣∣∣∣φnN

Δx

∣∣∣∣2],(3.28)

where E0 is the conserved energy of the solutions of the discrete system (3.24). Thisinequality implies by HUM a controllability property of the discrete analogue (3.23)of the control system (3.1). Of course, we seek for a positive constant C > 0 in (3.28),independent on Δt and Δx. This will yield a family of controls that will be boundedas Δt → 0, which constitutes a natural candidate to converge to the control of (3.1).

Inequality (3.28) is the discrete analogue of (3.5). In particular, note that, accord-ing to Taylor’s formula φx(1, t) ∼ (φ(1, t) − φ(1 − Δx, t))/Δx. Thus, at the discretelevel and taking into account that, according to the boundary conditions, φn

N+1 = 0,we obtain φx(1, t) ∼ −φn

N/Δx. Thus, the right-hand side of (3.28) represents a dis-crete version of the right-hand side term in the continuous observability inequality(3.5).

Inequality (3.28) may also be seen as a time-discretization of the semi-discreteobservability inequality (3.11). Note that, in fact, the semi-discrete case correspondsto taking μ = 0 in the fully discrete scheme.

According to Theorem 2.1, the spectral gap between two consecutive eigenvaluesplays a very important role in the analysis of the uniform observability inequality(3.28).


It is important to distinguish two cases:• In the particular case where Δt = Δx := h (μ = 1) we have

λk = sgn(k)2

harcsin

(sin

kπh

2

)= sgn(k)kπ.

Thus,

λk+1 − λk = γ = π.

But the condition (2.2) does not hold, because

maxk,l

|λk − λl| =2π − 2πΔt

Δt.

Note, however, that, in this particular case, due to the orthogonality properties ofthe family of complex discrete exponentials involved in the Fourier representation ofsolutions,

M∑n=0

einΔtπ(k−l) = (M + 1)δk,l,

where δk,l is Kronecker’s delta, an inequality of type (2.3) holds immediately and thediscrete Ingham inequality is not needed.

Indeed, denoting by mk = (−1)kak sin(kπΔx)/Δx, the energy of the solutions(3.24) concentrated on the extreme x = 1 can be written as

ΔtM∑n=0

∣∣∣∣φN

Δx

∣∣∣∣2

= Δt

M∑n=0

∣∣∣∣∣N∑

k=−N

mkeinΔtπk

∣∣∣∣∣2

(3.29)

and the total energy of the solutions is

E0 =1

2

N∑k=−N

|mk|2(3.30)

(see [20] for more details). Then, for T = 2 we have

hM∑n=0

∣∣∣∣φN

h

∣∣∣∣2

= h

M∑n=0

∣∣∣∣∣N∑

k=−N

mkeinhπk

∣∣∣∣∣2

= h

M∑n=0

N∑k=−N

|mk|2 + h

M∑n=0

N∑k=−N,k =l

mkmleinhπ(k−l) = 2

N∑k=−N

|mk|2 ,

and therefore

E0 =1

4

[h

M∑n=0

∣∣∣∣φnN

h

∣∣∣∣2].

A similar identity holds for the continuous wave equation (3.3) in the minimal ob-servability time T = 2. Namely

E =1

4

∫ 2

0

|φx(1, t)|2


for every solution φ of (3.3), where E is the energy of the solutions φ = φ(x, t).• In the case when μ < 1 the gap between two consecutive eigenfrequencies

decreases at high frequencies and it is of the order of Δx when Δx → 0. Indeed, wehave

|λk+1 − λk| =

∣∣∣∣ 2

Δt

[arcsin

(Δt

Δxsin

(k + 1)πΔx

2

)− arcsin

(Δt

Δxsin

kπΔx

2

)]∣∣∣∣≤∣∣∣∣π2 2

Δt

Δt

Δx

(sin

(k + 1)πΔx

2− sin

kπΔx

2

)∣∣∣∣=

∣∣∣∣π2 2

Δx

[sin

kπΔx

2

(cos

πΔx

2− 1

)+ sin

πΔx

2cos

kπΔx

2

]∣∣∣∣≤∣∣∣∣π2 2

Δx

∣∣∣∣1 − cosπΔx

2

∣∣∣∣+ π2

2cos

kπΔx

2

∣∣∣∣=

∣∣∣∣π2 2

Δx2 sin2 πΔx

4+

π2

2cos

kπΔx

2

∣∣∣∣ ≤∣∣∣∣π2

2

[πΔx

4+ sin

(((N + 1) − k)Δxπ

2

)]∣∣∣∣ .In particular, the gap for the highest frequencies satisfies

|λN − λN−1| ≤π2

2

(πΔx

4+

πΔx

2

)=

3π3Δx

8→ 0, when Δx → 0.

So the uniform gap condition (2.1) is not satisfied and we cannot directly applyTheorem 2.1 to prove inequality (3.28). Therefore, as soon as μ < 1, we are in thesame situation as for the semi-discrete equation (3.9) in which μ = 0: the lack ofspectral gap may produce the degeneracy of the observability constant.

To remedy this lack of uniform estimates, we need to introduce a subclass ofsolutions of system (3.24) where the high frequency components have been filtered.To do that, given α ∈ (0, 1), the so-called filtering parameter, we consider the classCα(Δx),

Cα(Δx) =

{φn sol. of (3.24) : φn =

αN∑k=−αN,k =0

akeiλknΔtϕ|k|

},(3.31)

of solutions involving the eigenvalues {λk}k∈[−αN,αN ], k �= 0:

φn =

αN∑k=−αN,k =0

akeiλknΔtϕ|k|.(3.32)

Let us first check the gap condition. We have

λk+1 − λk =2

Δt

[arcsin

(Δt

Δxsin

(k + 1)πΔx

2

)− arcsin

(Δt

Δxsin

kπΔx

2

)]

=π cos ξΔx

2√1 −(

ΔtΔx sin ξΔx

2

)2:= γk,

(3.33)


for every k ∈ [−αN,αN ] and for some ξ ∈ [kπ, (k + 1)π]. Therefore, in particular,

λk+1 − λk ≥π cos NαπΔx

2√1 −(

ΔtΔx sin ξΔx

2

)2≥ π cos

NαπΔx

2≥ π(1 − α).

Consequently, for any filtering parameter α ∈ (0, 1), the gap condition (2.1) holdswith

γα := min|k|≤αN

(γk) ≥ γ(α) = π cos

(NαπΔx

2

)≥ π(1 − α).(3.34)

On the other hand, by the mean value theorem,

|λk − λl| =

∣∣∣∣ 2

Δt

(arcsin

(Δt

Δxsin

(kπΔx

2

))− arcsin

(Δt

Δxsin

(lπΔx

2

)))∣∣∣∣=

∣∣∣∣∣∣2

Δt

ΔtΔx

πΔx2 cos

(ξπΔx

2

)(k − l)√

1 −(

ΔtΔx

)2sin2 ξπΔx

2

∣∣∣∣∣∣ ≤∣∣∣∣∣∣

2Nαπ cos(

ξπΔx2

)√

1 −(

ΔtΔx

)2sin2 ξπΔx

2

∣∣∣∣∣∣≤

∣∣∣∣∣∣2Nαπ cos

(ξπΔx

2

)√(

ΔtΔx

)2 − (ΔtΔx

)2sin2 ξπΔx

2

∣∣∣∣∣∣ =2NαπΔx

Δt=

2απ − 2απΔx

Δt

≤ 2πα(1 − Δt)

Δt.

(3.35)

In view of (3.35), by choosing conveniently the filtering parameter α such that

α ≤ α∗(Δt) :=2π − (Δt)p

2π(1 − Δt),(3.36)

with 0 ≤ p < 1/2, hypothesis (2.2) of Theorem 2.1 is verified.In practice it is convenient to fix the filtering parameter 0 < α < 1, independent

of Δt. In this way (3.36) is automatically satisfied for Δt small enough, which is therelevant case in numerical approximation problems. On the other hand the gap con-dition (3.34) is also automatically and uniformly satisfied for the truncated sequence{λk}|k|≤Nα.

Note that the gap γα (respectively, the minimal observability/ control time 2π/γα)tends to π (respectively, to 2) when α ↘ 0+ while it converges to zero (respectively,to infinity) when α ↗ 1−.

Note also that the minimal observability/control time can be taken to be anyT > 2π/γα since the minimal time T (α) = 2π/γα + ε(Δt) tends to 2π/γα as Δt tendsto zero.

More precisely, the following theorem holds.Theorem 3.1. For all Courant numbers 0 < μ < 1 and all values of the filtering

parameter 0 < α < 1, the observability inequality below holds that

E0 ≤ 1

2 cos2 απ2 C1(T, γα)

[Δt

M∑n=0

∣∣∣∣φN

Δx

∣∣∣∣2]

(3.37)

for every solution of (3.24) in the class Cα(Δx), uniformly as (Δt,Δx) → (0, 0) forany T > T (α) = 2π/γα, with C1(T, γα) given by (1.3). Moreover,


1. T (α) ↗ ∞ as α ↗ 1− and T (α) ↘ 2 as α ↘ 0+.

2. Cα(T ) :=1

2 cos2 απ2 C1(T, γα)

↘ C(T ) =1

2C1(T, γ)as α ↘ 0+ with C1(T, γ)

given by (1.3), where C(T ) is the constant of the continuous observabilityinequality (3.5).

Remark 3.2. This theorem allows the recovery of the uniform observability ofthe original system (3.3) as the limit when (Δt,Δx) → (0, 0) of the observability ofthe solutions of discrete one (3.24) in the classes (3.31) by means of Fourier filter-ing; the statements in this theorem coincide with the predictions one may deducefrom the analysis of the dispersion diagram of the numerical scheme [28], as we shallsee in the next section.

Proof (Sketch of the proof). The energy of the solutions (3.26) of the discretesystem (3.24), concentrated on x = 1 is given by (3.29) and the total energy (3.25) ofthe solutions is

E0 =2

(Δx)2

∑k

a2k sin2 kπΔx

2=

2

(Δx)2

∑k

a2k

sin2(kπΔx)

4 cos2 kπΔx2

=1

2

∑k

| mk |2 1

cos2 kπΔx2

,

where mk = sin(NkπΔx)/Δx.For all k ∈ [−αN,αN ] we have cos(απ/2) ≤ cos(αNπΔx/2) ≤ cos(kπΔx/2) ≤ 1

and, in this case,

1

2

∑k

| mk |2≤ E0 ≤ 1

2 cos2 NαπΔx2

∑k

| mk |2≤ 1

2 cos2 απ2

∑k

| mk |2.(3.38)

Applying Theorem 2.1 and the Fourier representation (3.32) of the solutions we obtainthat, for all T > 2π/γα + ε(Δt), there exist positive constants Cj(Δt, T, γα), j = 1, 2,such that

C1(Δt, T, γα)

αN∑k=−αN

|mk|2 ≤ Δt

M∑n=0

∣∣∣∣∣αN∑

k=−αN

mkeinΔtλk

∣∣∣∣∣2

≤ C2(Δt, T, γα)

N∑k=−N

|mk|2.

Therefore, for every α as in (3.36), by (3.38), the following inequalities hold:

2 cos2απ

2C1(Δt, T, γα)E0 ≤ Δt

M∑n=0

∣∣∣∣φN

Δx

∣∣∣∣2

≤ 2C2(Δt, T, γα)E0,(3.39)

with Cj(Δt, T, γα), j = 1, 2, defined by relations (2.4), for every truncated solution(3.32) of system (3.24) belonging to the class Cα(Δx).

The uniform observability inequality (3.39) implies uniform controllability results,as we shall prove in the next section, for the projection (over the subspace of unfilteredFourier components) of solutions of the dual controlled system (3.23). In the limit asΔt,Δx → 0 one recovers the sharp controllability results of the wave equation (3.1).For the details of the proof of convergence of controls we refer to [20] where the caseΔt = Δx was studied in detail. But, as mentioned above, for this particular one,because of the orthogonality of complex harmonic polynomials, the discrete Inghaminequality is not needed. We also refer to [16] where the convergence of controls forthe semi-discretizations of the beam equation was analyzed in detail.

The usual centered finite-difference approximation of the wave equation we haveconsidered here is only a simple example in which the discrete Ingham’s theoremcan be applied, together with some filtering mechanism, to get uniform observabilityinequalities. The discrete Ingham inequality can also be applied, for instance, to theimplicit fully finite difference approximation of the wave equation, introduced in [18].


4. Uniform controllability of the filtered solutions. In this section, weapply the uniform observability results obtained above to analyze the controllabilityproperties of the fully discrete system (3.23).

Let us define the Hilbert spaces of square summable sequences �1 and �

−1 asfollows:

�1 =

{{ak} ∈ �2 : ‖ak‖2

�1 =∑k∈N

| kπak |2< ∞},(4.1)

�−1 =

{{ak} ∈ �2 : ‖ak‖2

�−1 =∑k∈N

∣∣∣ akkπ

∣∣∣2 < ∞},(4.2)

where the discrete space �2 is given by (1.5).

For every α ∈ (0, 1), we introduce the space Sα generated by the eigenvectors(ϕk) involved in Cα(Δx) of the filtered solutions of the homogeneous system (3.24)with filtering parameter α:

Sα = span {ϕk : |k| ≤ αN} .(4.3)

For every s ∈ R, we denote by �sΔx,α the space Sα endowed with the norm

‖v‖2s,Δx =

∑|k|≤Nα

λsk |ak|

2, for v ∈ Sα : v =

∑|k|≤Nα

akϕk,

where λk are as in (3.27).

For every α ∈ (0, 1) and T > 0, we consider the partial controllability problemfor system (3.23) in the space �2 × �

−1, which consists of finding a control vn ∈ RM

such that, for all initial data (u0, u1) ∈ �2 × �−1, the solution un of (3.23) satisfies

(ΠαuM ,Παu

M+1) = (0, 0),(4.4)

where Πα is the orthogonal projection over Sα; i.e.,

(Παu

M ,ΠαuM+1)

=

( ∑|k|≤Nα

ckϕk,∑

|k|≤Nα

dkϕk

),

where (ck) and (dk) are the Fourier coefficients of (uM , uM+1) in the basis of theeigenvectors (ϕk)k. Observe that we only require to control uniformly the projectionΠα of the solutions of the discrete system (3.23) over subspaces in which the highfrequencies have been filtered.

As we shall see this result is a consequence of the partial observability results ofthe previous section in the class of filtered solutions Cα(Δx).

Multiplying the first equation in (3.23) by an arbitrary solution φn of (3.24) andadding in j and n, we get

Δt

M∑n=1

vnΔx

φnN

Δx+

1

μ

N∑j=0

[u1jφ

0j − u0

jφ1j

]=

1

μ

N∑j=0

[uM+1j φM

j − uMj φM+1

j

].(4.5)


The solution of system (3.23) may be characterized through a transposition argumentbased on the identity above. Indeed, given M, N ∈ N, vΔt ∈ R

M , and (u0, u1) ∈R

N × RN , {un} solves (3.23) if for every s ∈ [1,M ] it holds that

Ls(φ0, φ1) =

1

μ

N∑j=1

[usjφ

s+1j − us+1

j φsj

],

or equivalently

Ls(φ0, φ1) =

1

μ

(us, φs+1

)RN +

1

μ

(us+1,−φs

)RN ,(4.6)

for every solution {φn} of the discrete problem (3.24), where the functional Ls: RN ×

RN → R is such that

Ls(φ0, φ1) =

1

μ

N∑j=1

[u0jφ

1j − u1

jφ0j

]− Δt

s∑n=1

vnΔx

[φnN

Δx

].

The projection Παun may be characterized by the same variational formulation

(4.6), with the only difference being that the test functions in (4.6) are solutions of(3.24) in the class Cα(Δx) (3.31).

Remark 4.1. Identity (4.4) is equivalent, by (4.5), to

Δt

M∑n=1

vnΔx

[φnN

Δx

]=

1

μ

N∑j=0

[u0jφ

1j − u1

jφ0j

],(4.7)

where (φ0, φ1) are the initial data corresponding to the solution φn ∈ Cα(Δx) of thediscrete system (3.24).

Now let Δx = 1/q, Δt = μ/q, N = q− 1, for some q ∈ N and μ < 1. We have thefollowing uniform (with respect to (Δt,Δx) → (0, 0)) partial controllability property.

Theorem 4.2. Let 0 < μ < 1 and let us fix an arbitrary value of the filteringparameter 0 < α < 1. For every T > T (α) = 2π/γα, the system (3.24) is partiallycontrollable on �2×�

−1 with controls vnΔt ∈ RM when M = [Tq/μ−1]. Moreover, the

controls of minimal norm are uniformly bounded with respect to Δt. More precisely[Δt

M∑n=0

|vnΔx|2

]1/2≤ C‖(u1,−u0)‖�−1×�2 ,(4.8)

where C = C(T, γα) > 0 is a constant independent of Δt ∈ (0, 1).Proof. Let (φn) ∈ Cα(Δx) be the solution of (3.24) with initial data (φ0, φ1) ∈

Sα × Sα and define the convex quadratic functional JΔx : RN × R

N → R, by

JΔx(φ0, φ1) =Δt

2

M∑n=0

∣∣∣∣φnN

Δx

∣∣∣∣2

− 1

μ2Δx

N∑j=0

(u0j

φ1j − φ0

j

Δt−

u1j − u0

j

Δtφ0j

).(4.9)

For every φn ∈ Cα(Δx) we have∣∣∣∣∣∣N∑j=0

(u0jφ

1j − u1

jφ0j

)∣∣∣∣∣∣ =∣∣(Παu

1, φ0)

RN

∣∣+ ∣∣(Παu0,−φ1

)RN

∣∣≤ ‖Παu

1‖�−1‖φ0‖�1 + ‖Παu0‖�2‖φ1‖�2

≤ ‖Πα(u1,−u0)‖�−1×�2‖(φ0, φ1)‖�1×�2 .

(4.10)


According to (4.10) and the direct observability inequality (the right-hand side termin (3.39)) we deduce that JΔx is continuous.

On the other hand, according to the observability inequality (3.37), JΔx is uni-formly coercive in Cα(Δx),

∣∣JΔx(φ0, φ1)∣∣ ≥ ∥∥(φ0, φ1)

∥∥�1×�2

[C1(T, γα)

∥∥(φ0, φ1)∥∥

�1×�2− ‖Πα(u1,−u0)‖�−1×�2

].

(4.11)

Thus, there exists a unique minimizer (φ0, φ1) of JΔx,

JΔx(φ0, φ1) = min(φ0,φ1)∈Sα×Sα

JΔx(φ0, φ1).

Let φn ∈ Cα(Δx) be the solution of the adjoint problem (3.24) with this minimizeras initial datum.

The pair (φ0, φ1) satisfies the Euler–Lagrange equation

Δt

M∑n=0

φnN

Δx

φnN

Δx=

1

μ

N∑j=0

[u0jφ

1j − u1

jφ0j

],(4.12)

for every initial data (φ0, φ1) ∈ Sα × Sα associated to the solution φn ∈ Cα(Δx) of

(3.24). Therefore, according to (4.7), the control we were looking for is vnΔx = φnN/Δx.

To conclude the proof we check the uniform boundedness of the controls vnΔx. Wehave

JΔx((φ0, φ1)) ≤ JΔx(0, 0) = 0,

and, by (4.10), this implies

Δt

2

M∑n=0

∣∣∣∣∣ φnN

Δx

∣∣∣∣∣2

≤ ‖Πα(u1,−u0)‖�−1×�2‖(φ0, φ1)‖�1×�2 .(4.13)

The discrete energy E0 of a solution φn of (3.24) with initial data (φ0, φ1) satisfies

E0 =1

2‖(φ0, φ1)‖2

�1×�2 .

Now, using the Fourier development (3.32) of the solution φn and applying theobservability inequality (3.37) we get

‖(φ0, φ1)‖2�1×�2 = 2E0 ≤ 1

cos2 NαπΔx2 C1(T, γα,Δt)

Δt

M∑n=0

∣∣∣∣∣ φnN

Δx

∣∣∣∣∣2

≤ 1

cos2 απ2 C1(T, γα)

ΔtM∑n=0

∣∣∣∣∣ φnN

Δx

∣∣∣∣∣2

.

(4.14)

Therefore, in (4.13) we obtain⎡⎣Δt

M∑n=0

∣∣∣∣∣ φnN

Δx

∣∣∣∣∣2⎤⎦

1/2

≤ 1√cos2 απ

2 C1(T, γα)‖Πα(u1,−u0)‖�−1×�2 ,(4.15)


and then, the discrete controls vnΔx = φnN/Δx satisfy

[Δt

M∑n=0

|vnΔx|2

]1/2≤ C(T, γα)‖Πα(u1,−u0)‖�−1×�2 ,(4.16)

as stated above.Remark 4.3. Note that, with the notations (3.32), the controls (vnΔx) are of the

form

vnΔx = − μ

Δt

Nα∑k=−Nα

cos(kπ) sin(kπΔt/μ)akeiλknΔt,(4.17)

where (ak)k are the Fourier coefficients of the solution φn ∈ Cα(Δx) of the adjoint

problem (3.24), with initial data (φ0, φ1) being the minimizer of the functional JΔx.Now we show the convergence of the controls vnΔx of the discrete system (3.23) to

the HUM control of the continuous one (3.1), as Δt,Δx → 0.Given an initial state (u0, u1) ∈ L2(0, 1) × H−1(0, 1) of the continuous system

(3.1), we develop it in Fourier series

(u0, u1) =

∞∑k=1

(ck, dk)ϕk(x),(4.18)

with

∑k∈N

[|ck|2 +

∣∣∣∣ dkkπ∣∣∣∣2]< ∞.(4.19)

We now construct the initial states for the discrete system (3.23) by setting

(u0, u1) =

N∑k=1

(ck, ck cos(λknΔt) +dkλk

sin(λknΔt))ϕk,(4.20)

with λk given by (3.27). They may be rewritten as

(u0, u1) =

∞∑k=1

(cNk , cNk cos(λknΔt) +dNkλk

sin(λknΔt))ϕk,(4.21)

where

cNk = ckχN (k), dNk = dkχN (k),

χN being the characteristic function of the set {1, . . . , N}.In view of Theorem 4.2, there exists a HUM control (vnΔx) for the discrete system

(3.23), satisfying (4.4), with initial data (4.20).Let us now prove that the sequence (vnΔx) converges (in a sense to be more precise

below) to v ∈ L2(0, T ), which is the HUM control for system (3.1) with initial data(4.18).

To better analyze the convergence of controls, we define the continuous extensionof the discrete controls by setting

vΔx(t) = − μ

Δt

∑k∈Z

cos(kπ) sin(kπΔt/μ)akeiλkt,


where ak are taken to be zero for |k| > αN . This function, when restricted to themesh, coincides with (vnΔx) (recall that vnΔx is given by (4.17)).

The following convergence result holds.Theorem 4.4. Let μ = Δt/Δx ≤ 1. Consider M, N as in Theorem 4.2. Fix

(u0, u1) ∈ L2(0, 1) × H−1(0, 1) and consider the continuous and discrete controls vand vΔx as above, with the filtering parameter α ∈ (0, 1) and T > Tα. Then,

vΔx(·) → v(·) strongly in L2(0, T ) as Δt → 0.(4.22)

Proof (Sketch of the proof ). In view of (4.8) it is easy to see that

Δt

M∑n=0

|vnΔx|2 ≤ C,(4.23)

and therefore ∫ T

0

|vΔx|2 dt ≤ C.(4.24)

Then, up to the extraction of a subsequence that we still denote by {vΔx}Δt, wehave

vΔx(t) ⇀ v(t) in L2(0, T ) as Δt → 0.(4.25)

By a Γ-convergence argument it can also be seen that the limit v is given by

v(t) = −∂xφ(1, t),(4.26)

where φ is the solution of the adjoint problem (3.3) with initial data (φ0, φ1) ∈H1

0 (0, 1) × L2(0, 1), the unique minimizer of the functional

J(φ0, φ1) =1

2

∫ T

0

| ∂xφ(1, t) |2 dt−∫ 1

0

u0φ1− < u1, φ0 >−1,1(4.27)

in the energy space H10 (0, 1) × L2(0, 1). By taking limits in (4.7) and thanks to the

construction of the initial data to be controlled for the discrete system we obtain

0 =

∫ 1

0

[u1(x)φ0(x) − u0(x)φ1(x)dx

]+

∫ T

0

v(t)∂xφ(1, t)dt(4.28)

and this latter condition is equivalent to the fact that v, the limit in (4.25), is acontrol for system (3.1), driving the initial data (u0, u1) to rest; i.e., v ∈ L2(0, T ) isthe control of minimal L2-norm.

The limit v being identified in a unique way, we deduce that the whole sequencevΔx converges.

Moreover, by the hypotheses of Theorem 4.4, the linear term of the discretefunctional JΔx in (4.9) converges to the linear term of the functional defined in (4.27).Therefore, proving (4.22) is equivalent to proving that

JΔx(φ0Δx, φ

1Δx) → J(φ0, φ1), as Δx → 0,

where (φ0Δx, φ

1Δx) ∈ Sα × Sα minimizes (4.9) and (φ0, φ1) ∈ H1

0 (0, 1) × L2(0, 1) min-imizes (4.27). Indeed, taking into account the convergence of the linear terms in


this functional, and the structure of the functionals (4.9) and (4.27), we deduce theconvergence of the norms of the controls that, together with the weak convergence,ensure strong convergence.

Thus, the controls vΔx and the controlled discrete solutions uΔx converge to thecontrol and the controlled solution of the wave equation (3.1). It is important to notethat the projections of the solutions of the controlled system end up covering thewhole range of frequencies so that, in the limit, we recover the exact controllabilityproperty (3.2) of the continuous wave equation.

The details of the several steps of the proof are given in [19] and we omit themfor brevity.

5. Discrete Ingham inequalities and dispersion diagrams. In this sectionwe discuss the observability results obtained in section 3 applying discrete Inghaminequalities in connection with the dispersion diagrams of the equations and numericalschemes under consideration. We also discuss the optimality of these results. First ofall, we introduce and recall some classical concepts and notations.

Any time-dependent scalar, linear partial differential equation with constant co-efficients admits plane wave solutions

φ(x, t) = ei(ωt−ξx), ξ ∈ R, ω ∈ C,(5.1)

where ξ is the wave number and ω is the frequency . The relationship

ω = ω(ξ)(5.2)

is known as the dispersion relation for the equation.Any individual “monochromatic wave” (involving only one Fourier component)

of (5.1) moves at the phase velocity

c(ξ, ω) =ω(ξ)

ξ.(5.3)

When one superimposes two waves with nearby propagation velocities, there ap-pear wave packets which can propagate with different velocities. The energy of wavepackets propagates at the so-called group velocity

C(ξ, ω) =dω(ξ)

dξ.(5.4)

In general, the dispersion relation for a partial differential equation is a polynomialrelation between ξ and ω, while a discrete model amounts to a trigonometric approx-imation.

• Continuous problem. For the continuous wave equation (3.3) we have ω(ξ) = ξand therefore c(ξ) = C(ξ) = 1.

• Semi-discrete problem. For the semi-discrete scheme (3.9) the dispersion relationis

ω(ξ) =2

Δxsin

ξΔx

2, ξ ∈

[− π

Δx,π

Δx

].(5.5)

Note that, at the semi-discrete level, each dispersion relation is 2π/Δx–periodic in ξ,and it is natural to take ξ ∈ [−π/Δx, π/Δx] as a fundamental domain.


The phase velocity is in this case

c(ξ, ω) =2

ξΔxsin

ξΔx

2.(5.6)

The corresponding group velocity is

C(ξ, ω) =dω(ξ)

dξ= cos

ξΔx

2.(5.7)

• Discrete problem. The same analysis can be developed for fully discrete schemes.Considering numerical plane waves φn

j = ei(ωnΔt−ξjΔx), for system (3.24), one obtainsthe dispersion relation

ω(ξ) =2

Δtarcsin

(Δt

Δxsin

ξΔx

2

).(5.8)

It is 2π/Δx-periodic in ξ and 2π/Δt-periodic in ω.• When Δt = Δx we obtain

ω(ξ) = ξ.(5.9)

This case is particularly interesting since (5.9) coincides with the dispersion relationfor the continuous weave equation. In this case, c(ξ, ω) = C(ξ, ω) = 1 and the discretewaves propagate at a constant velocity identically equal to one, like in the continuouscase. But, as we shall see, this is a completely exceptional situation.

• When μ < 1, the phase velocity is given by

c(ξ, ω) =2

ξΔtarcsin

(Δt

Δxsin

ξΔx

2

)(5.10)

and the group velocity is

C(ξ, ω) =dω(ξ)

dξ=

cos ξΔx2√

1 −(

ΔtΔx sin ξΔx

2

)2.(5.11)

For Δt = 0 the phase and group velocities in (5.10) and (5.11), which depend on ξ,coincide with those of the semi-discrete case (5.6) and (5.7), respectively, as expected.

Note that, as Δx → 0, for all ξ we have

C(ξ, ω) ≤cos ξΔx

2√1 −(

ΔtΔx

)2 → 0

when ξ = π/Δx.In Figures 1–4 we describe the evolution of the group velocity diagrams starting

with the semi-discrete case (μ = 0) up to μ = 1, for fixed Δx = 0.001.In general, any discrete dynamics generates spurious high-frequency oscillations

that do no exist at the continuous level [23, 25]. Moreover, the interaction of waveswith the grid produces a dispersion phenomenon and the velocity of propagation ofthese high frequency numerical waves may converge to zero when the mesh-size tendsto zero. These spurious oscillations weakly converge to zero. Consequently, their


existence is compatible with the convergence of the numerical scheme for solving theinitial-value problem. However, when we are dealing with the exact controllabilityor observability problems, a uniform time for the control of all numerical waves isneeded. Since the velocity of propagation of some high frequency numerical wavesmay tend to zero as the mesh becomes finer and finer, uniform observability andtherefore controllability properties of the discrete model may fail for all T > 0.

According to Theorem 2.1, the uniform gap between two consecutive eigenvaluesis a sufficient (and actually also necessary) property for uniform (with respect to Δxand Δt) observability. On the other hand, the group velocity is the derivative ofthe eigenfrequencies λk and the spectral gap is, as we have seen, λk+1 − λk. Bothmagnitudes are similar, and they become closer as Δx → 0.

Thus, to efficiently observe at the point x = 1 a wave packet concentrated to theleft of x = 1 that moves to the left (in the space variable) as t increases, and bouncesback at x = 0 to eventually reach the observation point x = 1, the time needed is

T ≥ 2/minξ

{C(ξ, ω)}.(5.12)

In the continuous case, (5.12) reduces to the well-known condition for observabilityT ≥ 2 and it is uniform for all the frequencies. The minimal time T = 2 is the oneone obtains in view of Ingham’s theorem (1.2) because the gap is γ = π in this case.

For the semi-discrete case, the observation time is

T ≥ 2/minξ

(cos(ξΔx/2)).(5.13)

But minξ(cos(ξΔx/2)) is of the order of Δx, the same order as we have obtained in(3.18) for the spectral gap for the highest frequencies. Consequently, the observationtime (5.13) diverges, T → ∞, as Δx → 0.

These facts confirm the necessity of filtering the high frequencies. Relation (5.13)shows that the time grows with the high frequencies, in the points where cos ξΔx/2 ∼0 (ξ ∼ π/Δx) and the same result is obtained applying the Ingham inequality.

For the fully discrete problem (3.24) the time needed for observation is

T ≥ maxξ

2

√1 −(

ΔtΔx sin ξΔx

2

)2

cos ξΔx2

.(5.14)

Passing to the limit in (5.14) as Δt → 0 for fixed Δx, one obtains the same time as inthe semi-discrete case (5.13). The observation time grows with the high frequencies,except for the very particular case Δt = Δx, where the time obtained in the previoussection, using the orthogonality of the time exponentials, is T = 2, which coincideswith the observation time given by the group velocity (5.14).

Summarizing, when 0 < μ < 1, the sequence of eigenvalues has no uniform gapand the observability time (5.14) tends to infinity. Therefore, as in the semi-discretecase, a suitable filtering of the spurious numerical high frequencies is necessary. The-orem 2.1 provides a sharp result in this direction and its main result coincides withthe predictions one may do in view of the structure of the dispersion diagram.

6. Proof of the discrete Ingham inequality. The proof of Theorem 2.1 usesin an essential way some classical properties of the discrete Fourier transform. Werecall these properties in subsection 6.1 following [23].


Fig. 1. Group velocity for the semi-discrete (–) and discrete (- -) cases with μ = Δt/Δx = 0.1(left), μ = Δt/Δx = 0.3 (middle), μ = Δt/Δx = 0.5 (right), Δx = 0.001.

Fig. 2. Group velocity for the semi-discrete (–) and discrete (- -) cases with μ = Δt/Δx = 0.9(left), μ = Δt/Δx = 0.999 (middle), μ = Δt/Δx = 1 (right), Δx = 0.001.

6.1. The discrete Fourier transform. Let h > 0 be a real number and let. . . , x−1, x0, x1, . . . be defined by xj = jh. Thus {xj} = hZ, where Z is the set ofintegers. The �2h-norm of a discrete function {vj} is defined as

‖v‖h =

⎡⎣h ∞∑

j=−∞|vj |2⎤⎦

1/2

.

We denote by l2h the Hilbert space l2h = {v : ‖v‖h < ∞}, the space of discrete functionsof finite ‖ · ‖h norm.

For any v ∈ �2h, the discrete Fourier transform of v is the function v defined by

v(ξ) = h∞∑

j=−∞e−iξxjvj , ξ ∈

[−π

h,π

h

].(6.1)

This can be viewed as a discrete approximation of the continuous Fourier transform

u(ξ) =

∫ ∞

−∞e−iξxu(x)dx, ξ ∈ R,

if u = u(x) is a sufficiently smooth function such that u(xj) = vj .A priori, the sum in (6.1) defines a function v(ξ) for all ξ ∈ R. The function v(ξ)

is 2π/h periodic on R and therefore we analyze it only for ξ ∈ [−π/h, π/h] to avoidaliasing.

Let us recall a standard definition. A function u defined on R is said to havebounded variation if there is a constant M such that for any finite m and any points


Fig. 3. Dispersion relation for the continuous (-.-), semi-discrete (- -) and discrete (–) caseswith μ = Δt/Δx = 0.1 (left) μ = Δt/Δx = 0.3 (middle), μ = Δt/Δx = 0.5 (right), Δx = 0.001.

Fig. 4. Dispersion relation for the continuous (-.-), semi-discrete (- -) and discrete (–) caseswith μ = Δt/Δx = 0.9 (left), μ = Δt/Δx = 0.999 (middle), μ = Δt/Δx = 1 (right), Δx = 0.001.

x0 < x1 < · · · < xm,

m∑j=1

| u(xj) − u(xj−1) |≤ M.

Now we give a fundamental result (see [23, p. 96]) which describes the effect ofdiscretization in the Fourier transform.

Theorem 6.1. Suppose that u ∈ L2(R) is a sufficiently smooth function definedon R and let v ∈ �2h be the discretization obtained by sampling u at the grid points xj,i.e., u(xj) = vj.

Then, if u has p − 1 continuous derivatives in L2(R) for some p ≥ 1 and a pthderivative in L2 that has bounded variation, it follows that

|v(ξ) − u(ξ)| = o(hp+1), when h → 0,(6.2)

uniformly on ξ ∈ [−π/h, π/h] .Proof. Since u is continuous, apply Poisson formula

v(ξ) =∞∑

j=−∞u(ξ + 2πj/h), ξ ∈ [−π/h, π/h] .

Thus, for every u ∈ L2(R) and v ∈ �2, we obtain

v(ξ) − u(ξ) =

∞∑j=−∞

u(ξ + 2πj/h) − u(ξ) =

∞∑j=1

[u(ξ + 2πj/h) + u(ξ − 2πj/h)] ,

with u and v the Fourier transforms of u and v, respectively.


If u verifies the hypothesis of Theorem 6.1, then

|u(ξ)| ≤ C1 |ξ|−p−1, when ξ → ∞

for some constant C1. Therefore

|v(ξ) − u(ξ)| ≤ C1

∞∑j=1

(jπ/h)−p−1 = C2hp+1

∞∑j=1

j−p−1.

For every p ≥ 1 this sum converges, which implies (6.2), as required.

6.2. The discrete Fourier transform of Ingham’s cut-off function. Westudy some general properties of a discrete Fourier transform that we shall use in theproof of the discrete Ingham inequality.

Given M ∈ N and T > 0 we consider the function g : R → R

g(t) = sin

(tπ

T

)χ(0,T ),(6.3)

where χ(0,T ) is the characteristic function of the interval (0, T ). Function (6.3) isprecisely the same that Ingham [9] used in the proof of the continuous inequality(1.2). Its Fourier transform G : R →R is

G(τ) =

∫ ∞

−∞g(t)eitτdt = −2 cos

Tτ

2e

iTτ2

πT

(T 2τ2 − π2).(6.4)

We define the restriction of g to the grid

. . . < t−1 < t0 = 0 < t1 < · · · < tM+1 = T < . . . ,

with tn = nΔt, i.e.,

h(nΔt) = g(tn) = sin(nΔtπ/T )χM+1(n),

χM+1 being the characteristic function of the set {0, . . . ,M + 1}.For any τ ∈ R we define the discrete Fourier transform of the discrete function h

H(τ) := Δt∞∑

n=−∞h(nΔt)einΔtτ(6.5)

for all τ ∈ [−π/Δt, π/Δt].Lemma 6.2. For all Δt, T > 0 and k ∈ Z we have

H(τ) = −Δt cos Tτ

2 eiTτ2 sin Δtπ

T

2 sin(Δt2T (Tτ + π)) sin(Δt

2T (Tτ − π))(6.6)

for any τ �= (2kπ)/Δt± π/T with k ∈ Z and

H

(2kπ

Δt± π

T

)= ∓ T

2i, k ∈ Z.(6.7)


The function H defined in (6.5) is continuous and

limτ→ 2kπ

Δt ± πT

H(τ) = ∓ T

2i.(6.8)

Proof (Proof of Lemma 6.2). We divide the proof into two steps: first, we provethat the explicit expression (6.5) of H is (6.6) and then we study the continuity of H.

• Step 1. From the definition of the function H for all τ �= 2kπ/Δt± π/T, k ∈ Z, wehave

H(τ) = ΔtM∑n=0

sinnπΔt

TeinΔtτ = Δt

M∑n=0

einπΔt

T − e−inπΔt

T

2ieinΔtτ .

Hence

H(τ) =Δt

2i

M∑n=0

einΔt

T (Tτ+π) − Δt

2i

M∑n=0

einΔt

T (Tτ−π).(6.9)

In order to obtain identity (6.5), it is useful to prove it only for any |τ | < 2kπ/Δt−π/Twith k = 1. Then, taking the periodicity properties of the complex exponentials intoaccount, it is easy to obtain the same result for all k ∈ Z, τ �= 2kπ/Δt± π/T .

The first term on the right-hand side of (6.9) is

M∑n=0

einΔt

T (Tτ+π) =e

i(M+1)ΔtT (Tτ+π) − 1

eiΔtT (Tτ+π) − 1

=e

i(M+1)ΔtT (Tτ+π) − 1

cos(ΔtT (Tτ + π)) + i sin(Δt

T (Tτ + π)) − 1

=ei(Tτ+π) − 1

1 − 2 sin2(Δt2T (Tτ + π)) + 2i sin(Δt

2T (Tτ + π)) cos(Δt2T (Tτ + π)) − 1

=−eiTτ − 1

2i sin(Δt2T (Tτ + π))(cos(Δt

2T (Tτ + π)) + i sin(Δt2T (Tτ + 1π)))

=−(eiTτ + 1)e−

iΔt2T (Tτ+π)

2i sin(Δt2T (Tτ + π))

.

(6.10)

For the second one we have

M∑n=0

einΔt

T (Tτ−π) =e

i(M+1)ΔtT (Tτ−π) − 1

eiΔtT (Tτ−π) − 1

=e

i(M+1)ΔtT (Tτ−π) − 1

cos ΔtT (Tτ − π) + i sin Δt

T (Tτ − π) − 1

=ei(Tτ−π) − 1

−2 sin2 Δt2T (Tτ − π) + 2i sin Δt

2T (Tτ − π) cos πΔt2T (Tτ − π)

=−eiTτ − 1

2i sin Δt2T (Tτ − π)e

iΔt2T (Tτ−π)

=−(eiTτ + 1)e−

iΔt2T (Tτ−π)

2i sin Δt2T (Tτ − π)

.

(6.11)


Substituting (6.10) and (6.11) into (6.9) we obtain

H(τ)

=−Δt

4

[(eiTτ + 1)e−

iΔt2T (Tτ−π)

sin Δt2T (Tτ − π)

− (eiTτ + 1)e−iΔt2T (Tτ+π)

sin(Δt2T (Tτ + π))

]

=−Δt(eiTτ + 1)

4

(cos Δt

2T (Tτ − π) − i sin Δt2T (Tτ − π)


−cos Δt

2T (Tτ + π) − i sin(Δt2T (Tτ + π))


)

=−Δt(eiTτ + 1)

4

(cos Δt

2T (Tτ − π)


−cos Δt

2T (Tτ + π)


)

=−Δt(eiTτ + 1)

4

(cos Δt

2T (Tτ − π) sin Δt2T (Tτ + π) − cos Δt

2T (Tτ + π) sin Δt2T (Tτ − π)

sin Δt2T (Tτ − π) sin(Δt

2T (Tτ + π)

)

=−Δt(eiTτ + 1)

4 sin Δt2T (Tτ − π) sin πΔt

2T (Tτ + π)sin

πΔt

T.

(6.12)

Therefore, applying Euler’s formula in (6.12) we obtain the following expression forH:

H(τ) =−Δt (cos(Tτ) + i sin(Tτ) + 1) sin πΔt

T

4 sin Δt2T (Tτ − π) sin(Δt

2T (Tτ + π))

=−Δt

(2 cos2 Tτ

2 − 1 + 2i sin Tτ2 cos Tτ

2 + 1)sin πΔt

T


2T (Tτ + π))

=−2Δt cos Tτ

2

(cos Tτ

2 + i sin Tτ2

)sin πΔt

T


2T (Tτ + π)

=−Δt cos Tτ

2 eiTτ2 sin πΔt

T

2 sin Δt2T (Tτ − π) sin(πΔt

2T (Tτ + π)).

(6.13)

Moreover, if τ = 2kπ/Δt + π/T, with k ∈ Z, using the definition (6.5) of H, wededuce that

H(τ) = ΔtM∑n=0

sinnπΔt

TeinΔt( 2kπ

Δt + πT )

= Δt

M∑n=0

einΔtπ

T − e−inΔtπ

T

2ie2kπine

inΔtπT =

Δt

2i

M∑n=0

e2inΔtπ

T − Δt

2i

M∑n=0

1

=Δt

2i

e2i(M+1)Δtπ

T − 1

e2iΔtπ

T − 1− Δt

2i(M + 1) =

Δt

2i

e2πi − 1

e2iΔtπ

T − 1− T

2i= − T

2i.


For every τ = 2kπ/Δt− π/T, with k ∈ Z,

H(τ) = Δt

M∑n=0

sinnπΔt

TeinΔt( 2kπ

Δt − πT )

= Δt

M∑n=0

einΔtπ

T − e−inΔtπ

T

2ie2kπine

−inΔtπT

=Δt

2i

M∑n=0

1 − Δt

2ie

−2i(M+1)ΔtπT

M∑n=0

e2inΔtπ

T

=Δt

2i(M + 1) − Δt

2i

e2i(M+1)Δtπ

T − 1

e2iΔtπ

T − 1=

T

2i− Δt

2i

e2πi − 1

e2iΔtπ

T − 1=

T

2i.

• Step 2. It is easy to see that H is continuous on R\{τ : τ =2kπ/Δt± π/T}, k ∈ Z.We now study the continuity of H at the singularities τ = 2kπ/Δt± π/T . For everyτ → 2kπ/Δt± π/T, we have τ = 2kπ/Δt± π/T + επ, with ε → 0.

1. The case τ = 2kπ/Δt + π/T + επ with ε → 0.Using the definition of H we have

H

(2kπ

Δt+

π

T+ επ

)= Δt

M∑n=0

sinnπΔt

TeinΔtπ( 2k

Δt+ 1T +ε)

= Δt

M∑n=0

einΔtπ

T − e−inΔtπ

T

2ieinπ2keinΔtπ( 1

T +ε).

Hence,

H

(2kπ

Δt+

π

T+ επ

)=

Δt

2i

M∑n=0

einΔtπ

T (2+Tε) − Δt

2i

M∑n=0

einΔtπε.(6.14)

For every |x| < 2T/Δt, according to the classical formula for the sum of ageometric series, the following identity holds:

M∑n=0

einΔtπ

T x =e

i(M+1)ΔtπT x − 1

eiΔtπ

T x − 1=

eiπx − 1

eiΔtπ

T x − 1.(6.15)

For the first sum entering on the right-hand term of (6.14) we have

M∑n=0

einΔtπ

T (2+Tε) =e2πieiTπε − 1

eiΔtπ

T (2+Tε) − 1=

eiTπε − 1

eiΔtπ

T (2+Tε) − 1.(6.16)

For the second sum on the right-hand term (6.14), using (6.15) and the factthat Δtπ/T (2 + Tε) < 2π, we have

M∑n=0

einΔtπε =eiTπε − 1

eiΔtπε − 1.(6.17)


Finally, replacing (6.16) and (6.17) in (6.14) and taking the limit ε → 0 weobtain

limε→0

H

(2kπ

Δt+

π

T+ επ

)=

Δt

2ilimε→0

eiTπε − 1

eiΔtπ

T (2+Tε) − 1− Δt

2ilimε→0

eiTπε − 1

eiΔtπε − 1

= 0 − Δt

2ilimε→0

eiTπε − 1

eiΔtπε − 1= −Δt

2ilimε→0

iπTeiTπε

iΔtπeiΔtπε= − T

2i.

2. The case τ = 2kπ/Δt− π/T + επ with ε → 0.We have

H

(2kπ

Δt− π

T+ επ

)= Δt

M∑n=0

sinnπΔt

TeinΔtπ( 2k

Δt−1T +ε)

= ΔtM∑n=0

einΔtπ

T − e−inΔtπ

T

2ieinπ2keinΔtπεe−

inΔtπT .

Hence

H

(2kπ

Δt− π

T+ επ

)=

Δt

2i

M∑n=0

einΔtπε − Δt

2i

M∑n=0

einΔtπ

T (Tε−2).(6.18)

By (6.17), for the first sum entering on the right-hand term of (6.18) we have

M∑n=0

einΔtπε =eiTπε − 1

eiΔtπε − 1.(6.19)

Moreover, in (6.17), applying the identity (6.15) for the second sum on theright-hand term of (6.18), we obtain

M∑n=0

e−inΔtπ

T (2−Tε) = e−i(M+1)Δtπ

T (2−Tε)M∑n=0

einΔtπ

T (2−Tε)

= = e−2πieiTπε ei(M+1)Δtπ

T (2−Tε) − 1

eiΔtπ

T (2−Tε) − 1= eiTπε e

2πie−iTπε − 1

eiΔtπ

T (2−Tε) − 1

=1 − eiTπε

eiΔtπ

T (2−Tε) − 1→ 0, ε → 0.(6.20)

Hence,

limε→0

H

(2kπ

Δt− π

T+ επ

)=

T

Δt

Δt

2i=

T

2i.

This concludes the proof of Lemma 6.2.Remark 6.3. For τ = 0 in (6.6) we have

H(0) = −Δt sin Δtπ

T

2 sin πΔt2T sin

(−πΔt

2T

) = −Δt sin Δtπ

T

−2 sin2 πΔt2T

=Δt2 sin Δtπ

2T cos Δtπ2T

2 sin2 πΔt2T

=Δt cos Δtπ

2T

sin πΔt2T

= Δt cotΔtπ

2T.

(6.21)


Taking the limit Δt → 0 in (6.6), for every τ fixed, we obtain

limΔt→0

H(τ) = −2 cosTτ

2e

iTτ2

πT

T 2τ2 − π2= G(τ)(6.22)

and this is the classical Fourier transform of g given by (6.4).

6.3. Proof of Theorem 2.1. This section is devoted to the proof of the mainresult of this paper. The proof of the discrete inequality (2.3) follows the strategy usedin [26, (pp. 162–163)] to prove the classical Ingham inequality (1.2).

Proof (Proof of the first (so-called inverse) inequality in (2.3)). We prove the firstinequality in (2.3), namely,

C1(Δt, T, γ)

N∑k=−N

|ak|2 ≤ Δt

M∑n=0

∣∣∣∣∣N∑

k=−N

akeinΔtλk

∣∣∣∣∣2

.

Taking into account that sin(nΔtπ/T ) ≤ 1, we have

Δt

M∑n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

≥ Δt

M∑n=0

sinnΔtπ

T

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

= Δt

M∑n=0

sinnΔtπ

T

∑k

∑l

akaleinΔt(λk−λl).

The function H defined by (6.6) is continuous, hence

ΔtM∑n=0

sinnΔtπ

T

∑k

∑l

akaleinΔt(λk−λl)

=∑k

∑l

akalH(λk − λl) = H(0)∑k

|ak|2 +∑k

∑l,l =k

akalH(λk − λl)

≥ H(0)∑k

|ak|2 −1

2

∑k

∑l,l =k

(|ak|2 + |al|2

)|H(λk − λl)|

= H(0)∑k

|ak|2 −∑k

|ak|2∑l,l =k

|H(λk − λl)| .

(6.23)

In the last term in (6.23) we have

∑l,k =l

|H(λk − λl)| =∑l,k �=l

|λk−λl|≤πΔt

|H(λk − λl)| +∑l,k �=l

|λk−λl|>πΔt

|H(λk − λl)| .(6.24)

Moreover, the function H is periodic with period 2π/Δt. Consequently, for everyk, l ∈ Z with π/Δt < |λk − λl| < 2π/Δt, there exist mk,l ∈ [−π/Δt, π/Δt] such that|mk,l| = 2π/Δt− |λk − λl| with the property H(λk − λl) = H(mk,l). Therefore, usingthis periodicity property and applying (6.2) from Theorem 6.1 and (6.24) in (6.23),


we obtain

ΔtM∑n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

≥ H(0)∑k

|ak|2 −∑k

|ak|2

⎛⎜⎝ ∑

l,k �=l|λk−λl|≤π/Δt

|H(λk − λl)| +∑l,k �=l

|mk,l|≤π/Δt

|H(mk,l)|

⎞⎟⎠

≥ H(0)∑k

|ak|2 −∑k

|ak|2

⎛⎜⎝ ∑

l,k �=l|λk−λl|≤π/Δt

|G(λk − λl)| +∑l,k �=l

|mk,l|≤π/Δt

|G(mk,l)|

⎞⎟⎠

+CN(Δt)2).

(6.25)

On the other hand, as pointed out in [26, p. 162], for every sequence {λk} satis-fying the gap condition (2.1), the function G satisfies

N∑l =k,l=−N

|G(λk − λl)| ≤ 2πT

∞∑l=−∞,l =k

1

T 2 (λk − λl)2 − π2

≤ 2πT

∞∑l=−∞,l =k

1

T 2γ2 (k − l)2 − π2

= 4πT∑r≥1

1T 2γ2

4π2 4π2r2 − π2≤ 16π

Tγ2

∑r≥1

1

4r2 − 1

=8π

Tγ2

∑r≥1

(1

2r − 1− 1

2r + 1

)=

8π

Tγ2.

(6.26)

Further, for the terms of the sequence {λk} satisfying π/Δt <| λk − λl |< (2π −(Δt)p)/Δt, (and then, (Δt)p−1 ≤ |mk,l| ≤ π/Δt), k �= l, we have

|G(mk,l)| ≤ 2πT1

T 2 (mk,l)2 − π2

= 2πT1

T 2(

2πΔt − (λl − λl)

)2 − π2

≤ 2πT1

T 2(

2πΔt −

2π−(Δt)p

Δt

)2

− π2

= 2πTΔt2

T 2(Δt)2p − π2Δt2

and it follows that

N∑l =k,l=−N

|G(mk,l)| ≤ (NΔt)2πT(Δt)1−2p

T 2 − π2(Δt)2−2p.(6.27)

Using the relations (6.26) and (6.27) in (6.25) we obtain

Δt

M∑n=0

sinnΔtπ

T

∑k

∑l

akaleinΔt(λk−λl)

≥ H(0)∑k

|ak|2 −∑k

|ak|2[

8π

Tγ2+ NCΔt2 + CNΔt(Δt)1−2p

](6.28)


when Δt → 0. For the function H(0) given by (6.21) we have limΔt→0

H(0) = 2T/π,

which is equivalent to 2T/π − θ ≤ H(0) ≤ 2T/π + θ, with θ → 0 when Δt → 0. In

order to ensure the positivity of all the coefficients |ak|2 in (6.28) it is necessary andsufficient to have

C1(Δt, T, γ) := H(0) − 8π

Tγ2− (NCΔt2 + CNΔt(Δt)1−2p) > 0,(6.29)

which is equivalent to

T 2 − Tπ

2(θ + ε1) −

4π2

γ2> 0,

where ε1 = NCΔt2 + CNΔt(Δt)1−2p, C > 0. This condition holds for every

T (Δt) > T0(Δt) =

π2 (ε1 + θ) +

√π2

4 (θ + ε1)2

+ 16π2

γ2

2:=

2π

γ+ ε(Δt)

with ε(Δt) = C(Δt + NΔt(Δt)1−2p). Hence, the inequality (2.3) holds with the con-stant C1(Δt, T, γ) defined by the relation (6.29) where δ1(Δt) = −(ε1 + θ).

Proof of the second (so-called direct) inequality. We now prove the inequality

Δt

M∑n=0

∣∣∣∣∣N∑

k=−N

akeinΔtλk

∣∣∣∣∣2

≤ C2(Δt, T, γ)

N∑k=−N

|ak|2 .

We have

Δt

M∑n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

= Δt

[M2 ]∑

n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

+ Δt

M∑n=[M

2 ]+1

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

.

(6.30)

Consider the first term on the right-hand side of (6.30),

Δt

[M2 ]∑

n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

= Δt

[M2 ]+[M+1

4 ]+1∑n=[M+1

4 ]+1

∣∣∣∣∣∑k

akei(n−[M+1

4 ]−1)Δtλk

∣∣∣∣∣2

= Δt

[M2 ]+[M+1

4 ]∑n=[M+1

4 ]+1

∣∣∣∣∣∑k

akei(n−[M+1

4 ]−1)Δtλk

∣∣∣∣∣2

+ Δt

∣∣∣∣∣∑k

akei[M

2 ]Δtλk

∣∣∣∣∣2

.

(6.31)

Using the properties of the entire part of a real number we have[M + 1

4

]≤ M + 1

4≤[M + 1

4

]+ 1,

[M

2

]+

[M + 1

4

]≤[3M + 1

4

],

[3M + 1

4

]≤ 3M + 1

4≤[3M + 1

4

]+ 1.


For every n ∈ N with

[M + 1

4

]+ 1 ≤ n ≤

[3M + 1

4

], we have

M + 1

4≤ n ≤ 3M + 1

4(6.32)

and

π

4≤ nπΔt

T≤ 3π

4,

due to the fact that (M + 1)Δt = T .Therefore, for every n ∈ N as in (6.32) we have sin(nπΔt/T ) ≥

√2/2 and

Δt

[M2 ]+[M+1

4 ]∑n=[M+1

4 ]+1

∣∣∣∣∣∑k

akei(n−[M+1

4 ]−1)Δtλk

∣∣∣∣∣2

+ Δt

∣∣∣∣∣∑k

akei[M

2 ]Δtλk

∣∣∣∣∣2

≤ 2Δt

[M2 ]+[M+1

4 ]∑n=[M+1

4 ]+1

sinnπΔt

T

∣∣∣∣∣∑k

akei(n−[M+1

4 ]−1)Δtλk

∣∣∣∣∣2

+ Δt

∣∣∣∣∣∑k

akei[M

2 ]Δtλk

∣∣∣∣∣2

≤ 2Δt

M∑n=0

sinnπΔt

T

∣∣∣∣∣∑k

akeinΔtλke−i([M+1

4 ]+1)Δtλk

∣∣∣∣∣2

+ Δt

∣∣∣∣∣∑k

akei[M

2 ]Δtλk

∣∣∣∣∣2

= 2Δt

M∑n=0

sinnπΔt

T

∑k

∑l

akaleinΔt(λk−λl)e−i([M+1

4 ]+1)Δt(λk−λl)

+Δt∑k

∑l

akale2i[M

2 ]Δt(λk−λl)

= 2∑k

∑l

akalH(λk − λl)e−i([M+1

4 ]+1)Δt(λk−λl) + Δt∑k

∑l

akalei[M

2 ]Δt(λk−λl)

= 2H(0)∑k

|ak|2 + 2∑k

∑l,l =k

akalH(λk − λl)e−i([M+1

4 ]+1)Δt(λk−λk)

+Δt∑k

|ak|2 + Δt∑k

∑l,l =k

akalei[M

2 ]Δt(λk−λk)

≤ 2H(0)∑k

|ak|2 +∑k

∑l,l =k

(|ak|2 + |al|2

)|H(λk − λl)| + Δt

∑k

|ak|2

+NΔt

2

∑k

∑l,l =k

(|ak|2 + |al|2

)

≤ 2H(0)∑k

|ak|2 + 2∑k

|ak|2∑l,l =k

|H(λk − λl)| + Δt∑k

|ak|2 + 2NΔt∑k

|ak|2 .

(6.33)


Using the same argument (6.2) as in the proof of the inverse inequality, for everyC > 0, we have

∑l,k =l

|H (λk − λl)| ≤N∑

l =k,l=−N

|G (λk − λl)| + CN(Δt)2,(6.34)

when Δt is small enough, with G the Fourier transform (6.4) satisfying (6.26) and(6.27).

Therefore, for every k,

2H(0)∑k

|ak|2 +∑k

∑l,l =k

(|ak|2 + |al|2

)|H(λk − λl)| + Δt

∑k

|ak|2

+NΔt

2

∑k

∑l,l =k

(|ak|2 + |al|2

)≤ 2Δt cot

Δtπ

2T

∑k

|ak|2

+2∑k

|ak|2(

8π

Tγ2+ NCΔt2 + CNΔt(Δt)1−2p + 2Δt

)+ 2NΔt

∑k

|ak|2 .

Hence

Δt

[M2 ]∑

n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

≤∑k

|ak|2(

2Δt cotΔtπ

2T+

16π

Tγ2+ ε(Δt)

),(6.35)

with ε(Δt) = 2NCΔt2 + 2CNΔt(Δt)1−2p + 2Δt + 2NΔt.

For the second right-hand term of (6.30) we have

ΔtM∑

n=[M2 ]+1

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

= Δt

M−[M+14 ]−1∑

n=[M2 ]−[M+1

4 ]

∣∣∣∣∣∑k

akei(n+[M+1

4 ]+1)Δtλk

∣∣∣∣∣2

= Δt

M−[M+14 ]−1∑

n=[M2 ]+1−[M+1

4 ]

∣∣∣∣∣∑k

akei(n+[M+1

4 ]+1)Δtλk

∣∣∣∣∣2

+ Δt∑k

∣∣∣akei([M2 ]+1)Δtλk

∣∣∣2 .

Taking into account that

M −[M + 1

4

]− 1 ≤ 3M + 1

4and

[M

2

]−[M + 1

4

]+ 1 ≥ M + 1

4,

for every n ∈ N, with (M + 1)/4 ≤ n ≤ (3M + 1)/4 we have sin(nπΔt/T ) ≥√

2/2.


Thus,

ΔtM∑

n=[M2 ]+1

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

≤ 2Δt

M−[M+14 ]−1∑

n=[M2 ]+1−[M+1

4 ]

sinnπΔt

T

∣∣∣∣∣∑k

akei(n+[M+1

4 ]+1)Δtλk

∣∣∣∣∣2

+Δt∑k


∣∣∣2

≤ 2Δt

M∑n=0

sinnπΔt

T

∣∣∣∣∣∑k

akeinΔtλkei([

M+14 ]+1)Δtλk

∣∣∣∣∣2

+ Δt∑k


∣∣∣2

= 2Δt

M∑n=0

sinnπΔt

T

∑k

∑l

akaleinΔt(λk−λl)ei([

M+14 ]+1)Δt(λk−λk)

+Δt∑k


∣∣∣2

and we obtain the estimate

Δt

M∑n=[M

2 ]+1

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

≤ 2H(0)∑k

|ak|2 + 2∑k

∑l,l =k

|ak|2 |H(λk − λl)| + 2NΔt∑k

|ak|2

≤(

2Δt cotΔtπ

2T+

16π

Tγ2+ NCΔt2 + CNΔt(Δt)1−2p + 2NΔt

)∑k

|ak|2 .

(6.36)

For the function H(0) defined by the relation (6.21) we have limΔt→0

H(0) = 2T/π.

From (6.30) and (6.35) we get

Δt

M∑n=0

∣∣∣∣∣∑k

akeinΔtλk

∣∣∣∣∣2

≤(

8T

π+

32π

Tγ2+ δ2(Δt)

)∑k

|ak|2 ,(6.37)

with

δ2(Δt) = 4ε(Δt) + θ,

(2T/π − θ ≤ H(0) ≤ 2T/π + θ, with θ → 0 when Δt → 0).

This concludes the proof of Theorem 2.1.


Proof (Proof of Theorem 2.2). Following the same steps of the above proofwe obtain the discrete version (2.5) of the L1 Ingham’s inequality (1.6) given byTheorem 2.2.

More precisely, we have

Δt

M∑n=0

sinnΔtπ

T

(∑k

akeinΔtλk

)e−inΔtλk =

∑l

alH(λk − λl),(6.38)

where function H is defined by (6.6).Taking l = ν in (6.38), where |aν | is the greatest |an|, we deduce

∣∣∣∣∣Δt

M∑n=0

sinnΔtπ

T

(∑k

akeinΔtλk

)e−inΔtλν

∣∣∣∣∣ ≥ |aνH(0)| − |aν |∑

k,k =ν

|H(λk − λν)| .

(6.39)

Function H is exactly the discrete Fourier transform (6.5) used in the proof of discreteIngham’s inequality (2.3). By (6.24) and the estimates for |H(λk − λν)| used in theproof of Theorem 2.1, i.e.,

∑k,k =ν

|H(λk − λν)| ≤[

8π

Tγ2+ NCΔt2 + CNΔt(Δt)1−2p

],

as in (6.28), we deduce that

ΔtM∑n=0

∣∣∣∣∣N∑

k=−N

akeinΔtλk

∣∣∣∣∣ ≥ C1(Δt, T, γ) max |an| .(6.40)

The direct inequality in (2.5) may be obtained using the same arguments and estimatesand we omit the details.

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