AN INGHAM – MÜNTZ TYPE THEOREM AND SIMULTANEOUS OBSERVATION PROBLEMS

We establish a theorem combining the estimates of Ingham and Müntz–Szász. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz–Szász part. A very special case of this theorem allows us to prove the simultaneous observability of some string–heat and beam–heat systems.

In the case of reversible linear evolutionary systems these methods are often based on various generalizations of a classical theorem of Ingham [11], itself a generalization of Parseval's equality, see, e.g., [10], [14], [16] and their references.See also [7] for a generalization allowing for complex exponents.
In this paper we establish a theorem combining the estimates of Ingham and Müntz-Szász.Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz-Szász part.
In formulating our theorem we use henceforth Vinogradov's notation: f (t) g(t) (t ∈ E) means that the real or complex quantities f (t) and g(t) satisfy for all t ∈ E where C E is a constant depending at most on the set E and possibly on various parameters to be specified.

VILMOS KOMORNIK AND G ÉRALD TENENBAUM
Theorem 1.1.Consider four real sequences Λ := (λ n ) n∈Z , E := (ε n ) n∈Z , M := (µ k ) k∈N , H := (η k ) k∈N and corresponding complex sequences (z n ) n∈Z , (w k ) k∈N , defined by the formulae Let γ > 0, and assume that the the following conditions hold for some α > 1 and p ∈ N * : Here, the implied constant depends at most on α, γ 1 , γ, p, T , and on the implicit constants in the assumptions.
In the second part of this paper we apply Theorem 1.1 to some observability problems Simultaneous observability of string-string, string-beam and beam-beam systems have been investigated in [2], [3], and [23] by applying some weakened Ingham type theorems.A very special case of Theorem 1.1 allows us to prove the simultaneous observability of some string-heat and beam-heat systems.We note that for a different kind of wave-heat systems observability estimates have been obtained by different approaches in [1], [20], [26], [27].
Let us consider a vibrating string of length 1 and a heated rod of length 2 , both with homogeneous Dirichlet boundary conditions.We assume that they have a common endpoint, where we may observe only the cumulative action of them during some time T .A natural question is whether this observation allows us to determine the unknown initial data for both equations.
We may model this problem in the following way.For some given real number κ we consider the following two independent problems: It is well-known that for any given initial data Furthermore, the Fourier series representation of the solutions shows that for any fixed T > 0 the linear maps are well defined and continuous from H 1 0 (0, 1 ) × L 2 (0, 1 ) to L 2 (0, T ) and from L 2 (0, 2 ) to L 2 (0, T ), respectively.
We ask whether the linear map is one-to-one on H 1 0 (0, 1 ) × L 2 (0, 1 ) × L 2 (0, 2 ).Since there is a finite propagation speed for the wave equation, this cannot hold unless T is sufficiently large, more precisely unless T 2 1 ; see, e.g., [13, Remark 3.6] for a simple proof even in higher dimension.
In order to formulate our result we expand the initial data into Fourier series: Proposition 1.2.If |κ| < π/ 1 and T > 2 1 , then the linear map (1.9) is one-toone.More precisely, there exists a positive constant c T such that the solutions of (1.7) and (1.8) satisfy the following estimate: Next we investigate the observability problem when the string is replaced by a hinged beam, modelled by the following system: We recall that for any given initial data 0 ∈ H 1 0 (0, 1 ) and 1 ∈ H −1 (0, 1 ) the system (1.10) has a unique solution satisfying )) Furthermore, for any fixed T > 0 the linear map is well defined and continuous from H 1 0 (0, 1 ) × H −1 (0, 1 ) to L 2 (0, T ).
We ask whether the linear map is one-to-one on H 1 0 (0, 1 ) × H −1 (0, 1 ) × L 2 (0, 2 ).Since the propagation speed is infinite for both our beam and heat conduction model, we may expect observability for arbitrarily small T > 0. Indeed, we obtain the following result.
More precisely, there exists a positive constant c T such that the solutions of (1.10) and (1.8) satisfy the following estimate: Our next applications illustrate the flexibility provided by Theorem 1.1 regarding the complex sequences of the frequencies.We fix two real or complex numbers α, β and we consider the following coupled wave-heat system on some bounded interval (0, ): (1.12) Since the parameters α, β represent a bounded perturbation of the uncoupled system, the problem is well posed.More precisely, given any initial data ( 0 , 1 , σ 0 ) ∈ H 1 0 (0, ) × L 2 (0, ) × L 2 (0, ), the system has a unique solution satisfying Given T > 0, we may then ask whether the linear maps are one-to-one.
Since we only observe one of the two unknown functions, these properties cannot hold in the uncoupling case α = β = 0.
We shall prove the following results, where we use the Fourier coefficients of the initial data defined by changing 1 and 2 to in the above formulae.
More precisely, there exists a positive constant c T = c T (α, β) such that the solutions of (1.12) satisfy the estimates Given an interior point x 0 ∈ (0, ), we may also ask whether the linear maps are one-to-one.These problems may be solved by a simple adaptation of the proof of Proposition 1.4, combined with some Diophantine approximation results as, e.g., in [2], [3] or [14].We leave the details for the interested reader.
The same questions may be asked for the following coupled beam-heat system: Since the parameters α, β represent a bounded perturbation of the uncoupled system, for any given initial data Proposition 1.5.Consider the solutions of the system (1.15) and assume that we have 0 < |αβ| π 6 /{6 5 (π + )}.Then the linear maps (1.13) and (1.14) are one-to-one for any fixed T > 0.
More precisely, there exists a positive constant c T = c T (α, β) such that the solutions of (1.15) satisfy the estimates The next two sections are devoted to the proof of Theorem 1.1.The remainder of the paper is devoted to the proof of Propositions 1.2-1.5.

2.
A lemma from complex analysis.The following result is a variant (and actually an extension) of Proposition 2 in [12].We provide a very simple proof, analogous to that of Lemma 3.3 in [25].
We systematically write a complex number as z = x + iy and let denote the Fourier transform of a function h, extended to suitable complex values of the variable z.
Here the implicit constants depend at most upon β and ε. .
We shall see that, for a sufficiently large p, the function t → h(t) := (L/ε)H(t/ε) meets our requirements.We have and so, for any integer j 0 and all z = x + iy ∈ C, In order to estimate H (j) ∞ , we consider some t ∈ [0, 1] and put := 1 − t.For δ ∈]0, 1  2 ], w This implies, for each fixed p, up to selecting δ = δ p sufficiently small.Cauchy's formula then yields a bound clearly also valid, with now := 1 + t, for −1 t 0 by symmetry.Taking the supremum in , assumed at = 1 3 (p/j) 1/p , we get, for some suitable constant K p , H (j) ∞ K j p j j(p+1)/p (j 1).Inserting into (2.2) and choosing j equal to some approximate optimum, for instance when |z| is sufficiently large, we obtain the expected upper bound in (2.1) by selecting p sufficiently large.The lower bound in (2.1) follows immediately from the formula 3. Proof of Theorem 1.1.The basic idea of the proof is the construction of a suitable biorthogonal sequence by using complex analysis tools.
Let g ∈ (2γ 1 /3, γ).By definition, each interval of length pg contains at most p values of the sequence Λ.Up to modifying Λ by inserting some new points, we may assume that each interval J R := [rpg, (r + 1)pg), r ∈ Z, contains exactly p terms from Λ and that inf Indeed, this may be performed in two steps.First, for each n, we define m n by m n γ 1 /3 λ n+1 − λ n < (m n + 1)γ 1 /3, and we add the points λ n + jγ 1 /3, j = 1, . . ., m n − 1.Then we get a sequence with gaps between γ 1 /3 and 2γ 1 /3, and hence each of the disjoint intervals J r contains at least pg/(2γ 1 /3) > p elements of the sequence Λ.We conclude by deleting as many points as necessary to reach the required goal.
For fixed p, we thus have We put The convergence of the infinite products on the right is immediate, since each term is 1 + O(1/k α ).That of the infinite products on the left follows from the above alteration of the sequence Λ, as explained in [5, lemma 7], provided that these products be defined as limits as R → ∞ of the finite products for |λ n | R.
We immediately see that As a first step, we observe that, still writing z = x + iy, we have for some suitable c 1 depending only on p and on the implicit constants of our statement, Indeed, with an obvious reindexing, we may write the product in the form n 1 T n T −n with Z := (z − z m )/g and Write Z = X + iY , with, say, X 0 (the case X 0 may be dealt with symmetrically) and let q := X .For n 1, n = q, (q + 1), we have We claim that the product, say P q , over n 1, n = q, (q + 1), of the terms inside curly brackets satisfies the upper bound P q (1 + |Z|) b for some constant b depending on our parameters.
First, if X 1, then 1) , since the last sum does not exceed If X 1, we consider in turn the ranges 1 n 1 2 X, 1 2 X < n 3 2 X, and n > 3  2 X.We have .
Combining these estimates we obtain the claimed upper bound.Invoking Euler's infinite product formula for sin(πZ), and observing that sin πZ πZ we readily get (3.3).Indeed, T q T −q and T q+1 T −q−1 are both bounded above by fixed a power of 1 + |Z|.Now, we have for z ∈ C, where we used the fact that the last inner sum is trivially t 1/α−1 .Hence there exists a constant K = K(Λ, H) such that Φ m (z) e π|y|/g+K|z−zm| 1/α (z ∈ C).
Next, we give ourselves a parameter ε > 0 and recall the definition of the function h from Lemma 2.1.We then put Select β := 2/(1 + α) in Lemma 2.1.For any integer m ∈ Z, we have with T ε := 2ε + π/g.In the last two upper bounds, implicit constants only depend on α, γ 1 , γ, ε, p, and the implicit constants in the statement.Since F m belongs to L 1 (R) ∩ L 2 (R) and has exponential type at most T , we infer from the Paley-Wiener theorem that it is the Fourier transform of some function In order to obtain an analogous result for Ψ j , we first observe that relation (3.3) with z m = 0 enables us to write Indeed, it may be readily checked that the infinite product of the denominators converges and is bounded from below independently of j: this follows from the estimates where we used the fact that, since µ j > 0 for all j and η j remains bounded, we have |n Furthermore, we have, uniformly with respect to z ∈ C, where we have put M j (s) := |w k −wj | s 1 s 1/α .Hence the inner integral is t 1/α−1 and so Therefore, there exists a constant C = C(M, E) such that For any integer j ∈ N, we have with T ε := 2ε + π/g, and where, as previously, implied constants only depend on α, γ 1 , γ, ε, p, and the implicit constants in the statement.Since, for each j ∈ N, the function G j belongs to L 1 (R) ∩ L 2 (R) and has exponential type at most T ε , it is the Fourier transform of a function ψ j supported in

Now let us consider the functions
for t ∈ R. We have, employing the Cauchy-Schwarz inequality, Moreover, by Plancherel's formula, still with the notation β := 2/(1 + α), we have where we used the fact that the penultimate integral is e −Bε|zn−zm| β e −Bεγ1|n−m| β /3 β for some constant B depending only on β.
Combining the above two inequalities we obtain that Replacing a n by a n e iznTε and b k by b k e −w k Tε , we get Now (1.6) follows easily.Indeed, given T > 2π/γ arbitrarily, we may choose g ∈ (2γ 1 /3, γ) such that T > 2π/g, and then select ε > 0 such that T = 2T ε .
4. Proof of Propositions 1.2 and 1.3.In order to simplify the formulae, we consider only the case 1 = 2 = π.The proofs may be extended without any difficulty to the general case.
Proof of Proposition 1.2.Using the Fourier series of the initial data and writing n κ := √ n 2 − κ 2 , we may write the solutions as The proposition follows by applying Theorem 1.1 with z 0 := 0, and with for n ∈ N * .Indeed, the assumptions of the theorem are satisfied with γ = 1 and α = 2.
Proof of Proposition 1.3.Using the same Fourier series (4.1) again, but redefining n κ := √ n 4 − κ 2 we may write the solutions as c n e −n 2 t sin nx, whence The proposition follows by applying Theorem 1.1 with z 0 := 0, and for n ∈ N * .Indeed, the assumptions of the theorem are satisfied with arbitrarily large γ and α = 2.
5. Proof of Propositions 1.4 and 1.5.First we prove Proposition 1.4.Writing n := πn/ (n 1), we may expand the solutions of (1.12) into Fourier series where the functions u n (t), v n (t) are solutions, for each n, of the linear initial value problem Lemma 5.1.If |αβ| π 3 /(8 3 ), then, for each positive integer n, the characteristic equation z 3 + n 2 z 2 + n 2 z + (n 4 − αβ) = 0 of (5.1) has three distinct complex roots iz −n , iz n and −w n , satisfying If In the remaining of the proof, we assume = π for notational simplicity.The proof of the general case is the same: we only have to change the coefficients n to n everywhere.
It follows from the lemma that the above sequences (z n ) and (w n ) satisfy the hypotheses of Theorem 1.1, and that, for each n 1, we have with suitable complex coefficients α n , β n , γ n , δ n .Substituting these expressions into the equations (5.1) we may express these coefficients through a n , b n and c n : Expressing γ −n , γ n , and δ n from the first three equations and substituting their expressions into the last equation, the last three equations become Since or equivalently Combining this with (5.2), the first estimate of Proposition 1.4 follows: The proof of the second estimate is similar.Considering the same linear system of nine equations as above, now we start by expressing α n , α −n and β n from the three middle equations, and we substitute the results into the last three equations to obtain Similarly to above, using also the relation w n − n 2  1, we infer that we deduce from Theorem 1.1, for each T > 2π, the validity of the estimate Combining this with (5.3) the second estimate of Proposition 1.4 follows: Now we turn to the proof of Proposition 1.5.We retain the notations s := π/ and n := ns .Expanding the solutions of (1.15) into Fourier series we see that, for each n, the functions u n (t), v n (t) are solutions of the linear initial value problem (5.4) The following proof is a variant of Lemma 5.1.
Lemma 5.2.If |αβ| < s 6 /(6 + 6s ), then, for each positive integer n, the characteristic equation of (5.4) has three distinct real or complex roots iz −n , iz n , and −w n , satisfying Moreover, the triplets of roots corresponding to distinct values of n are disjoint.For the completion of the proof, let us assume again for notational simplicity that = π.
It follows from the lemma that the above sequences (z n ) and (w n ) satisfy the hypotheses of Theorem 1.1 for any fixed T > 0, and that, for each n 1, we have u n (t) = α −n e iz−nt + α n e iznt + β n e −wnt , v n (t) = γ −n e iz−nt + γ n e iznt + δ n e −wnt , with suitable complex coefficients α n , β n , γ n , δ n .Substituting these expressions into the equations (5.4) we may express these coefficients in terms of a n , b n and c n : Adapting the proof of the previous proposition we now obtain:

5 )n∈Z a n e iznt + k∈N b k e −w k t 2 dt n∈Z |a n | 2 +
) inf n∈Z, k∈N |iz n ± w k | > 0.(1.Then the following estimate holds for all T > 2π/γ and all square summable sequences (a n ) n∈Z and (b k ) k∈N :T 0 k∈N |b k | 2 e −µ k T .(1.6) Moreover, the triplets of roots corresponding to distinct values of n are disjoint.Proof.Put s := π/ .Rewriting the equation in the formf (z) := (z − in )(z + in )(z + n 2 ) = αβ,it is sufficient by Rouché's theorem to show that |f (z)| > s 3 /8 on each of the three circles of radius s /2, centered at in , −in and −n 2 .