QUANTITATIVE UNCERTAINTY PRINCIPLE FOR STURM-LIOUVILLE TRANSFORM

In this paper we consider the Sturm-Liouville transform F(f) on R+. We analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given.


Introduction
The uncertainty principle says that a function and its transform cannot concentrate both on small sets. Depending on the precise way to measure "concentration" and "smallness" this principle can assume different forms. This paper focuses on studying different uncertainty principles for the Sturm-Liouville transform, by following the procedures for similar transforms, such as the Fourier transform (the classical setting) we refer to the book [10] and the surveys [4,7] for further references. The concept of concentration has taken different interpretations in different contexts. For example: Benedicks [2], Slepian and Pollak [18], Landau and Pollak [13], and Donoho and Stark [6] paid attention to the supports of functions and gave quantitative uncertainty principles for the Fourier transforms. Qualitative uncertainty principles are not inequalities, but are theorems that tell us how a function (and its Fourier transform) behave under certain circumstances. For example: Hardy [11], Cowling and Price [5], Beurling [3], Miyachi [15] theorems enter within the framework of the quantitative uncertainty principles. The quantitative and qualitative uncertainty principles have been studied by many authors for various Fourier transforms, for examples (cf. [1,11,14,16]).
Our aim here is to consider uncertainty principles in which concentration is measured in sense of smallness of the support and when the transform under consideration is the Sturm-Liouville transform.
The first principle that is studied is a Donoho-Stark-type inequality. One can write the classical uncertainty principle in the following way: If a function f (t) is essentially zero outside an interval of length ∆t and its Fourier transform f (w) is essentially zero outside an interval of length ∆w, then ∆t∆w ≥ 1. In [6], Donoho and Stark show that it is not necessary to assume that the support and the spectrum are concentrated on intervals and one can replace intervals by measurable sets, and then the length of the interval is naturally replaced by the measure of the set. In Section 2, a version of this inequality for the Sturm-Liouville transform is given, and, as it appears in [6] it is explained how to reconstruct a signal f from a noisy measurement, knowing that the signal is supported on a set S.
The second principle, studied in Section 3, is a Benedicks-type result which shows that two measurable sets (S, Σ) with finite measure form a strong annihilating pair. This means that a function supported in S cannot have an spectrum in Σ giving a quantitative information of the mass of a function whose spectrum is contained in Σ. The approach is based on the corresponding version of this type of principle for the integral operators transform, studied in [8]. A version of Benedicks type-inequality for integral operators transform with bounded and homogeneous kernel has been proved in [8]. In this paper, we consider a transform of a different nature where in particular the kernel is not homogeneous.
We recall that, Soltani in [19] study what is the relation between the measure and the spectrum of a function f that is ε-concentrated in measurable sets giving. Concentration in support means that the part of the function that is not supported on a set is at least an ε part of the total mass. The analogous version for spectrum states that the part of the spectrum not supported on a set is an ε part of the total spectrum. It is shown that if a function is ε-concentrated in space and frequency, then the product of the measures of the support and spectrum is lower bounded by a number close to one.
In order to describe our results, we first need to introduce some facts about harmonic analysis related to Sturm-Liouville transform. We cite here, as briefly as possible, some properties. For more details we refer to [19].
The Sturm-Liouville operator ∆ defined on R + by where ρ is a nonnegative real number and A(x) = x 2α+1 B(x), α > − 1 2 , where B is a positive, even, infinitely differentiable function on R such that B(0) = 1. Moreover, we assume that A and B satisfy the following conditions: A(x) = 2ρ; • there exists a constant δ > 0 such that where D is an infinitely differentiable function on ]0, ∞[, bounded and with bounded derivatives on all intervals [x 0 , ∞[ for x 0 > 0. For all λ ∈ C the equation admits a unique solution denoted ϕ λ , with the following properties: • for x ≥ 0 the function λ → ϕ λ (x) is analytic on C; • for λ ∈ C the function λ → ϕ λ (x) is even and infinitely differentiable on R; Moreover, there exist positive constants k 1 , k 2 and k such that for all λ such that Imλ ≤ 0 and |λ| ≥ k. Let us introduce the dilation operator D ρ , ρ > 0, defined by We denote by Let ν the measure defined on [0, Then we get the Hausdorff-Young inequality (see [20]) where S c = R + \S and suppf = {x : f (x) = 0}.

The Donoho-Strak's Uncertainty Principle
The classical uncertainty principle says that if a function f (t) is essentially zero outside an interval of light ∆t and its Fourier transform f (w) is essentially zero outside an interval of length ∆w, then ∆t∆w ≥ 1. In this section we will prove a quantitative uncertainty inequality about the essential supports of a nonzero function f ∈ L 2 (R + , µ) and its Sturm-Liouville transform.
The first such inequality for the usual Fourier transform was obtained by Donoho-Stark [6].
We consider a pair of orthogonal projections on , where S and Σ are measurable subsets of R + , and χ S denote the characteristic function of S.
Let 0 < ε S , ε Σ < 1 and let f ∈ L 2 (R + , µ) be a nonzero function. We say that We denote by P S ∩ Q Σ for the orthogonal projection onto the intersection of the ranges of P S and Q Σ , we will write ImT for the range of a linear operator T . We denote by T HS the Hilbert-Schmidt norm of the linear operator T . The definition of this norm [21, page 262] implies that for any pair of projections E, F one has Theorem 2.1. Let Σ, S ⊂ R + be a pair of measurable subsets and let ε S , ε Σ > 0 such that We will need the following well-known lemma.
Lemma 2.1. Let (S, Σ) be two measurable subsets of R + . Then the following assertions are equivalent.
ii) (S, Σ) is strongly annihilating pair for the Sturm-Liouville transform. Moreover, Proof. Firstly we show the following implication i)⇒ii). The identity operator I satisfies I = P S + P S c = P S Q Σ + P S Q Σ c + P S c , we have from the orthogonality of P S and P S c It follows, by P S = 1, that On the other hand, we have It follows, from inequality (2.2), As P S Q Σ < 1, then we obtain the desired result.

Let us now show the second implication ii)⇒i). Recall that
We suppose that P S Q Σ = 1. Then we can find a bandlimited sequence f n ∈ L 2 (R + , µ) on Σ of norm 1 (in particular f n = Q Σ f n ) such that P S f n L 2 (R + ,µ) → 1 as n → ∞.
By the orthogonality of S, we have
HS < ∞. The following example is prototypical. A signal f is transmitted to a receiver who know that f is bandlimited on S for the Sturm-Liouville transform, meaning that f is synthesized using only frequency on S; equivalently f = Q Σ f. Suppose that the observation of f is corrupted by a noise n ∈ L 2 (R + , µ) (which is nonetheless assumed to be small) and an unregistered values on S. Thus, the observable function r satisfies Here, we have assumed without loss of generality that n = 0 on S. Equivalently, r = (I − P S )f + n. We say that f can be stably reconstructed from r, if there exists a linear operator K and a constant C such that The estimate (2.7) shows that the noise n is at most amplified by a factor C. Proof. If µ(S)ν(Σ) < 1, using (2.4), P S Q Σ < 1. Hence, So, that The constant C in equation (2.7) is therefore not larger than 1 − µ(S)ν(Σ) −1 .
The identity K = (I − P S Q Σ ) −1 = ∞ k=0 (P S Q Σ ) k suggests an algorithm for computing Kr. Put f (n) = n k=0 (P S Q Σ ) k r, then As f is bandlimited on Σ we deduce that Algorithms of this type have applied to a host of problems in signal recovery (see for examples [12,17]).
Proof. If the function f ∈ L 2 (R + , µ) has non empty support, by the Cauchy-Schwartz inequality and (1.1), we have . Using Plancherel's theorem 1.2 we have the following quantitative uncertainty inequality connecting the support of f and the support of its Sturm-Liouville transform F It follows that if µ(suppf )ν(suppF) < 1, then f = 0.
Since F(f n ) is the scalar product of f n and χ S ϕ λ (·), it follows that F(f n ) converge to F(f ). Since |F(f n )| as bounded by µ(S), it follows from Lebesgue's theorem that F(f n )χ Σ converges to F(f ) in L 2 (R + , ν) and the limit f has norm 1. But the function f has support in S and spectrum in Σ, since (S, Σ) is a weak annihilating pair, it follows that f = 0, which gives a contradiction.