Keywords: Espaces de Hardy, fonctions analytiques, approximation, problèmes extrémaux bornés, formules de Carleman

Problèmes extrémaux contraints dans l'espace de Hardy H 2 et formules de Carleman Résumé : Nous étudions des problèmes d'approximation sur un strict sousensemble du cercle par des fonctions analytiques de l'espace de Hardy H 2 du disque unité (de C), soumises en module à une contrainte ponctuelle sur la partie complémentaire du cercle. Des résultats d'existence, d'unicité et de saturation de la contrainte sont établis, ainsi qu'une équation aux points critiques qui permet de proposer une formule de dualité. Nous considérons enfin une version polynômiale (de dimension finie) de ces problèmes extrémaux bornés.

Introduction

If D is a finitely connected plane domain with rectifiable boundary ∂D, a holomorphic function f in the Smirnov class E 1 (D) can be recovered from its boundary values by the Cauchy formula [START_REF] Duren | Theory of H p spaces[END_REF]. When the boundary values are only known on a strict subset I of ∂D having positive linear measure, they still define f uniquely but the recovery cannot be achieved in closed form. In fact, it becomes a special case of a classical ill-posed issue namely the Cauchy problem for the Laplace equation. This issue is quite important in physics and engineering [START_REF] Isakov | Inverse problems for partial differential equations[END_REF][START_REF] Lavrentiev | Some improperly posed problems of mathematical physics[END_REF][START_REF] Partington | Interpolation, identification and sampling[END_REF]. Following an original idea of Carleman, one approach to the recovery of f from its knowledge on I is to introduce an auxiliary "quenching" function ϕ, holomorphic and bounded in D, such that |ϕ| ≡ 1 a.e. on ∂D \ I and |ϕ| > 1 in D; such a function is easily constructed by solving a Dirichlet problem for log |ϕ|. In [START_REF] Goluzin | Generalized carleman formula and its application to analytic extension of functions[END_REF], it was proven by Goluzin and Krylov that

f (z) = lim n→∞ f n (z), where f n (z) ∆ = 1 2π I ϕ(ξ) ϕ(z) n f (ξ) ξ -z dξ, z ∈ D, (1)
the convergence being locally uniform in D. Cauchy integrals like those defining f n in (1) are called Carleman's formulas [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF]. On the unit disk D where E p (D) coincides with the Hardy class H p , see [START_REF] Duren | Theory of H p spaces[END_REF], it is proved in [START_REF] Patil | Representation of H p functions[END_REF] that if f ∈ H p with 1 < p < ∞, then the convergence actually holds in H p . Two questions arise naturally, namely what is the meaning of f n for fixed n, and what is its asymptotic behaviour if f ∈ L p (I) is not the trace of a Hardy function? On D, when f ∈ L 2 (I) and ϕ is a quenching function with the additional property that |ϕ| is constant a.e. on I, it was proven in [START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF] that f n is closest to f in L 2 (I)-norm among all g ∈ H 2 such that g L 2 (T\I) ≤ f n L 2 (T\I) , where T denotes the unit circle. In the present paper, among other things, we will see that if ϕ is holomorphic and bounded on D together with its inverse, then f n is closest to f w.r.t. the weighted L 2 (|ϕ |I | 2 , I)-norm among all g ∈ H 2 such that |g| ≤ |f n | a.e. on T \ I. These extremal properties of f n are all the more remarkable than Carleman's formulas were originally defined without reference to optimization. Still one point is unsatisfactory, namely the extremal properties of f n we just mentioned are implicit in that the level of the pointwise constraint on T \ I is |f n | itself. This is why we make a slight twist and we rather investigate on D the following extremal problem. Let I ⊂ T be a subset of positive Lebesgue measure and set J = T \ I for the complementary subset. The question we raise is the following.

BEP : Given f ∈ L 2 (I) and M ∈ L 2 (J), M ≥ 0, find g 0 ∈ H 2 such that |g 0 (e iθ )| ≤ M (e iθ ) a.e. on J and fg 0 L 2 (I) = min g∈H 2

|g|≤M a.e. on J fg L 2 (I) .

(

) 2
This should be compared with the so-called bounded extremal problems BEP p . studied in [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF][START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF][START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF] for 1 ≤ p ≤ ∞.

BEP p : Given f ∈ L p (I), ψ ∈ L p (J) and a positive constant C, find g 0 ∈ H p such that g 0ψ L p (J) ≤ C and fg 0 L p (I) = min

g∈H p g-ψ L p (J) ≤C f -g L p (I) . (3)
Note that in problem BEP (2), we did not introduce a reference function ψ on J as was done in BEP p [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF]. While it is straightforward to handle such a generalization when ψ is the trace on J of a H 2 function, the general case conceals further difficulties that are left here for further research. When I is of full measure, both problems (2) and (3) reduce to classical extremal problems, see e.g. [START_REF] Duren | Theory of H p spaces[END_REF][START_REF] Garnett | Bounded analytic functions[END_REF]. Therefore we limit our discussion to the case where J has positive measure. The first reference dealing with bounded extremal problems seems to be [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF], where BEP 2 is studied for f = 0 and I an interval on the half-plane rather than the disk. The case ψ = 0 is solved in [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF] using Toeplitz operators, and error rates when C goes large and I is an arc can be found in [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF]. Weighted versions of BEP 2 for L 2 (|ϕ| 2 )-norms as those discussed above were also solved in [START_REF] Leblond | Weighted H 2 approximation of transfer functions[END_REF].

The general version BEP p in the range 1 ≤ p < ∞ is taken up in [START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF] where the link with Carleman's formulas is pointed out, while existence and uniqueness results are also presented. Reformulations of BEP p in abstract Hilbert or smooth Banach space settings were carried out in [START_REF] Chalendar | Constrained approximation and invariant subspaces[END_REF][START_REF] Chalendar | Approximation in reflexive Banach spaces and applications to the invariant subspace problem[END_REF][START_REF] Leblond | Constrained approximation and interpolation in hilbert function spaces[END_REF][START_REF] Smith | Constrained approximation in banach spaces[END_REF], leading to the construction of backward minimal vectors and hyperinvariant subspaces for certain classes of operators that need not be compact nor quasinilpotent, thereby generalizing [START_REF] Ansari | Extremal vectors and invariant subspaces[END_REF]. Versions of BEP 2 where the constraint bears on the imaginary part rather than the modulus, useful among other things to approach inverse Dirichlet-Neumann problems, are presented in [START_REF] Jacob | A constrained approximation problem arising in parameter identification[END_REF][START_REF] Leblond | Solution of inverse diffusion problems by analytic approximation with real constraints[END_REF]. Together with meromorphic generalizations, problem BEP p was studied in [START_REF] Baratchart | An L p analog to AAK theory for p ≥ 2[END_REF] for p ≥ 2, while problem BEP ∞ was studied in [START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF][START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF], with related completion issues.

A major incentive to study BEP p came from engineering problems, more precisely from questions pertaining to system identification and design. This motivation is quite explicit in [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF], and all-pervasive in [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF][START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of Toeplitz operators[END_REF][START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF][START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF][START_REF] Seyfert | Problèmes extrémaux dans les espaces de Hardy[END_REF] whose results have been used effectively to identify hyperfrequency filters [START_REF] Baratchart | Identification d'un filtre hyperfréquences par approximation dans le domaine complexe[END_REF]. The connection with identification is more transparent on the half plane, where f represents the so-called transfer-function of a linear dynamical system as measured in the frequency bandwidth I using harmonic identification techniques. Recall that a linear dynamical system is just a convolution operator, and that its transfer function is the Fourier-Laplace transform of its kernel [START_REF] Doyle | Feedback Control Theory[END_REF]. Now, by the Paley-Wiener and Hausdorff-Young theorems, causality and L r → L s stability of the system cause f to belong to the Hardy class H p of the half plane with 1/p = 1/r -1/s, as soon as the latter is less than or equal to 1/2. Because f is only known up to modelling and measurement errors, one is led to approximate the data on I by a H p function while controlling its deviation from some reference behaviour ψ outside I, which is precisely the analog of (3) on the half-plane.

It is mapped to BEP p via the isometry g → (1 + w) -2/p g((w -1)/(w + 1)) from H p onto H p . More on the relations between Hardy spaces, system identification and control can be found in [START_REF] Fuhrmann | Linear systems and operators in Hilbert spaces[END_REF][START_REF] Nikolskii | Operators, functions, and systems: an easy reading[END_REF][START_REF] Partington | Interpolation, identification and sampling[END_REF]. Note that in BEP p , it is indeed essential to bound the behaviour of g 0 on J, for traces of Hardy functions are dense INRIA in L p (I) (in C(I) if p = ∞) so that BEP p has no solution if C = ∞ unless f is already the trace of a Hardy function. In practice, since modelling and measurement errors will prevent this from ever happening, the error fg L p (I) can be made arbitrarily small at the cost of g L p (J) becoming arbitrarily large, which does not make for a valid identification scheme. The present paper deals with a mixed situation, where an integral criterion is minimized on I under a pointwise constraint on J. Here again, the motivation of the authors stems from system identification. Indeed, the L 2 norm on I has a probabilistic interpretation, being the variance of the output when the system is fed by noise whose spectrum is uniformly distributed in the bandwidth, that suits some classical framework. On another hand, it is often the case that the transfer function has to meet uniform bounds for physical reasons. For instance when identifying a passive device, it should be less than 1 at all frequencies. This way one is led to consider problem (2) with M ≡ 1 (which is BEP 2,∞ below). Such issues and motivations are also the topics of the recent work [START_REF] Schneck | Constrained optimization in hardy spaces[END_REF][START_REF] Schneck | Constrained optimization in hardy spaces II: Numerics[END_REF], where BEP -like problems are considered in L p (I), for 1 ≤ p ≤ ∞, with a pointwise constraint acting on the whole T, when the approximated function f and the constraint M are assumed to be continuous functions (on T and I, respectively), with M > 0 and |f | ≤ M on I. Problem ( 2) is considerably more difficult to analyze than BEP 2 , due to the fact that pointwise evaluation is not smooth -actually not even defined-in L 2 (J). Its solution depends in a rather deep fashion on the multiplicative structure of Hardy functions and all our results, beyond existence and uniqueness, will hold under the extra-assumption that the boundary of I has measure zero. We do not know the extend to which this assumption can be relaxed.

The organization of the paper is as follows. In section 2 we set up some notation and recall standard properties of H p -spaces, BM O and conjugate functions. Section 3 deals with existence and uniqueness issues, as well as pointwise saturation of the constraint. In section 4 we establish an analog, in this nonsmooth and infinite-dimensional context, of the familiar critical point equation from convex analysis. It gives rise to a saddle-point characterization of the optimal value that yields a dual formulation of the problem. The latter is connected in section 5 to Toepliz operators and Carleman's formulas, and used to compute the concave dual functional whose maximization is tantamount to solve the problem. We further compute directional derivatives for this functional as a means to approach the issue from a computational point of view. Section 6 is devoted to a finite-dimensional polynomial version of (2), valid when I is a union of arcs, which is of interest in its own right and provides an alternative way to constructively approximate the solution to the original problem.

Notations and preliminaries

Let T be the unit circle endowed with the normalized Lebesgue measure ℓ, and I a subset of T such that ℓ(I) > 0 with complementary subset J = T \ I. To avoid dealing with trivial instances of problem (2) we assume throughout that ℓ(J) > 0.

If h 1 (resp. h 2 ) is a function defined on a set containing I (resp. J), we use the notation h 1 ∨ h 2 for the concatenated function, defined on the whole of T, which is h 1 on I and h 2 on J.

For E ⊂ T, we let ∂E and

• E denote respectively the boundary and the interior of E when viewed as a subset of T; we also write χ E for the characteristic function of E and h |E to mean the restriction to E of a function h defined on a set containing E. When 1 ≤ p ≤ ∞, we write L p (E) for the familiar Lebesgue space of (equivalence classes of a.e. coinciding) complex-valued measurable functions on E with finite L p norm, and we indicate by L p R (E) the real subspace of real-valued functions. Likewise C(E) stands for the space of complex-valued continuous functions on E, while C R (E) indicates real-valued continuous functions. The norm on L p (E) is denoted by L p (E) , and if h is defined on a set containing E we write for simplicity h

L p (E) to mean h |E L p (E) . When E is compact the norm of C(E) is the sup norm.
Recall that the Hardy space H p is the closed subspace of L p (T) consisting of functions whose Fourier coefficients of strictly negative index do vanish. These are the nontangential limits of functions analytic in the unit disk D having uniformly bounded L p means over all circles centered at 0 of radius less than 1. The correspondence is one-to-one and, using this identification, we alternatively regard members of H p as holomorphic functions in the variable z ∈ D. This extension is obtained from the values on T through a Cauchy as well as a Poisson integral [35, ch. 17, thm 11], namely if g ∈ H p then, for z ∈ D:

g(z) = 1 2 i π T g(ξ) ξ -z dξ and g(z) = 1 2 π T Re e iθ + z e iθ -z g(e iθ ) dθ . (4)
Because of this Poisson representation, g(re iθ ) converges to g(e iθ ) in L p (T) as soon as 1 ≤ p < ∞. Moreover, (4) entails that, for 1 ≤ p ≤ ∞, a Hardy function g is uniquely determined, up to a purely imaginary constant, by its real part h on T:

g(z) = iImg(0) + 1 2 π T e iθ + z e iθ -z h(e iθ ) dθ , z ∈ D. (5)
The integral in the right-hand side of ( 5) is called the Riesz-Herglotz transform of h and, whenever h ∈ L 1 R (T), it defines a holomorphic function in D which is real at 0 and whose nontangential limit exists a.e. on T with real part equal to h. However, only if 1 < p < ∞ is it guaranteed that g ∈ H p when h ∈ L p R (T). In fact, the Riesz-Herglotz transform assumes the form h(e iθ ) + i h(e iθ ) a.e. on T, where the real-valued function h is said to be conjugate to h, and the property that h ∈ L p R (T) whenever h ∈ L p R (T) holds true for 1 < p < ∞ but not for p = 1 nor p = ∞. The map h → h is called the conjugation operator, and for

1 < p < ∞ it is bounded L p R (T) → L p R (T)
K 2 = 1, but it is rather subtle that K p = tan(π/(2p)) for 1 < p ≤ 2 while K p = cot(π/(2p)) for 2 ≤ p < ∞ [33].
A sufficient condition for h to be in L 1 (T) is that h belongs to the the socalled Zygmund class L log + L, consisting of measurable functions φ such that φ log + |φ| ∈ L 1 (T) where we put log + t = log t if t ≥ 1 and 0 otherwise. More precisely, if we denote by m h the distribution function of h defined on R + with values in [0, 1] according to the formula

m h (τ ) = ℓ ({ξ ∈ T; |h(ξ)| > τ }) ,
and if we further introduce the non-increasing rearrangement of h given by

h * (t) = inf{τ ; m h (τ ) ≤ t}, t ≥ 0,
it turns out that h ∈ L log + L if and only if the quantity

h L log + L ∆ = 1 0 h * (t) log(1/t) dt (6)
is finite [11, lem. 6.2.], which makes L log + L into a Banach function space. Then, it is a theorem of Zygmund [11, cor. 6.9.] that

h L 1 (T) ≤ C 0 h L log + L (7)
for some universal constant C 0 . A partial converse, due to M. Riesz, asserts that if a real-valued h is bounded from below and if moreover h ∈ L 1 (T), then h ∈ L log + L [11, cor. 6.10].

We mentioned already that h needs not be bounded if h ∈ L ∞ R (T). In this case all one can say in general is that h has bounded mean oscillation, meaning that h ∈ L 1 (T) and

h BMO ∆ = sup E 1 ℓ(E) E | h -h E | dθ < ∞, with h E ∆ = 1 ℓ(E) E h dθ,
where the supremum is taken over all subarcs E ⊂ T. Actually [19, chap. VI, thm 1.5], there is a universal constant C 1 such that

h BMO ≤ C 1 h L ∞ (T) .
The subspace of L 1 (T) consisting of functions whose BM O-norm is finite is called BM O for short. Notice that BMO is a genuine norm modulo additive constants only. A theorem of F. John and L. Nirenberg [19, ch. VI, thm. 2.1] asserts there are positive constants C, c, such that, for each real-valued ϕ ∈ BM O, every arc E ⊂ T, and any x > 0,

ℓ ({t ∈ E : |ϕ(t) -ϕ E | > x}) ℓ(E) ≤ C exp -cx ϕ BMO . (8)
Conversely, if (8) holds for some finite A > 0 in place of ϕ BMO , every arc E and any x > 0, then ϕ ∈ BM O and A ∼ ϕ BMO . The John-Nirenberg theorem easily implies that BM O ⊂ L p for all p < ∞. The space of H

ω e h (ρ) ≤ C 2 ρ 0 ω h (t) t dt + ρ π ρ ω h (t) t 2 dt (9)
where C 2 is a constant independent of f . From (9) it follows easily that h is continuous if h is Dini-continuous, and moreover that

h L ∞ (T) ≤ ω e h (π) ≤ C 2 π 0 ω h (t) t dt, (10)
where the first inequality comes from the fact that h is continuous on T and therefore vanishes at some point since it has zero-mean. We now turn to multiplicative properties of Hardy functions. It is well-known (see e.g. [START_REF] Duren | Theory of H p spaces[END_REF][START_REF] Garnett | Bounded analytic functions[END_REF][START_REF] Koosis | Introduction to H p spaces[END_REF]) that a nonzero f ∈ H p can be uniquely factored as f = jw where

w(z) = exp 1 2π 2π 0 e iθ + z e iθ -z log |f (e iθ )| dθ (11)
belongs to H p and is called the outer factor of f , while j ∈ H ∞ has modulus 1 a.e. on T and is called the inner factor of f . The latter may be further decomposed as j = bS µ , where

b(z) = e iθ0 z k z l =0 -z l |z l | z -z l 1 -zl z (12)
is the Blaschke product, with order k ≥ 0 at the origin, associated to the sequence z l ∈ D \ {0} and to the constant e iθ0 ∈ T, while

S µ (z) = exp - 1 2π 2π 0 e iθ + z e iθ -z dµ(θ) (13)
is the singular inner factor associated with µ, a positive measure on T which is singular with respect to Lebesgue measure. The z l are of course the zeros of f in D \ {0}, counted with their multiplicities, while k is the order of the zero at 0. If there are infinitely many zeros, the convergence of the product b(z) in D is ensured by the condition l (1 -|z l |) < ∞ which holds automatically when f ∈ H p \ {0}. If there are only finitely many z l , we say that ( 12) is a finite Blaschke product; note that a finite Blaschke product may alternatively be defined as a rational function of the form q/q R , where q is an algebraic polynomial whose roots lie in D and q R indicates the reciprocal polynomial given by q R (z) = z n q(1/z) if n is the degree of q. The integer n is also called the degree of the Blaschke product. That w(z) in [START_REF] Bennett | Interpolation of operators[END_REF] Hp of C \ D can be given a treatment parallel to H p using the conformal map z → 1/z. Specifically, Hp consists of L p functions whose Fourier coefficients of strictly positive index do vanish; these are, a.e. on T, the complex conjugates of H p -functions, and they can also be viewed as nontangential limits of functions analytic in C \ D having uniformly bounded L p means over all circles centered at 0 of radius bigger than 1. We also set BM OA = H1 ∩ BM O. We further single out the subspace Hp 0 of Hp , consisting of functions vanishing at infinity or, equivalently, having vanishing mean on T. Thus, a function belongs to Hp 0 if, and only if, it is a.e. on T of the form e -iθ g(e iθ ) for some g ∈ H p . For G ∈ Hp 0 , the Cauchy formula assumes the form:

G(z) = 1 2 i π T G(ξ) z -ξ dξ , z ∈ C \ D. ( 15
)
If E is a measurable subset of T, we set

< f, g > E = 1 2π E f (e iθ )g(e iθ ) dθ (16)
whenever f ∈ L p (E) and g ∈ L q (E) with 1/p+1/q = 1. If f and g are defined on a set containing E, we often write for simplicity < f, g > E to mean < f |E , g |E >.

The duality product < , > T makes H p and Hq 0 orthogonal to each other, and reduces to the familiar scalar product on L 2 (T) × L 2 (T). We note in particular the orthogonal decomposition:

L 2 (T) = H 2 ⊕ H2 0 . (17)
For f ∈ C(T) and ν ∈ M, the space of complex Borel measures on T, we set

ν.f = T f (e iθ ) dν(θ) (18)
and this pairing induces an isometric isomorphism between M (endowed with the norm of the total variation) and the dual of C(T) [35, thm 6.19]. If we let A ⊂ H ∞ designate the disk algebra of functions analytic in D and continuous on D, and if A 0 indicates those functions in A vanishing at zero, it is easy to see that A 0 is the orthogonal space under [START_REF] Fuhrmann | Linear systems and operators in Hilbert spaces[END_REF] to those measures whose Fourier coefficients of strictly negative index do vanish. Now, it is a fundamental theorem of F. and M. Riesz that such measures have the form dν(θ) = g(e iθ ) dθ with g ∈ H 1 , so the Hahn-Banach theorem implies that H 1 is dual via [START_REF] Fuhrmann | Linear systems and operators in Hilbert spaces[END_REF] to the quotient space C(T)/A 0 [19, chap. IV, sec. 1]. Equivalently, H1 0 is dual to C(T)/A under the pairing arising from the line integral:

( ḟ , F ) = 1 2iπ T f (ξ)F (ξ) dξ , (19)
where F belongs to H1 0 and ḟ indicates the equivalence class of f ∈ C(T) modulo A. This entails that, contrary to L 1 (T), the spaces H 1 and H1 0 enjoy a weak-* compactness property of their unit ball. Finally, we define the analytic and anti-analytic projections P + and P -on Fourier series by:

P + ∞ n=-∞ a n e inθ = ∞ n=0 a n e inθ , P - ∞ n=-∞ a n e inθ = -1 n=-∞
a n e inθ .

Equivalent to the M. Riesz theorem is the fact that P + : L p → H p and P -: L p → Hp 0 are bounded for 1 < p < ∞, in which case they coincide with the Cauchy projections:

P + (h)(z) = 1 2iπ T h(ξ) ξ -z dξ, P -(h)(s) = 1 2iπ T h(ξ) s -ξ dξ, (20)
for z ∈ D, s ∈ C \ D. When restricted to L 2 (T), the projections P + and P -are just the orthogonal projections onto H 2 and H2 0 respectively. Likewise P + : L ∞ → BM OA and P -: L ∞ → BM OA are also bounded. Although P ± (h) needs not be the Fourier series of a function when h is merely in L 1 (T), it is nevertheless Abel summable almost everywhere to a function lying in L s (T) for 0 < s < 1, and it can still be interpreted as the trace of an analytic function in some Hardy space of exponent s that we did not introduce [17, cor. to thm 3.2]. To us it will be sufficient, when h ∈ L 1 , to regard P ± (f ) as the Fourier series of a distribution. Finally, we record for further reference the following elementary fact:

Lemma 1 Let v ∈ L 1 (J) be such that P + (0 ∨ v) ∈ L 2 (T). Then, whenever g ∈ H 2 is such that g ∈ L 2 (I) ∨ L ∞ (J), it holds that < P + (0 ∨ v) , g > T = < v , g > J .
Proof. Since by hypothesis P + (0 ∨ v) is a square summable function on T, we can define a function u ∈ L 1 (T) by the formula:

u = (0 ∨ v) -P + (0 ∨ v) ,

INRIA

and by the very definition of u all its Fourier coefficients of non-negative index do vanish hence u ∈ H1 0 . In addition it is clear that u |I ∈ L 2 (I) and consequently, if g ∈ H 2 is such that g ∈ L 2 (I) ∨ L ∞ (J), we have upon checking summability on I and J separately that ug ∈ H1 0 . Therefore we get:

< v , g > J = < vg , 1 > J = < (0 ∨ v) g , 1 > T = < u g , 1 > T + < P + (0 ∨ v) g , 1 > T = < P + (0 ∨ v) g , 1 > T = < P + (0 ∨ v) , g > T
where the next-to-last equality uses that the mean of the H1 0 -function ug is zero.

3 Well-posedness of the bounded extremal problem BEP

We first reduce problem BEP (2) to a standard form BEP 2,∞ where M ≡ 1.

As the log-modulus of a nonzero Hardy function is integrable, we will safely assume that log M ∈ L 1 (J) for otherwise the zero function is the only candidate approximant. Then, letting w M be the outer function with modulus 1 on I and M on J, we have that g belongs to H 2 and satisfies |g| ≤ M a.e. on J if, and only if g/w M lies in H 2 and satisfies g/w M ≤ 1 a.e. on J; it is so because g/w M lies by construction in the Nevanlinna class N + whose intersection with L 2 (T) is H 2 . Altogether, upon replacing f by f /w M and g by g/w M , we see that Problem ( 2) is equivalent to the following normalized case which is the one we shall really work with.

BEP 2,∞ : Given f ∈ L 2 (I), find g 0 ∈ H 2 such that |g 0 (e iθ )| ≤ 1 a.e. on J and f -g 0 L 2 (I) = min g∈H 2 |g|≤1 a.e. on J f -g L 2 (I) . (21)
Let us begin with a basic existence and uniqueness result:

Theorem 1 Problem BEP 2,∞ (21
) has a unique solution g 0 , and necessarily

g 0 L 2 (I) ≤ f L 2 (I) . Moreover g 0 L ∞ (J) = 1 unless f = g |I for some g ∈ H 2 such that g L ∞ (J) < 1.
Corollary 1 Problem BEP (2) has a unique solution.

Proof of Theorem 1. Define a convex subset of L 2 (I) by putting

C := {g |I ; g ∈ H 2 , g L ∞ (J) ≤ 1}. We claim that C is closed. Indeed, let {g n } be a sequence in H 2 , with g n L ∞ (J) ≤ 1, that converges in L 2 (I) to some φ. Clearly {g n } is bounded in L 2 (T)
, therefore some subsequence g kn converges weakly to g ∈ H 2 . Since |g kn | ≤ 1 on J, we may assume upon refining the subsequence further that it converges weak-* in L ∞ (J) to a limit which can be none but g |J . By weak-* compactness of balls in L ∞ (J), we get g L ∞ (J) ≤ 1, hence g |I ∈ C. But g kn |I a fortiori converges weakly to g |I in L 2 (I), thus φ = g |I ∈ C as claimed. By standard properties of the projection on a non-empty (for 0 ∈ C) closed convex set in a Hilbert space, we now deduce that the solution g 0 to (21) uniquely exists, and is characterized by the variational inequality [13, thm V.2.]

g 0 |I ∈ C and Re < f -g 0 , φ -g 0 > I ≤ 0, ∀φ ∈ C. ( 22
)
Using φ = 0 in [START_REF] Jacob | A constrained approximation problem arising in parameter identification[END_REF] and applying the Schwarz inequality yields

g 0 L 2 (I) ≤ f L 2 (I) . Assume finally that g 0 L ∞ (J) < 1. Given h ∈ H ∞ , g 0 + th is a candidate ap- proximant for small t ∈ R hence the map t → f -g 0 -th 2 L 2 (I)
has a minimum at t = 0. Differentiating under the integral sign and equating the derivative to zero yields 2Re < fg 0 , h > I = 0 whence < fg 0 , h > I = 0 upon replacing h by ih. Letting h = e ikθ for k ∈ N we see that (fg 0 ) ∨ 0 lies in H2 0 , hence it is identically zero because it vanishes on J. Thus f = g 0| I as was to be shown.

INRIA

Theorem 1 entails that the constraint g L ∞ (J) ≤ 1 in Problem ( 21) is saturated unless f = g 0 |I . If the boundary of I has measure zero, much more in fact is true:

Theorem 2 Assume that ℓ(∂I) = 0 and let g 0 be the solution to Problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF].

Then |g 0 | = 1 a.e. on J unless f = g |I for some g ∈ H 2 such that g L ∞ (J) ≤ 1.
It would be interesting to know how much the assumption ℓ(∂I) = 0 can be relaxed in the above statement. Reducing Problem BEP (2) to Problem BEP 2,∞ [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] as before, we obtain as a corollary:

Corollary 2 Assume that ℓ(∂I) = 0 and let g 0 be the solution to Problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF].

If log M ∈ L 1 (J), then |g 0 (e iθ )| = M (e iθ ) a.e. on J unless f = g |I for some g ∈ H 2 such that |g(e iθ )| ≤ M (e iθ ) a.e. on J.
To prove Theorem 2 we establish three lemmas, the second of which will be of repeated use in the paper.

Lemma 2 Let E ⊂ T be infinite and K 1 ⊂ T be a compact set such that E ∩ K 1 = ∅. If we define a collection R of rational functions in the variable z by R = c 0 + i n k=1 c k e i ψ k + z e i ψ k -z ; ( 23
)
c 0 , c k ∈ R , e i ψ k ∈ E , 1 ≤ k ≤ n, n ∈ N , then R |K 1 is uniformly dense in C R (K 1 ).
Proof. It is elementary to check that members of R are real-valued a.e. on T. Also, it is enough to assume that E consists of a sequence {e iψ k } k∈N that converges in T to some e iψ∞ . We work over the real axis where computations are slightly simpler, and for this we consider the Möbius transform:

ϕ(z) = i e iψ∞ + z e iψ∞ -z , that maps T onto R ∪ {∞} with ϕ(e iψ∞ ) = ∞. Set K 2 = ϕ(K 1 )
, and note that it is compact in R since e iψ∞ / ∈ K 1 . Let R R denote the collection of all functions r • ϕ -1 as r ranges over R. We are now left to prove that the restrictions to K 2 of functions in R R are uniformly dense in C R (K 2 ). For this, we put t k = ϕ(e iψ k ) and, denoting by t = ϕ -1 (z) the independent variable in R, we compute from [START_REF] Koosis | Introduction to H p spaces[END_REF] that

R R = {a 0 + n k=1 b k t -t k , a 0 , b k ∈ R , 1 ≤ k ≤ n , n ∈ N },
that is to say R R is the set of real rational functions bounded at infinity, each pole of which is simple and coincides with some t k . Thus if P R,n stands for the space of real polynomials of degree at most n, we get

R R = p n (t) n k=1 (t -t k ) , p n ∈ P R,n , 1 ≤ k ≤ n, n ∈ N , RR n 7087
where the empty product is 1. We claim that to each ǫ > 0 and p ∈ P R,n there exists r ∈ R R such that ||r -p|| L ∞ (K2) ≤ ǫ, and this will achieve the proof since P R,n is dense in C R (K 2 ) by the Stone-Weierstrass theorem. To establish the claim, let U be a neighborhood of 0 in R n such that

∀(x 1 . . . x n ) ∈ U, 1 - 1 n k=1 (1 -x k ) ≤ ǫ 1 + ||p|| L ∞ (K2)
.

Next, pick n distinct numbers t k1 , . . . , t kn so large in modulus that t/t kj ∈ U for t ∈ K 2 and 1 ≤ j ≤ n; this is certainly possible since K 2 is compact whereas

|t k | tends to ∞ because e iψ k → e iψ∞ . Finally, set r(t) = p(t) n j=1 (1 -t t k j
) .

Clearly r belongs to R R , and

||p -r|| L ∞ (K2) ≤ ||p|| L ∞ (K2) 1 - 1 n j=1 (1 -t t k j ) L ∞ (K2)
≤ ǫ as claimed.

Lemma 3 Let f ∈ L 2 (I) and g 0 be the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]. For h a real-valued Dini-continuous function on T supported on the interior

• I of I, let b(z) = 1 2 π I e it + z e it -z h(e it ) dt , z ∈ D, (24)
be the Riesz-Herglotz transform of h. Then b is continuous on D, and moreover

Re < (f -g 0 ) g 0 , b > I = 0 . (25)
Proof. It follows from ( 9) that b continuous on D.

For λ ∈ R, consider the function ω λ (z) = exp λ b(z) , z ∈ D, which is the outer function in H ∞ whose modulus is equal to exp λ h. Since |ω λ | = 1 on J, the function g 0 ω λ is a candidate approximant in problem (21) thus λ → f -g 0 ω λ 2 L 2 (I)
reaches a minimum at λ = 0. By the boundedness of b, we may differentiate this function with respect to λ under the integral sign, and equating the derivative to 0 at λ = 0 yields [START_REF] Yosida | Functional Analysis[END_REF].

Lemma 4 Let f ∈ L 2 (I) and g 0 be the solution to Problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]. Then (fg 0 ) g 0 has real mean on I:

Re < (f -g 0 ) g 0 , i > I = 0. ( 26
)

INRIA

Proof. For each α ∈ [-π, π], the function g 0 e iα belongs to H 2 and is a candidate approximant in [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] since it has the same modulus as g 0 . Hence the function α → fg 0 e iα L 2 (I) reaches a minimum at α = 0, and differentiating under the integral sign yields [START_REF] Lavrentiev | Some improperly posed problems of mathematical physics[END_REF].

Proof of Theorem 2. Since ∂J = ∂I has measure zero, it is equivalent to show that |g 0 | = 1 a.e. on

• J . Let E = {e iθ ∈ • J , |g 0 (e iθ )| < 1} ,
and assume for a contradiction that ℓ(E) > 0. By countable additivity, there is

ε > 0 such that E ε = {e iθ ∈ • J , |g 0 (e iθ )| ≤ 1 -ε}
has strictly positive measure. Hence by inner regularity of Lebesgue measure, there is a compact set K ⊂ E ε such that ℓ(K) > 0, and since K ⊂

•
J it is at positive distance from I, say, η. For λ ∈ R and F a measurable subset of K, let w λ,F be the outer function whose modulus is exp λ on F , and 1 on T \ F . By definition w λ,F (z) = exp {λ A F (z)}, where

A F (z) = 1 2 π F e it + z e it -z dt , z ∈ D (27)
is the Riesz-Herglotz transform of χ F . For λ < log(1/(1ε)) the function g 0 w λ,F belongs to H 2 and satisfies |g 0 w λ,F | ≤ 1 a.e. on J so that, by definition of g 0 , the function λ → fg 0 w λ,F L 2 (I) reaches a minimum at λ = 0. From [START_REF] Leblond | Solution of inverse diffusion problems by analytic approximation with real constraints[END_REF], we see that A F is uniformly bounded on I because |e ite iθ | ≥ η > 0 whenever e it ∈ F and e iθ ∈ I. Therefore we may differentiate under the integral sign to compute the derivative of fg 0 w λ,F 2 L 2 (I) with respect to λ, which gives us

-2Re < f -g 0 exp{λ A F } , g 0 A F exp{λ A F } > I .
Since the latter must vanish at λ = 0 we obtain

Re < f -g 0 , g 0 A F > I = Re < (f -g 0 ) g 0 , A F > I = 0 . (28)
Let e it0 be a density point of K and I l denote the arc centered at e it0 of length l, so that ℓ(I l ∩ K)/l → 1 as l → 0. In particular ℓ(I l ∩ K) = 0 for sufficiently small l. Noting that

e it + e iθ e it -e iθ - e it0 + e iθ e it0 -e iθ ≤ 2l/η 2 for e it ∈ I l ∩ K, e iθ ∈ I, (29)
and observing that (fg 0 )g 0 ∈ L 1 (I), we get from ( 28)-( 29) that Re < (fg 0 )g 0 , e it0 + e iθ e it0e iθ > I

= lim

l→0 Re < (f -g 0 ) g 0 , 2π ℓ(I l ∩ K) A I l ∩K > I = 0 .
Thus, if we let D K denote the set of density points of K, we may capsulize [START_REF] Nikolskii | Operators, functions, and systems: an easy reading[END_REF] and ( 26) by saying that (fg 0 ) g 0 is orthogonal to the real vector space

S K = {i c 0 + n k=1 c k e i φ k + z e i φ k -z , c 0 , c k ∈ R , e i φ k ∈ D K , 1 ≤ k ≤ n, n ∈ N }
for the real scalar product Re < , > I . Since ℓ(∂I) = 0 we can replace I by Ī in this product:

Re < (f -g 0 ) g 0 , r > Ī = 0 , ∀ r ∈ S K . ( 31
)
As ℓ(K) > 0 and almost every point of K is a density point by Lebesgue's theorem [35, sec. 7.12], the set D K is certainly infinite. Moreover, since K ⊂

• J, we have that

I ∩ D K = ∅. Now, Lemma 2 with E = D K and K 1 = I implies in view of (31) that Re < (f -g 0 ) g 0 , iφ > Ī = 0 , ∀φ ∈ C R ( Ī). ( 32
)
By Riesz duality it follows that (fg 0 ) g 0 is real-valued a.e. on Ī. In particular, if h is a Dini-continuous real function supported on

• I, (32) holds with φ = h | Ī . Hence by Lemma 3, where I can be replaced by Ī,

< (f -g 0 ) g 0 , h > Ī = 0. (33)
However, by regularization, Dini-continuous -even smooth-functions are uniformly dense in the space of continuous functions with compact support on • I thus also on I. This implies that either g 0 = f a.e. on I or g 0 = 0 on a set of positive measure, in which case g 0 = 0. In any case, by Theorem 1, f is the trace on I of a H 2 -function with modulus at most 1 on J.

We now turn to the continuity of the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] with respect to the data. Theorem 3 Let f ∈ L 2 (I) and g 0 be the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]. Assume that

f {n} converges to f in L 2 (I) as n → ∞,
g {n} 0 L 2 (I) ≤ f {n} L 2 (I) , hence g {n} 0
is a bounded sequence in H 2 . Let g ∞ be a weak accumulation point and g {kn} 0 a subsequence converging weakly to g ∞ in H 2 ; a fortiori g

f -g ∞ L 2 (I) ≤ lim inf n→∞ f {kn} -g {kn} 0 L 2 (I) .
In particular g ∞ is a candidate approximant, so one has the series of inequalities:

f -g 0 L 2 (I) ≤ f -g ∞ L 2 (I) ≤ lim inf n→∞ f {kn} -g {kn} 0 L 2 (I) (34)
≤ lim sup

n→∞ f {kn} -g {kn} 0 L 2 (I) .
If one of these were strict, there would exist ε > 0 such that

f -g 0 L 2 (I) + ε ≤ f {kn} -g {kn} 0 L 2 (I) (35)
for infinitely many n. But ff {kn} L 2 (I) < ε/2 for large n, thus for infinitely many n (35) yields

f {kn} -g 0 L 2 (I) + ε/2 ≤ f {kn} -g {kn} 0 L 2 (I)
contradicting the definition of g {kn} 0

. Therefore equality holds throughout in [START_REF] Rivlin | Chebyshev polynomials[END_REF], whence g ∞ = g 0 by the uniqueness part of Theorem 1. Equality in [START_REF] Rivlin | Chebyshev polynomials[END_REF] is also to the effect that

lim n→∞ f {kn} -g {kn} 0 = f -g 0 in L 2 (I)
because the norm of the weak limit is not less than the limit of the norms. Refining k n if necessary, we can assume in addition that g {kn} 0 |J converges weak-* to some h in L ∞ (J), and since we already know that it converges weakly to g 0 |J in L 2 (J) we get h = g 0 |J . Finally if ℓ(∂I) = 0, we deduce from Theorem 2 that |g 0 | = 1 a.e. on J hence g {kn} 0 |J converges to g 0| J in L 2 (J) for again the norm of the weak limit is not less than the limit of the norms. Altogether we have shown that any sequence meeting the assumptions contains a subsequence satisfying the conclusions, which is enough to prove the theorem.

To conclude this section, we prove that if f has more summability than required, then so does g 0 .

Proposition 1 Assume that f ∈ L p (I) for some finite p > 2. If g 0 denotes the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] and if ℓ(∂I) = 0, then g 0 ∈ H p and g 0 L p (I) ≤

(1 + K p/2 ) f L p (I) .
Proof. Let h be a Dini-continuous real-valued function supported in

< |g 0 | 2 , h > I = Re < f g 0 , b > I . ( 36
)
Using Hölder's inequality in [START_REF] Schneck | Constrained optimization in hardy spaces[END_REF] and observing that g 0 L 2 (I) ≤ f L 2 (I) ≤ f L p (I) in view of Theorem 1 and the fact that p > 2 while ℓ(I) < 1, we obtain, with 1/p + 1/2 + 1/s 0 = 1:

< |g 0 | 2 , h > I ≤ f L p (I) g 0 L 2 (I) b L s (I) ≤ f 2 L p (I) b L s 0 (I) .
Thus, because the conjugation operator has norm K s0 on L s0 (T) while h is supported on I, we get a fortiori

< |g 0 | 2 , h > I ≤ (1 + K s0 ) f 2 L p (I) h L s 0 (I) . (37)
Now, Dini-continuous functions supported on

• I are dense in L s0 ( • I)
, hence also in L s0 (I) as ℓ(∂I) = 0. Therefore [START_REF] Schneck | Constrained optimization in hardy spaces II: Numerics[END_REF] implies by duality

g 0 L p 1 (I) ≤ (1 + K s0 ) 1/2 f L p (I) , 1/p 1 = (1/p + 1/2)/2. ( 38
)
Hölder's inequality in [START_REF] Schneck | Constrained optimization in hardy spaces[END_REF], using this time (38) instead of g 0 L 2 (I) ≤ f L p (I) , strengthens [START_REF] Schneck | Constrained optimization in hardy spaces II: Numerics[END_REF] to

< |g 0 | 2 , h > I ≤ (1+K s0 ) 1/2 (1+K s1 ) f 2 L p (I) h L s 1 (I) , 1/p+1/p 1 +1/s 1 = 1,
which gives us by duality

g 0 L p 2 (I) ≤ (1 + K s0 ) 1/4 (1 + K s1 ) 1/2 f L p (I) , 1/p 2 = (1/p + 1/p 1 )/2.
Set 1/p k = (1/p + 1/p k-1 )/2 and 1/p + 1/p k + 1/s k = 1. Iterating this reasoning yields by induction

g 0 L p k (I) ≤ f L p (I) Π k-1 j=0 (1 + K sj ) 1/2 k-j . (39)
As k goes large p k increases to p and K s k = K p k+1 /2 decreases to K p/2 . Hence the product on the right of (39) becomes arbitrarily close to 1 + K p/2 , and the result now follows on letting k → +∞.

In problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF], it would be interesting to know whether g 0 ∈ BM OA when f ∈ L ∞ (I) and ℓ(∂I) = 0.

INRIA 4 The critical point equation

In any convex minimization problem, the solution is characterized by a variational inequality saying that the criterium increases with admissible increments of the variable. If the problem is smooth, infinitesimal increments span a halfspace whose boundary hyperplane is tangent to the admissible set, and the variational inequality becomes an equality asserting that the derivative of the objective function is zero on that hyperplane. This equality, sometimes called a critical point equation, expresses that the vector gradient of the objective function in the ambient space lies orthogonal to the constraint; this vector is an implicit parameter of the critical point equation, known as a Lagrange parameter.

In problem ( 21) the variational inequality is [START_REF] Jacob | A constrained approximation problem arising in parameter identification[END_REF]. However, the non-smoothness of the L ∞ -norm makes it a priori unclear whether a critical point equation exists.

It turns out that it does, at least when ℓ(∂I) = 0.

Theorem 4 Assume that f ∈ L 2 (I) is not the trace on I of a H 2 -function of modulus less than or equal to 1 a.e on J, and suppose further that ℓ(∂I) = 0. Then, g 0 ∈ H 2 is the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] if, and only if, the following two conditions hold:

(i) |g 0 (e iθ )| = 1 for a.e. e iθ ∈ J, (ii) there exists a non-negative function λ ∈ L 1 R (J) such that,

(g 0| I -f ) ∨ λ g 0| J ∈ H1 0 . ( 40
)
Moreover, if f ∈ L p (I) for some p such that 2 < p < ∞, then λ ∈ L p R (J). Remark: Note that (40) is equivalent to saying that (g 0| If ) ∨ λ g 0| J ∈ L 1 (T) and

P + (g 0 |I -f ) ∨ λ g 0 |J = 0 (41)
which is the critical point equation proper, with Lagrange parameter λ. Observe that log λ ∈ L 1 R (J), otherwise the H1 0 -function (g 0| If ) ∨ (λ g 0| J ) would be zero hence f = g 0 |I , contrary to the hypothesis.

To prove Theorem 4, we need two lemmas the first of which stands somewhat dual to Lemma 3: Lemma 5 Let f ∈ L 2 (I) and g 0 be the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF].

If h is a non-negative function in L ∞ (T) which is supported on • J, and if a(z) = 1 2 π J e iθ + z e iθ -z h(e iθ ) dθ , z ∈ D, (42)
denotes its Riesz-Herglotz transform, then a is continuous on I and we have that Re < (fg 0 ) g 0 , a > I ≥ 0 .

Proof. Since h is supported in • J, it is clear from the definition that a is continuous on I. For t ∈ R, let us put

w t (z) = exp t a(z) , z ∈ D,
which is the outer function in H ∞ whose modulus is equal to exp{t h}. As h ≥ 0, the function g 0 w t is a candidate approximant in problem (21) when t ≤ 0. Since t → fg 0 w t 2 L 2 (I) can be differentiated with respect to t under the integral sign by the boundedness of a on I, its derivative at t = 0 must be non-positive by the minimizing property of g 0 . But this derivative is just -2Re < (fg 0 ) g 0 , a > I .

Our second preparatory result is of technical nature: Lemma 6 Assume that f ∈ L 2 (I) and let g 0 be the solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF]. If f = g 0 |I and ℓ(∂I) = 0, then there exists a unique λ ∈ L 1 R (J) such that

(g 0| I -f ) g 0| I ∨ λ ∈ H1 0 . ( 44
)
Necessarily λ ≥ 0 a.e. on J, and if

f ∈ L ∞ (I) then λ ∈ L p (J) for 1 < p < ∞. If f {n} ∈ L ∞ (I) converges to f in L 2 (I) while g {n} 0
is the corresponding solution to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF], and if we write by (44)

g 0 {n} |I -f {n} g 0 {n} |I ∨ λ {n} ∈ H1 0 , with λ {n} ∈ L 1 R (J), (45)
then the sequence of concatenated functions in (45) converges weak-* in H1 0 to the function (44).

Proof. The uniqueness of λ is clear because if λ ′ ∈ L 1 R (J) satisfies (44), then 0 ∨ (λλ ′ ) ∈ H1 0 so that λ = λ ′ . To prove the existence of λ, assume first that f ∈ L ∞ (I) and fix p ∈ (2, ∞). By proposition 1 and Hölder's inequality, we know that (g 0f ) g 0 ∈ L p (I). For h a real-valued function in L q (J) where 1/q = 1 -1/p, let a be the Riesz-Herglotz transform of 0 ∨ h given by (42) and put

L(h) = Re < (f -g 0 ) g 0 , a > I . (46)
As 0 ∨ h vanishes on I by construction, it is clear that

L(h) = Re < (f -g 0 ) g 0 , 0 ∨ h > I ,
and since the conjugation operator is bounded by K q on L q R (T), we obtain from Hölder's inequality

|L(h)| ≤ K q (f -g 0 ) g 0 L p (I) h L q (J) .
Thus L is a continuous linear form on L q R (J) and there exists λ

∈ L p R (J) such that L(h) =< λ , h > J , h ∈ L q (J). ( 47
)
By Lemma 5, L is a positive functional on bounded functions supported on

L(h) = Re < λ , a > J , h ∈ L q (J), (48)
INRIA and therefore, substracting (46) from (48), we get

Re < (g 0| I -f ) g 0| I ∨ λ , a > T = 0 ( 49
)
whenever a is the Riesz-Herglotz transform of some h ∈ L q R (J). By regularization Dini-continuous functions are dense in continuous functions with compact support in • I, so they are dense in L q (I) since ℓ(∂I) = 0. Hence it follows from Lemma 3 and the boundedness of the conjugation operator in

L q R (T) that Re < (g 0 -f ) g 0 , b > I = 0 . ( 50
)
whenever b is the Riesz-Herglotz transform of some φ ∈ L q R (I). As λ is realvalued and Re b = 0 a.e. on J, we may rewrite (50) in the form

Re < (g 0| I -f ) g 0| I ∨ λ , b > T = 0. ( 51
)
Now, by [START_REF] Baratchart | Identification d'un filtre hyperfréquences par approximation dans le domaine complexe[END_REF], every H q -function is the sum of three terms: a pure imaginary constant, the Riesz-Herglotz transform of φ ∨ 0 for some φ ∈ L q R (I), and the Riesz-Herglotz transform of 0 ∨ h for some h ∈ L q R (J). Therefore by ( 51), ( 49), [START_REF] Lavrentiev | Some improperly posed problems of mathematical physics[END_REF] and the realness of λ, we obtain

Re < (g 0 |I -f ) g 0| I ∨ λ , g > T = 0 , ∀g ∈ H q .
Changing g into ig we see that the real part is superfluous and letting g(e iθ ) = e ikθ for k ∈ N we get

(g 0 |I -f ) g 0| I ∨ λ ∈ Hp 0 . ( 52
)
If f is now an arbitrary function in L 2 (I) and f {n} , g {n} 0 are as indicated in the statement of the lemma, we know from (52), since f {n} ∈ L ∞ (I), that there is a unique λ {n} meeting (45). By Theorem 3 we have that g

{n} 0 → g 0 in H 2 , hence by the Schwarz inequality lim n→∞ g {n} 0 -f {n} g {n} 0 -(g 0 -f ) g 0 L 1 (I) = 0. (53)
Besides, since λ {n} ≥ 0 and the mean on T of a H1 0 -function is zero, (45) implies

λ {n} L 1 (J) = J λ {n} (t) dt = I f {n} -g {n} 0 g {n} 0 (t) dt ≤ g {n} 0 -f {n} g {n} 0 L 1 (I)
, and in view of (53) we deduce that λ {n} L 1 (J) is bounded independently of n. Consequently the sequence

g {n} 0 |I -f {n} g {n} 0 |I ∨ λ {n} (54)
has a weak-* convergent subsequence to some F in H1 0 , regarding the latter as dual to C(T)/A under the pairing < , > T . Checking this convergence on continuous functions supported on the interior of I, we conclude from (53) that

F |• I = (g 0 |I -f ) g 0| I a.e.

on

• I thus also on I. Therefore if we let λ = F |J , we RR n 7087 meet (44). Checking the same convergence on positive functions supported on • J, we deduce since λ {n} ≥ 0 that F |J is non-negative. Finally, since F is determined by its trace (g 0 |If ) g 0| I on I, there is a unique weak-* accumulation point of the bounded sequence (54) which is thus convergent.

Proof of Theorem 4. To prove sufficiency, assume that g 0 ∈ H 2 satisfies (i)-(ii), and let u ∈ H 2 be such that u L ∞ (J) ≤ 1. From (41) we get

P + 0 ∨ λ g 0| J = P + (f -g 0| I ) ∨ 0 ∈ H 2 ,
thus applying Lemma 1 with v = λ g 0| J and g = ug 0 , we obtain

< λg 0 , u -g 0 > J = -< P + (f -g 0| I ) ∨ 0 , u -g 0 > T (55) = -< f -g 0 , u -g 0 > I .
Since Re < λg 0 , ug 0 > J = Re < λ , uḡ 0 -1 > J is non-negative because λ ≥ 0 and Re(uḡ 0 ) ≤ |u| ≤ 1, we see from (55) that ( 22) is met.

Proving necessity is a little harder. For this, let g 0 solve problem 21 and observe from Theorem 2 that (i) holds. Thus we are left to prove (ii); in fact, we will show that the function λ from Lemma 6 meets (40). Assume first that f ∈ L ∞ (I). From Proposition 1 we get in particular g 0 ∈ H 4 , and by Lemma 6 there is λ ≥ 0 in L 2 R (J) such that (44) holds with H1 0 replaced by H2 0 . Using (i), we may rewrite this as

(g 0| I -f ) ∨ λ g 0| j g 0 = F, F ∈ H2 0 . (56)
Let g 0 = jw be the inner-outer factorization of g 0 . We will show that F ∈ j H2 0 , and this will achieve the proof when f ∈ L ∞ (I). Indeed, dividing (56) by ḡ0 then yields

(g 0 |I -f ) ∨ λ g 0 |j ∈ w-1 H2 0 ( 57
)
which means that the concatenated function in (57) is of the form:

e -iθ g(e iθ )/w(e iθ ) for some g ∈ H 2 . However, g/w belongs to the Nevanlinna class N + by definition, and it also lies in L 2 (T) because so does the function on the left-hand side of (57) (recall |g 0 | = 1 a.e. on J). Hence g/w ∈ H 2 , implying that e -iθ g(e iθ )/w(e iθ ) ∈ H2 0 ⊂ H1 0 , as desired. Let j = bS µ where b is the Blaschke product defined by [START_REF] Borwein | Convex Analysis and Nonlinear Optimization[END_REF] and S µ the singular inner factor defined by [START_REF] Brézis | Analyse fonctionnelle[END_REF]. To prove that F ∈ j H2 0 , it is enough by uniqueness of the inner-outer factorization to establish separately that F ∈ b H2 0 and F ∈ Sµ H2 0 . To establish the former, it is sufficient to show that F ∈ b1 H2 0 whenever b 1 is a finite Blaschke product dividing b, i.e. such that b = b 1 b 2 with b 2 a Blaschke product. Pick such a b 1 and put for simplicity γ 0 = b 2 S µ w, so that g 0 = b 1 γ 0 . We can write b 1 = q/q R , where q is an algebraic polynomial and q R = z n q(1/z) its reciprocal. We may assume that q is monic and deg q > 0:

q(z) = z n + α n-1 z n-1 + α n-2 z n-2 + . . . + α 0 ,
for some n ∈ N \ {0}.

INRIA

When the set of monic polynomials of degree n gets identified with C n , taking as coordinates all the coefficients except the leading one, the subset Ω of those polynomials whose roots

lie in D is open. Now, if Q ∈ Ω and b Q = Q/Q R denotes the associated Blaschke product, the function g = b Q γ 0 is a candidate approximant in Problem (21) since |g| = |g 0 | on T, thus the map Q → f -γ 0 b Q 2 L 2 (I) (58)
reaches a minimum on Ω at Q = q. Let us write a generic Q ∈ Ω as

Q(z) = z n + a n-1 z n-1 + a n-2 z n-2 + . . . + a 0 . Because b Q (e iθ
) is a rational function in the variables a j whose denominator is locally uniformly bounded away from 0 on T, we may differentiate (58) under the integral sign with respect to Re a j , Im a j . Since q is a minimum point, equating these partial derivatives to zero at (a l ) = (α l ) yields

-2Re < (f -g 0 ) γ 0 , x j ∂b Q (e iθ ) ∂Re a j + y j ∂b Q (e iθ ) ∂Im a j a l = α l 0≤l≤n-1 > I = 0 ,
for all x j , y j ∈ R and every j ∈ {0, . . . , n -1}. After a short computation, this gives us

Re < (fg 0 ) γ 0 , z j e ijθ q R (e iθ ) -(x jiy j )e i(n-j)θ q(e iθ ) (q R (e iθ ))

2

> I = 0 ,
for all z j = x j , +i y j ∈ C, where the second argument in the above scalar product is a function of e iθ ∈ I. Multiplying both arguments of this product by the unimodular function b 1 (e iθ ) = q R /q(e iθ ) does not affect its value, thus

Re < (fg 0 ) g 0 , z j e ijθ q(e iθ ) -zj e i(n-j)θ q R (e iθ )

> I . ∈ R , (59)
for all z j ∈ C. In another connection, by the very definition of q R , we have that e i(n-j)θ q R (e iθ ) = e i(n-j)θ e inθ q(e iθ ) = e ijθ q(e iθ ) hence the second argument of < , > I in (59) is pure imaginary on T, and since λ is real a.e. on J Re < λ , z j e ijθ q(e iθ ) -zj e i(n-j)θ q R (e iθ )

> J = 0 , ∀ z j ∈ C . (60)
Therefore, substracting (59) from (60), we obtain from (i) and (56) that Re < F , z j e ijθ q(e iθ ) -zj e i(n-j)θ q R (e iθ )

> T = 0 , ∀ z j ∈ C . ( 61
)
The roots of q R are reflected from those of q across T, thus lie outside D. Hence e i(n-j)θ /q R (e iθ ) ∈ H 2 , and since F ∈ H2 0 we see from ( 17) that (61) simplifies to Re < F , z j e ijθ q(e iθ ) > T = 0 , ∀ z j ∈ C .

As z j is an arbitrary complex number, the symbol "Re" is redundant in this equation, therefore < F , e ijθ /q(e iθ ) > T = 0 for all j ∈ {0, . . . , n -1} and combining linearly these n equations gives us < F , p(e iθ ) q(e iθ ) > T = 0 , ∀p ∈ P n-1 ,

where P n-1 is the space of algebraic polynomials of degree at most n -1. Now, it is elementary that b1

H2 0 = q R q H2 0 = P n-1 q ⊥ in H2 0 , (63)
and consequently from ( 62) and (63), we see that F ∈ b1 H2 0 as desired. We turn to the proof that F ∈ Sµ H2 0 , assuming that µ is not the zero measure otherwise it is trivial. We need introduce the inner divisors of S µ which, by uniqueness of the inner-outer factorization, are just the singular factors S µ0 where µ 0 is a positive measure on T such that µµ 0 is still positive. Pick such a µ 0 , and set β 0 = bS µ-µ0 w so that g 0 = S µ0 β 0 . For a ∈ D, consider the function

j a (z) = S µ0 (z) + a 1 + āS µ0 (z) , z ∈ D.
It is elementary to check that j a is inner, so that β 0 j a is a candidate approximant in problem ( 21) because |β 0 j a | = |g 0 | a.e. on T. Therefore the map

a → f -β 0 j a 2 L 2 (I) (64)
reaches a minimum on D at a = 0. Since

∂j a (z) ∂Re a = 1 1 + āS µ0 (z) - S µ0 (z)(S µ0 (z) + a) (1 + āS µ0 (z)) 2 , ∂j a (z) ∂Im a = i 1 + āS µ0 (z) + iS µ0 (z)(S µ0 (z) + a) (1 + āS µ0 (z)) 2 ,
are bounded for z ∈ T, locally uniformly with respect to a ∈ D, we may differentiate (64) under the integral sign with respect to Re a and Im a, and equating both partial derivatives to zero at a = 0 yields

Re < (f -g 0 ) β 0 , (x + iy) -(x -iy)S 2 µ0 > I = 0 , ∀x , y ∈ R .
Multiplying both arguments of < , > I by the unimodular function S µ0 we get Re < (fg 0 ) g 0 , (x + iy) S µ0 -(xiy)S µ0 > I = 0 , ∀x , y ∈ R . (65)

In another connection, as (x+ iy) S µ0 -(x-iy)S µ0 is pure imaginary on T while λ is real-valued, Re < λ , (x + iy) S µ0 -(xiy)S µ0 > J = 0 , ∀x , y ∈ R .

Substracting (65) from (66), we deduce from (i) and (56) that Re < F , (x + iy) S µ0 -(xiy)S µ0 > T = 0 , ∀x , y ∈ R.

INRIA

Since F ∈ H2 0 while S µ0 ∈ H 2 , this simplifies to Re < F , (x + iy) S µ0 ) > T = 0 , ∀x , y ∈ R .

But x + iy is arbitrary in C, so the symbol "Re" is redundant in the above equation and we obtain < F , S µ0 ) > T = 0.

(67)

Put F (e iθ ) = e -iθ g(e iθ ) with g ∈ H 2 , and conjugate (67) after multiplying both arguments by e iθ : < g , e -iθ S µ0 > T = 0.

(68)

As S µ is a nontrivial singular inner factor, it follows from [1, cor. 6.1.] that the closed linear span of the functions P + (e -iθ S µ0 ) when S µ0 ranges over all inner divisors of S µ is equal to (S µ H 2 ) ⊥ in H 2 . Hence (68) implies that g ∈ S µ H 2 , and therefore F ∈ S µ H2 0 as announced. Having completed the proof of necessity when f ∈ L ∞ (I), we now remove this restriction. Let f ∈ L 2 (I) and f {n} ∈ L ∞ (I) converge to f in L 2 (I). Adding to f {n} a small L 2 (I)-function that goes to zero with n if necessary, we may assume that f {n} / ∈ H 2 |I . With the notations of Lemma 6, let us put for simplicity

F {n} ∆ = g 0 {n} |I -f {n} g 0 {n} |I ∨ λ {n} , F ∆ = g 0 |I -f g 0| I ∨ λ. (69)
By the first part of the proof, we can write

F {n} = ḡ{n} 0 G {n} , where G {n} ∆ = (g 0 {n} |I -f {n} ) ∨ λ {n} g 0 {n} |J ∈ H1 0 . (70) Note that G {n} L 1 (T) is bounded since f {n} -g {n} 0 L 2 (I) ≤ f {n} L 2 (I) (
for the zero function is a candidate approximant) and λ {n} g {n} 0

L 1 (J) = λ {n} L 1 (J)
is bounded by Lemma 6. Thus, extracting a subsequence if necessary, we may assume that G {n} converges weak-* to some G ∈ H1 0 , and then G {n} (z) → G(z) for fixed z ∈ C \ D by [START_REF] Chalendar | Approximation in reflexive Banach spaces and applications to the invariant subspace problem[END_REF]. Moreover, still from Lemma 6, we know that F {n} converges to F weak-* in H1 0 , so we get by [START_REF] Chalendar | Approximation in reflexive Banach spaces and applications to the invariant subspace problem[END_REF] again that F {n} (z) → F (z) for fixed z ∈ C \ D. Finally Theorem 3 entails that g {n} 0 → g 0 in H2 , hence using (15) once more we get that ḡ{n} 0 (z) → ḡ0 (z) for fixed z ∈ C \ D. Altogether, in view of (70), this implies

F (z) = lim n→∞ F {n} (z) = ḡ0 (z) G(z) , z ∈ C \ D ,
showing that F/ḡ 0 = G ∈ H1 0 . By (i) and the definition (69) of F , this yields (40) and achieves the proof.

Using Theorem 4 it is easy to characterize the solution to problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF]. For this, we write L 1 (M 2 dθ, J) to mean those functions h on J such that hM 2 ∈ L 1 (J).

Corollary 3 Assume that M ∈ L 2 (J) is non-negative with log M ∈ L 1 (J), and that f ∈ L 2 (I) is not the trace on I of an H 2 -function of modulus less than or equal to M a.e on J; suppose further that ℓ(∂I) = 0. Then, for g 0 ∈ H 2 to be the solution to problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF], it is necessary and sufficient that the following two properties hold: (i) |g 0 (e iθ )| = M (e iθ ) for a.e. e iθ ∈ J, (ii) there exists a non-negative measurable function λ ∈ L 1 (M 2 dθ, J), such that:

(g 0| I -f ) ∨ λ g 0| J ∈ w-1 M H1 0 , ( 71
)
where w M designates the outer function with modulus 1 a.e. on I and modulus M a.e. on J. In particular if 1/M ∈ L ∞ (J) (more generally if λM ∈ L 1 (J)), then (71) amounts to:

(g 0 |I -f ) ∨ λ g 0 |J ∈ H1 0 . ( 72
)
Remark: We observe that, of necessity, log λ ∈ L 1 (J).

Proof. Clearly (i) is equivalent to |g 0 /w M | = 1 a.e. on J, and since |w M | 2 = 1 ∨ M 2 we see on multiplying (71) by wM that it is equivalent to

g 0 |I w M - f w M ∨ λ M 2 g 0 |J w M ∈ H1 0 .
The conclusion now follows from Theorem 4 and the reduction of problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF] to problem [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] given in section 3. If λM ∈ L 1 (J) so does λg 0| J by (i), and the function (71) lies in e -ıθ N + ∩ L 1 (T) = H1 0 .

Relation (72) can be recast as a spectral equation for a Toeplitz operator, which should be compared with those in [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF][START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF] that form the basis of a constructive approach to BEP 2 . There, λ is a constant and the operators involved are continuous. In our case we consider the Toeplitz operator φ 0∨(λ-1)

φ 0∨(λ-1) (g) = P + 0 ∨ (λ -1)g |J , having symbol 0 ∨ (λ -1), with values in H 2 and domain Corollary 4 Let M ∈ L 2 (J) be non-negative and 1/M ∈ L ∞ (J). Assume f ∈ L 2 (I) is not the trace on I of a H 2 -function of modulus less than or equal to M a.e on J; suppose further that ℓ(∂I) = 0. If g 0 is the solution to problem (2) and λ is as in (71), then

D = {g ∈ H 2 ; λg |J ∈ L 1 (J), P + (0 ∨ λg |J ) ∈ H 2 }
g 0 = I + φ 0∨(λ-1) -1 P + (f ∨ 0) . ( 73
)
Proof. From (72) we see that λg 0| J ∈ L 1 (J) and that

P + (0 ∨ λg 0| J ) = P + ((f -g 0| I ) ∨ 0) ∈ H 2 ,
INRIA hence g 0 ∈ D. Using that g 0 = P + (g 0 ), we now obtain (73) on rewriting (72) as

P + g 0 + 0 ∨ (λ -1)g 0| J -f ∨ 0 = 0.
Further smoothness properties of λM 2 ∈ L 1 (J) follow from the next representation formula.

Proposition 2 Let M ∈ L 2 (J) be non-negative with log M ∈ L 1 (J), and assume that f ∈ L 2 (I) is not the trace on I of an H 2 -function of modulus less than or equal to M a.e. on J. Suppose also that ℓ(∂I) = 0. If g 0 denotes the solution to problem (2) and λ ∈ L 1 (M 2 dθ, J) is the non-negative function such that (71) holds, then λM 2 extends across

• J to a holomorphic function F on C \ I satisfying F (1/z) = F (z), z ∈ C \ I. ( 74
)
Moreover, we have the Herglotz-type representation:

F (z) = 1 2iπ I e iθ + z e iθ -z Im f (e iθ ) g 0 (e iθ ) dθ , z ∈ C \ I. (75)
Proof. By (i) of Corollary 3 we know that |g 0 | = M a.e. on J, hence multiplying (71) by ḡ0 we get

|g 0| I | 2 -f ḡ0| I ∨ λ M 2 ∈ e -iθ N + ∩ L 1 (T) = H1 0 . ( 76
)
Call F the concatenated function on the left of (76), so that H(z) = i F (1/z) lies in H The interpretation of λ as a Lagrange parameter is justified by the duality relation below. For convenience, we write L 1 + (M 2 dθ, J) for the set of nonnegative functions in L 1 (M 2 dθ, J) whose logarithm lies in L 1 (J).

Proposition 3 Assume that M ∈ L 2 (J) is non-negative with log M ∈ L 1 (J), and that f ∈ L 2 (I) is not the trace on I of an H 2 -function of modulus less than or equal to M a.e on J. Suppose further that ℓ(∂I) = 0, and let g 0 ∈ H 2 be the solution to Problem 2 with λ as in (71). Then, it holds that

f -g 0 2 L 2 (I) = max µ∈L 1 + (M 2 dθ,J) min g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ (77) RR n 7087 = min g∈H 2 max µ∈L 1 + (M 2 dθ,J) f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ.
Moreover, the max min and the min max are simultaneously met for g = g 0 and µ = λ.

Proof. Let A, B respectively denote the max min and the min max in (77). Setting g = g 0 for each µ, we get fg 0 2 L 2 (I) ≥ A from Corollary 3-(i). For the reverse inequality, we fix µ = λ and we show that min

g∈H 2 f -g 2 L 2 (I) + J λ (|g| 2 -M 2 ) dθ
is attained at g 0 . Clearly, it is enough to minimize over those g ∈ H 2 such that λ|g| 2 ∈ L 1 (J). Pick such a g, and for t ∈ R let g t = g 0 + t(gg 0 ). The function

Ψ(t) = f -g t 2 L 2 (I) + J λ (|g t | 2 -M 2 ) dθ,
is convex and continuously differentiable on R. Differentiating under the integral sign, we get

Ψ ′ (t) = 2Re (< g t -f, g -g 0 > I + < λg t , g -g 0 > J ) ,
and in particular

Ψ ′ (0) = 2Re < (g 0| I -f ) ∨ λg 0| J , g -g 0 > T (78) = 2Re < (g 0| I -f ) ∨ λg 0| J (g -g 0 ) , 1 > T .
Now (g 0f ) ∨ λg 0 ∈ e -iθ N + by (71), and since gg 0 ∈ H 2 it also holds that gg 0 ∈ N + . Therefore

(g 0| I -f ) ∨ λg 0| J (g -g 0 ) ∈ e -iθ N + ,
and since it belongs to L 1 (T) because λ 1/2 g 0| J and λ 1/2 g |J both lie in L 2 (J), we deduce that it is also in H1 0 . Consequently it has zero mean on T, and we see from (78) that Ψ ′ (0) = 0, hence Ψ meets a minimum at 0 by convexity. Expressing that fg 0 2

L 2 (I) = Ψ(0) ≤ Ψ(1) for each g ∈ H 2 such that λ|g| 2 ∈ L 1 (J) leads us to f -g 0 2 L 2 (I) ≤ A,
as desired. Thus we have proven the first equality in (77) and we also have shown it is an equality for g = g 0 and µ = λ.

To establish that f -g 0 2 L 2 (I) = B, observe first that max µ∈L 1 + (M 2 dθ,J) f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ = +∞
unless |g| ≤ M a.e. on J; indeed if |g| > M on a set E ⊂ J of strictly positive measure, we can set µ = ρχ E + ε for fixed ε > 0 and arbitrarily large ρ. Thus we may restrict the minimization in the second line of (77) to those g such that |g| ≤ M a.e. on J. For such g the maximum is attained when µ = 0, and by definition g 0 minimizes fg 0 2 L 2 (I) among them. Moreover, when g = g 0 , we see from Corollary 3-(i) that µ is irrelevant in the criterion and can be chosen INRIA to be λ. This achieves the proof.

Note that Proposition 3 would still hold if we dropped the log-integrability requirement in the definition of L 1 (M 2 dθ, J), for the latter was never needed in the proof. However, this requirement conveniently restricts the maximization space in (77) to a class of µ for which one can form the outer function w µ , and this will be of use in what follows.

The dual functional and Carleman's formulas

For M ∈ L 2 (J) a non-negative function such that log M ∈ L 1 (J) and f ∈ L 2 (I) which is not the trace on I of a H 2 -function of modulus less than or equal to M a.e. on J, we denote by Φ M the dual functional in problem (2) acting on L 1 + (M 2 dθ, J) as follows (compare [12, sec. 4.3]):

Φ M (µ) = min g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ, µ ∈ L 1 + (M 2 dθ, J). ( 79
)
As an infimum of affine functions, Φ M is concave and upper semi-continuous with respect to µ. In view of (77), solving problem (2) amounts to maximize Φ M over the convex set L 1 + (M 2 dθ, J). As we shall see momentarily (cf. Proposition 4), the true nature of Carleman-type formulas in this context is that they solve for the optimal g in (79) whenever the min is attained. We begin with a theorem showing how Carleman's formula solves for g 0 in (71) as a function of f and λ.

Theorem 5 Let M ∈ L 2 (J) be non-negative with log M ∈ L 1 (J), and assume that f ∈ L 2 (I) is not the trace on I of a H 2 -function of modulus less than or equal to M a.e. on J. Suppose that ℓ(∂I) = 0, and let g 0 be the solution to problem (2) while λ ∈ L 1 (M 2 dθ, J) denotes the non-negative function such that (71) holds. Write w λ 1/2 for the outer function with modulus λ 1/2 a.e. on J and modulus 1 a.e. on I. Then

g 0 (z) = 1 2iπ w λ 1/2 (z) I w λ 1/2 (ξ) f (ξ) ξ -z dξ, z ∈ D. (80)
Conversely, if λ is a positive function on J such that log λ ∈ L 1 (J) and if g 0 defined by (80) lies in H 2 , then g 0 is the solution to problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF] where

M = |g 0|J |.
In this case λ is the function appearing in (71).

Proof. Assume g 0 is the solution to problem (2) so that (i) and (ii) of Corollary 3 hold. Dividing (71) by wλ 1/2 and using that |w λ 1/2 | 2 = 1 ∨ λ, we deduce

w λ 1/2 (g 0 -(f ∨ 0)) ∈ w-1 λ 1/2 w-1 M H1 0 .
Since λ ∈ L 1 (M 2 dθ, J), the left-hand side lies in L 2 (T) and therefore it belongs to H2 0 because the right-hand side is in e -iθ N + by construction. In particular

P + (w λ 1/2 (g 0 -(f ∨ 0))) = 0. ( 81
)
But w λ 1/2 g 0 ∈ H 2 because it clearly belongs to N + ∩ L 2 (T), so that (81) implies

w λ 1/2 g 0 = P + (w λ 1/2 g 0 ) = P + (w λ 1/2 (f ∨ 0)) .
Now (80) follows from this and [START_REF] Goluzin | Generalized carleman formula and its application to analytic extension of functions[END_REF]. Conversely, assume that g 0 defined by (80) lies in H 2 and set M = |g 0 | |J . Since f w λ 1/2 ∈ L 2 (I), we see from (80) and ( 20) that g 0 w λ 1/2 ∈ H 2 and that

g 0 w λ 1/2 = P + (f w λ 1/2 ∨ 0)
which implies (81). Thus w λ 1/2 (g 0 -(f ∨0)) ∈ H2 0 and multiplying by wM wλ 1/2 ∈ H2 yields

wM |w λ 1/2 | 2 (g 0 -(f ∨ 0)) = wM (g 0 |I -f ) ∨ λg 0 |J ∈ H1 0 INRIA
from which (71) follows. As (i) of Corollary 3 is met by definition, g 0 indeed solves for problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF].

Theorem 5 can be used as follows to compute the function Φ M (µ) introduced in (79).

Proposition 4 Let M ∈ L 2 (J) be non-negative with log M ∈ L 1 (J), and assume that f ∈ L 2 (I) is not the trace on I of an H 2 -function of modulus less than or equal to M a.e. on J. Suppose further that ℓ(∂I) = 0 and let µ ∈ L 1 + (M 2 dθ, J). Then, the function Φ M (µ) defined by ( 79) can be expressed as

Φ M (µ) = P -f w µ 1/2 ∨ 0 2 L 2 (T) -µ 1/2 M 2 L 2 (J) . ( 82
)
Moreover, if we set

g µ (z) = 1 2iπ w µ 1/2 (z) I w µ 1/2 (ξ) f (ξ) ξ -z dξ, z ∈ D, (83)
then the infimum in the right-hand side of ( 79) is attained at g = g µ whenever the latter belongs to H 2 . In particular, this is the case when 1/µ ∈ L ∞ (J).

Proof. Assume first that µ is such that g µ ∈ H 2 ; this holds in particular when 1/µ ∈ L ∞ (J), because then 1/w µ 1/2 ∈ H ∞ while [START_REF] Goluzin | Generalized carleman formula and its application to analytic extension of functions[END_REF] shows that the integral in (83) lies in H 2 . From Theorem 5 it follows that g µ is the solution to problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF] where M gets replaced by |g µ |, and µ plays the role of λ in (71). Hence Proposition 3 implies that g µ is an infimizer of min

g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -|g µ | 2 ) dθ,
and since µ is kept fixed g µ is clearly also an infimizer of min

g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ
which is just the right-hand side of (79). This proves the second assertion of the proposition. By (83)) and ( 20), taking into account that

|w µ 1/2 | = 1 ∨ µ 1/2
, what precedes can be reformulated as

Φ M (µ) = f -g µ 2 L 2 (I) + J µ (|g µ | 2 -M 2 ) dθ = (w µ 1/2 f ∨ 0) -w µ 1/2 g µ 2 L 2 (T) - J µ M 2 dθ = P H2 0 f w µ 1/2 ∨ 0 2 L 2 (T) -µ 1/2 M 2 L 2 (J)
.

This proves (82) when g µ ∈ H 2 . To get it in general we apply what we just did to the sequence

µ n = µ + 1/n, observing that g µn ∈ H 2 because 1/µ n ∈ L ∞ (J).
By monotone convergence we obtain

lim n→∞ µ 1/2 n M -µ 1/2 M L 2 (J) = 0. ( 84
)
RR n 7087

Moreover, as log µ n decreases to log µ, we certainly have on putting log -(x) = max{-log x, 0} and log

+ (x) = max{log x, 0} that log -µ n ≤ log -µ ≤ | log µ| ∈ L 1 (J), log + µ n ≤ log + µ n M 2 + log M 2 ≤ µ n M 2 -1 + 2| log M | ≤ (µ + 1)M 2 + 1 + 2| log M | ∈ L 1 (J),
and therefore, by dominated convergence as applied to log

µ n = log + µ n - log -µ n , we obtain lim n→∞ exp 1 4 π J e it + z e it -z log µ n dt = exp 1 4 π J e it + z e it -z log µdt , z ∈ • I,
in other words w µ 1/2 n converges pointwise to w µ 1/2 on • I and therefore almost everywhere on I since ℓ(∂I) = 0. Thus, appealing to dominated convergence once more, we get lim

n→∞ f w µ 1/2 n -f w µ 1/2 L 2 (I) = 0, (85)
and from (84), (85), and (82) which is known to hold with µ replaced by µ n , we see that

lim n→∞ Φ M (µ n ) = P H2 0 f w µ 1/2 ∨ 0 2 L 2 (T) -µ 1/2 M 2 L 2 (J) . (86)
In another connection, it is plain that

lim sup n→∞ Φ M (µ n ) ≤ Φ M (µ) ≤ lim inf n→∞ Φ M (µ n ), (87)
where the first inequality comes from (84) and the upper semi-continuity of Φ M in L 1 + (M 2 dθ, J) while the second inequality is obvious from (79), (84), and the fact that µ ≤ µ n . Now (82) follows from (86) and (87).

Being concave on the convex set L 1 + (M 2 dθ, J), the functional Φ M has a directional derivative at every point in each admissible direction. Here, a direction h is said to be admissible at µ ∈ L 1 + (M 2 dθ, J) if µ + th ∈ L 1 + (M 2 dθ, J) as soon as t ≥ 0 is small enough. From a constructive viewpoint, computing this derivative is important when designing ascent algorithms to maximize Φ M and thus numerically solve for problem [START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF]. The next proposition does it, under mild assumptions on f , in those directions h such that h/µ ∈ L ∞ (J). Note since µ = 0 a.e.(for log µ ∈ L 1 (J)) that such directions are dense in the set of all admissible directions, hence this result allows one indeed to find a direction of ascent for Φ M . Proposition 4, suppose in addition that |f | 2 lies in the Zygmund class L log + L. Let further h be a real function on J such that h/µ L ∞ (J) < 1. Then µ + h ∈ L 1 + (M 2 dθ, J) and h ∈ L 1 (M 2 dθ, J). Moreover, defining g µ as in (83), it holds that h|g µ | 2 ∈ L 1 (J) and that

Proposition 5 Assumptions and notations being as in

Φ M (µ + h) -Φ M (µ) - J h(|g µ | 2 -M 2 ) dθ = o h/µ L ∞ (J) , (88)
where the function o, which depends on f and µ only, is a little o of its argument near 0.

INRIA

Proof. Clearly µ + h = µ(1 + h/µ) ∈ L 1 + (M 2 dθ, J) whenever h/µ L ∞ (J) < 1, which in turn entails h ∈ L 1 (M 2 dθ, J). In another connection, we see from ( 20) that (83) can be rewritten as

w µ 1/2 g µ = P + (f w µ 1/2 ∨ 0), (89)
and since

|w µ 1/2 | 2 = 1 ∨ µ we see that w µ 1/2 g µ ∈ H 2 hence h|g µ| J | 2 = (h/µ)µ|g µ| J | 2 ∈ L 1 (J)
for h/µ ∈ L ∞ (J). Thus the integral in the left-hand side of (88) is well-defined.

Next, multiplying the H2 0 -function w µ 1/2 g µ -(w µ 1/2 f ∨ 0) by the H2 -function

w µ 1/2 g µ yields (|g µ| I | 2 -f ḡµ| I ) ∨ µ|g µ| J | 2 ∈ H1 0 . Therefore the conjugate function of (|g µ |I | 2 -Re( f g µ| I )) ∨ µ|g µ |J | 2 lies in L 1 (T),
and by Zygmund's theorem so does the conjugate function of |f | 2 ∨ 0 since the latter lies in L log + L by assumption. Adding up yields

|g µ | 2 |I + |f | 2 2 + |g µ| I -f | 2 2 ∨ µ|g µ | 2 |J ∈ L 1 (T),
and since the function under brace is positive it lies in L log + L by the M. Riesz theorem. A fortiori then,

P -(f w µ 1/2 |I ∨ 0) 2 = (f w µ 1/2 |I ∨ 0) -w µ 1/2 g µ 2 (90) = |g µ| I -f | 2 ∨ µ|g µ | 2 |J ∈ L log + L. Now, let us write w (µ+h) 1/2 (z) = w µ 1/2 (z) exp 1 4 π J e iθ + z e iθ -z log(1 + h/µ)(e iθ ) dθ = w µ 1/2 (z) e ∆ h (z) ,
where we have put for simplicity

∆ h (z) = 1 4 π J e iθ + z e iθ -z log(1 + h/µ)(e iθ ) dθ, z ∈ D. (91)
Note that ∆ h ∈ BM OA since log(1 + h/µ) ∈ L ∞ (J). With this notation, it is straightforward that

P -f w (µ+h) 1/2 ∨ 0 2 L 2 (T) -P -f w µ 1/2 ∨ 0 2 L 2 (T) (92) = P -f w µ 1/2 (e ∆ h -1) ∨ 0 2 L 2 (T) +2Re < P -f w µ 1/2 ∨ 0 , P -f w µ 1/2 (e ∆ h -1) ∨ 0 > T ,
and our next goal is to prove that

2Re < P -f w µ 1/2 ∨ 0 , P -f w µ 1/2 (e ∆ h -1) ∨ 0 > T - J h|g µ | 2 dθ RR n 7087 = o h/µ L ∞ (J) . (93)
For this, since P + + P -= id, we first observe from ( 17) that

< P -f w µ 1/2 ∨ 0 , P -f w µ 1/2 (e ∆ h -1) ∨ 0 > T =< P -f w µ 1/2 ∨ 0 , (e ∆ h -1)(f w µ 1/2 ∨ 0) > T =< P -f w µ 1/2 ∨ 0 , (e ∆ h -1)P -f w µ 1/2 ∨ 0 > T =< P -f w µ 1/2 ∨ 0 2 , e ∆ h -1 > T
where we used in the second equality that (e ∆ h -1)

P + f w µ 1/2 ∨ 0 ∈ H 2 for e ∆ h -1 ∈ H ∞ . Besides, P -f w µ 1/2 ∨ 0 + P + f w µ 1/2 ∨ 0 = 0 a.e. on J
which implies in view of (89) that

J h|g µ | 2 = J h µ P + f w µ 1/2 ∨ 0 2 =< P -f w µ 1/2 ∨ 0 2 , 0 ∨ h/µ > T .
Altogether, the expression inside absolute values on the left-hand side of (93) is thus

< P -f w µ 1/2 ∨ 0 2 , Re 2(e ∆ h -1) -(0 ∨ h/µ) > T .
Now, if we remark from (91) that, on T, we have 2∆ h = 0 ∨ log(1 + h/µ) + iϕ where ϕ denotes the conjugate function of 0 ∨ log(1 + h/µ), the above quantity becomes

Q 1 + Q 2 with Q 1 ∆ = 2 < P -f w µ 1/2 ∨ 0 2 , (cos(ϕ/2) -1) 1 ∨ (1 + h/µ) 1/2 > T , Q 2 ∆ = 2 < P -f w µ 1/2 ∨ 0 2 , (1 + h/µ) 1/2 -1 -h/(2µ) > J .
We prove separately that Q 1 and Q 2 are both o h/µ L ∞ (J) ; here and thereafter, we use the same symbol o for different functions as this causes no confusion. On the one hand, since there is an absolute constant C such that

(1 + h/µ) 1/2 -1 -h/(2µ) < C h/µ 2 L ∞ (J) for h/µ L ∞ (J) < 1, we have that |Q 2 | ≤ 2C f 2 L 2 (T) h/µ 2 L ∞ (J) (94)
which is indeed o h/µ L ∞ (J) , where "o" is independent of µ. On the other hand, as cos(ϕ/2) -1 ≤ 0, it holds for h/µ L ∞ (J) < 1 that

|Q 1 | ≤ 2 √ 2 < P -f w µ 1/2 ∨ 0 2 , 1 -cos(ϕ/2) > T . (95)
Put for simplicity

B h ∆ = (1 -cos(ϕ/2))/ϕ and u n ∆ = min P -f w µ 1/2 ∨ 0 2 , n ,
so that B h is uniformly bounded (independently of h) and so is u n for fixed n.

By monotone convergence we get

< P -f w µ 1/2 ∨ 0 2 , 1 -cos(ϕ/2) > T ( 96
) INRIA = lim n→∞ < u n , 1 -cos(ϕ/2) > T = lim n→∞ < B h u n , ϕ > T .
Being the product of two functions in BM OA, the function

(B h u n + i B h u n )((0 ∨ log(1 + h/µ)) + iϕ)
certainly lies in H 1 and since it is real at 0 we deduce from the Cauchy formula and ( 7) that

|< B h u n , ϕ > T | = < B h u n , log(1 + h/µ) > J (97) ≤ B h u n L log + L log(1 + h/µ) L ∞ (J) .
As

|B h u n | ≤ B h P -f w µ 1/2 ∨ 0 2 the
< P -f w µ 1/2 ∨ 0 2 , 1 -cos(ϕ/2) > T ≤ 2 B h P -f w µ 1/2 ∨ 0 2 L log + L h/µ L ∞ (J)
as soon as h/µ L ∞ (J) < 1/2. Therefore, to that (95

) is o( h/µ L ∞ (J) ), it is enough to show that lim h/µ L ∞ (J) →0 B h P -f w µ 1/2 ∨ 0 2 L log + L = 0. ( 98
)
It is easily checked that B h L ∞ (T) ≤ 8, say, which entails for the decreasing rearrangements the inequality

B h P -f w µ 1/2 ∨ 0 2 * ≤ 8 P -f w µ 1/2 ∨ 0 2 * . ( 99
)
Since the right-hand side of (99) is independent of h and lies in L log + L by (90), we deduce from the definition (6) of the L log + L-norm that to each ε > 0 there is η > 0 such that

E ⊂ [0, 1] and ℓ(E) < η =⇒ E B h P -f w µ 1/2 ∨ 0 2 * (t) log(1/t) dt < ε.
Thus, (98) will hold if only

B h P -f w µ 1/2 ∨ 0 2 *
tends to 0 in measure as h n /µ L ∞ (J) → 0. Since a function and its decreasing rearrangement have the same distribution function, this is equivalent to

lim h/µ L ∞ (J) →0 B h P -f w µ 1/2 ∨ 0 2 = 0 in measure on T. ( 100
)
Because |w µ 1/2 | = 1 on I and P -is a contraction in L 2 (T), we have the Kolmogorov estimate

ℓ{ξ ∈ T; P -f w µ 1/2 ∨ 0 2 > x} ≤ f 2 L 2 (I)
x ,

RR n 7087 hence (100) will hold if B h alone converges to 0 in measure. As |B h | ≤ C ′ |ϕ| for some absolute constant C ′ it is sufficient to establish that ϕ in turn converges to 0 in measure on T. But this follows from the fact that ϕ tends to 0 in L p (T) when h/µ L ∞ (J) → 0 for 1 < p < ∞, since by the M. Riesz theorem

ϕ L p (T) ≤ K p log(1 + h/µ) L p (T) ≤ 2K p h/µ L ∞ (T)
as soon as h/µ L ∞ (J) < 1/2. This completes the proof of (93). In the same vein we show that

P -f w µ 1/2 (e ∆ h -1) ∨ 0 2 L 2 (T) = o h/µ L ∞ (J) . (101)
Indeed, since P -is a contraction in L 2 (T) and |w µ 1/2 | ≡ 1 on I, we have that

P -f w µ 1/2 (e ∆ h -1) ∨ 0 2 L 2 (T) ≤ < |f | 2 , e ∆ h -1 2 > L 2 (I) = 2 < |f | 2 , (1 -cos(ϕ/2)) > L 2 (I)
which can be treated like the right-hand side of (95) to obtain (101), granted that |f | 2 ∨ 0 ∈ L log + L. In view of (82), ( 92), ( 93) and ( 101), the proof is complete once we have observed that

(µ + h) 1/2 M 2 L 2 (J) -(µ) 1/2 M 2 L 2 (J) = J hM 2 . (102)
Remark: It would be interesting to know whether Proposition 5 holds true as soon as f ∈ L 2 (I), without having to assume that |f | 2 ∈ L log + L. In this case, it is easy to check using (10), (94), (95), and (102) that

Φ M (µ + h) -Φ M (µ) - J h(|g µ | 2 -M 2 ) dθ = O h/µ 2 L ∞ (J) + π 0 ω 0∨h/µ (t) t dt 2 ,
which is a weak substitute to (88) under the (much stronger) assumption that 0 ∨ h/µ is Dini-continuous.

INRIA

A constructive polynomial approach

We now establish a finite dimensional (polynomial) analogous of Theorem 4 in order to constructively approach problem BEP 2,∞ [START_REF] Isakov | Inverse problems for partial differential equations[END_REF], assuming I to be a finite union of closed disjoint sub-arcs of T. This is done from the point of view of convex optimization theory and somehow independently of the previous results, and allows us to get an alternative proof of Theorem 4. Let T n be the space of algebraic polynomials in the variable z = e iθ of degree less or equal to n with coefficients in C. We introduce the following finite dimensional bounded extremal problem F BEP n 2,∞ .

F BEP n 2,∞ : For f ∈ L 2 (I), find k n ∈ T n such that |k n (e iθ )| ≤ 1 for a.e e iθ ∈ J and such that f -k n L 2 (I) = min g∈Tn |g|≤1 a.e. in J f -g L 2 (I) . (103)
The existence of k n is ensured by the compactness of the approximation set {g ∈ T n , ||g|| L ∞ (J) ≤ 1} whereas uniqueness follows from the convexity of this set combined with the strict convexity of the L 2 norm. For g ∈ T n define

E(g) = {x ∈ J, |g(x)| = ||g|| L ∞ (J) }
the so called set of critical points of g. The solution k n is caracterised by the following result.

Theorem 6 The element g ∈ T n is the optimal solution of F BEP n 2,∞ (103) if, and only if, the following two conditions hold:

||g|| L ∞ (J) ≤ 1,
there exists a set of at most 2(n + 1) distinct points x i ∈ E(g) and associated positive real numbers λ i such that, for r ≤ 2(n + 1):

∀h ∈ T n , < g -f, h > I + r i=1 λ i g(x i )h(x i ) = 0 . ( 104
)
Moreover the (λ i ) verify the following boundedness equation:

r i=1 |λ i | ≤ 2||f || 2 L 2 (I) . (105)
Remark: The subset of extremal points {x i , i = 1, . . . , r} is possibly empty (i.e. r = 0).

Proof. Suppose g verifies the two latter conditions and differs from k n . Set h = k ng and observe that,

Re g(x i )h(x i ) = Re g(x i )k n (x i ) -1 ≤ 0 i = 1 . . . r. (106)
From the uniqueness and optimality of k n we deduce,

||k n -f || 2 L 2 (I) = ||g -f + h|| 2 L 2 (I) = ||g -f || 2 L 2 (I) + ||h|| 2 L 2 (I) + 2Re < g -f, h > I < ||g -f || 2 L 2 (I)
The latter leads to:

Re < gf, h > I < 0 which combined with (106) contradicts (104).

Conversely, suppose that g solves F BEP n 2,∞ and let φ 0 be the Rlinear forms defined on T n by ∀h ∈ T n , φ 0 (h) = Re < gf, h > I .

For each critical point x of g define the following Rlinear form φ x , ∀h ∈ T n , φ x (h) = Re g(x)h(x) .

Finally define K as follows:

K = {φ 0 } ∪ {φ x , x ∈ E(g)}.
The set K can be seen seen as a subset of the dual of T R n which is defined to be the real vector space formed by the elements of T n . Simple inspection shows that K is a closed bounded set of (T R n ) * , hence compact, as well as its convex hull denoted by K (note that the finite dimensional setting is crucial here). In order to get a contradiction, suppose that 0 ∈ K, so that by the Hahn Banach theorem there exists h 0 ∈ T n such that:

∀φ ∈ K, φ(h 0 ) ≥ τ > 0.
The latter and the continuity of g and h 0 ensure the existence of a neighborhood V of E(g) such that for x in U = J ∩ V , then Re g(x)h 0 (x) ≥ τ 2 , and for

x in J\U , then |g(x)| ≤ 1 -δ (δ > 0). Observe first that for ǫ > 0 such that ǫ||h 0 || L ∞ (J) < δ we have sup x∈J\U |g(x) -ǫh 0 (x)| ≤ 1. ( 107
)
For all x ∈ U :

|g(x) -ǫh 0 (x)| 2 = |g(x)| 2 -2Re g(x)h(x) + ǫ 2 |h 0 (x)| 2 ≤ |g(x)| 2 -2Re ǫg(x)h(x) + ǫ 2 |h 0 (x)| 2 ≤ 1 -ǫτ + ǫ 2 ||h 0 || 2 L ∞ (J) ,
which combined with (107) shows that for ǫ sufficiently small we have,

||g -ǫh|| L ∞ (J) ≤ 1. ( 108
) INRIA But ||f -g -ǫh|| 2 L 2 (J) = ||f -g|| 2 L 2 (J) -2ǫφ 0 (h) + ǫ 2 ||h|| 2 L 2 (J) ≤ ||f -g|| 2 L 2 (J) -2ǫτ + ǫ 2 ||h|| 2 L 2 (J) ,
which, with (108), indicates that for ǫ small enough, the function (gǫh) provides a better candidate for (103) than g. Hence 0 ∈ K. Carathéodory's theorem [START_REF] Rivlin | Chebyshev polynomials[END_REF] is now to the effect that there exists r ′ elements γ k of K with r ′ ≤ 2(n + 1) + 1 such that:

r ′ i=1 α i γ i = 0 , (109)
where the α i are positive real numbers such that

α i = 1. Now suppose that φ 0 = γ k , k = 1, • • • , r ′ . Evaluating the sum (109) on the element g yields, r ′ i=1 α i γ i (g) = r ′ i=1 α i |g(x i )| 2 = 1.
Equation (109) can therefore be written as:

∀h ∈ T n , α 1 Re < f -g, h > I + r ′ i=2 α i Re(g(x i )h(x i )) = 0.
Dividing the latter by α 1 and noting that the latter equation is also true for the element (ih) leads to (104). Finally replacing h by g in (104) we obtain:

r i=1 λ i = r i=1 |λ i | =< f -g, g > I ≤ < f -g, f -g > I +| < f -g, f > I | ≤ ||f || 2 L 2 (I) + ||f -g|| L 2 (I) ||f || L 2 (I) ≤ 2||f || 2 L 2 (I) ,
where the last two inequalities are obtained by observing that 0 is a valid candidate for (103).

The next result is mainly concerned with the behavior of F BEP n 2,∞ when n goes towards infinity.

Theorem 7

The sequence (k n ) of polynomials solutions to F BEP n 2,∞ (103) converges as n → ∞ towards the solution g 0 ∈ H 2 to BEP 2,∞ [START_REF] Isakov | Inverse problems for partial differential equations[END_REF] with respect to the L 2 (T) norm. On J, the sequence (k n ) converges w.r.t. the weak-* topology of L ∞ (J) and w.r.t. the L p (J) norm, for 1 ≤ p < ∞. In other words we have:

lim n→∞ ||g 0 -k n || L 2 (T) = 0 , ∀h ∈ H 1 , lim n→∞ < k n , h > J =< g 0 , h > J , ∀p , 1 ≤ p < ∞, lim n→∞ ||g 0 -k n || L p (J) = 0 .
Proof. Our first objective is to show (constructively, below) that g 0 can be approximated in the L 2 sense on I by polynomials that remain bounded by 1 on J (thus belong to the approximating class of F BEP n 2,∞ for large enough n). By hypothesis, I is the union of N disjoint closed sub-arcs of T and can therefore be written as,

I = N i=1 (e iai , e ibi )
where without loss of generality we can impose

0 = a 1 ≤ b 1 ≤ a 2 • • • ≤ b N ≤ 2π.
The inner-outer decomposition of g 0 therefore takes the form [START_REF] Duren | Theory of H p spaces[END_REF][START_REF] Garnett | Bounded analytic functions[END_REF]:

g 0 (z) = B(z) exp N i=1 1 2π bi ai e it + z e it -z log(|g 0 |)dt .
Let (ǫ n ) be a decreasing of positive real numbers converging towards 0. We define a sequence (v n ) of H 2 as: We claim that (v n ) converges to g 0 in the L 2 sense on I. To prove this we first show that v n converges pointwise to g 0 a.e. in I. Let e iψ be a point of the interior of I and such that g 0 admits a radial limit at e iψ . For n sufficiently large e iψ is contained in none of the sub-arcs (a i , a i + ǫ n ) nor (b i , b i + ǫ n ). This is to the effect that Hence, the αq furnish the desired polynomials.

v n (z) = B(z) exp 1 2π N i=1 bi-ǫn ai+ǫn e it + z e it -z log(|g 0 |)dt = B(z)× exp - 1 2π
Because they belong to the approximating class in F BEP n 2,∞ , for large enough n, the above inequality is to the effect that: (111)

As a bounded sequence of elements of H 2 , (k n ) admits a weak convergent subsequence. The traces on J of this sub-sequence are bounded in the L ∞ sense on J, hence up to another sub-sequence we obtain a sequence (k ′ n ) converging in addition in the weak-* sense on J. Let g be the weak limit (H 2 sense) of k ′ n . As the balls are weak-* closed in L ∞ we have ||g|| L ∞ (J) ≤ 1, and it follows from (111) that ||f -g|| L 2 (I) = ||fg 0 || L 2 (I) . The uniqueness of g 0 leads to g = g 0 . Now (111) and the constraint's saturation are to the effect that lim sup ||k ′ n || L 2 (T) ≤ ||g 0 || L 2 (T) which in turn proves that k ′ n converges strongly in the L 2 sense to g 0 . The same kind of remark on the trace on J of k ′ n leads to the strong convergence in the L p (for the reflexive cases 1 < p < ∞) sense on J. Finally we remark that the preceding arguments are also true when k n is replaced by any subsequence of the latter; hence k n contains no sub-sequence not converging to g 0 .

Remark: A discretization on T n is also taken up in [START_REF] Schneck | Constrained optimization in hardy spaces II: Numerics[END_REF], for approximation issues of BEP type in L p (I) and constrained on T, with smooth data [START_REF] Schneck | Constrained optimization in hardy spaces[END_REF]. This issues might themselves be normalized and formulated as BEP p,∞ type problems, g being this time a Schur function.

When I is a finite union of closed disjoint arcs of T, Theorem 4 may now be viewed as a corollary to Theorem 6, of which it is a infinite dimensional analogous. We detail below this alternative proof.

We define H 2,∞ and H 2,1 to be the following vector spaces:

H 2,∞ = {h ∈ H 2 , ||h|| L ∞ (J) < ∞}, H 2,1 = {h ∈ H 1 , ||h|| L 2 (I) < ∞}.
We begin with a technical lemma.

Lemma 7 Let v ∈ L 1 (J) such that P + (0 ∨ v) ∈ H 2,1 , then the following holds: ∀h ∈ H 2,∞ , < P + (0 ∨ v), h > T =< v, h > J .

Proof. Let u be the function defined on T by u = (0 ∨ v) -P + (0 ∨ v).

By this very definition all the Fourier coefficients of u of non-negative index vanish, u is L 2 integrable on I and L 1 integrable on J. Hence we conclude that u ∈ H 2,1 and that u(0) = 0 (u has now a canonical extension to the disc). Let h ∈ H 2,∞ . We have:

< vχ J , h > T = < u, h > T + < P + (0 ∨ v), h > T = u(0)h(0)+ < P + (0 ∨ v), h > T = < P + (0 ∨ v), h > T , (112)
where the second equality occurs because (uh) ∈ H 1 .

Proof of Theorem 4. In view of (41), point (ii) of Theorem 4 and equation (40) can be equivalently stated as: there exists a non-negative function λ ∈ L 1 R (J) such that, ∀h ∈ H 2,∞ , < gf, h > I + < λg, h > J = 0.

(113)

Suppose that g ∈ H 2 verifies |g(e iθ )| = 1 for a.e. e iθ ∈ J and that (113) holds, while g = g 0 , the solution to BEP 2,∞ . Set h = (g 0g) ∈ H 2,∞ and observe that, Re < λg, h > J = 1 2π J λ(Re(gg 0 ) -1) ≤ 0 (114)

Uniqueness and optimality of g 0 lead (as in the proof of Theorem 6)) to Re < gf, h > I < 0 , which combined with (114) contradicts (113). Suppose now that g is the optimal solution of BEP 2,∞ . The property |g| = 1 on J has already been proved in Theorem 2. In order to let n go to infinity rewrite (104) with self explaining notations as: We define (Λ n ) to be a family of linear forms on C(J) defined in the following way:

∀u ∈ C(J), Λ n (u) = r(n) i=1
λ n i k n (e iθ n i )u(e θ n i ).

INRIA Equation (105) shows now that (Λ n ) is a bounded sequence of elements in C(J) * which by the Banach-Alaoglu theorem [START_REF] Brézis | Analyse fonctionnelle[END_REF] admits a weak-* converging subsequence whose limit we call Λ. Now the Riesz representation theorem ensures the existence of a complex measure µ associated to Λ, so that appealing to Theorem 7 we obtain ∀m ∈ N, < g 0f, e imθ > I + J e imθ dµ = 0 (116) by taking the limit in (115). Now F. and M. Riesz Theorem asserts that µ is absolutely continuous with respect to the Lebesgue measure so that there exists v ∈ L 1 (J) such that:

∀m ∈ N, < g 0f, e imθ > I + < v, e imθ > J = 0 , which is equivalent to ∀m ∈ N, < g 0f, e imθ > I + < λg 0 , e imθ > J = 0 ,

where we have defined ∀z ∈ J, λ(z) = v(z)g 0 (z). Equation ( 117) is to the effect that, P + ((g 0f )χ I ) = -P + (0 ∨ λg 0 ) which indicates that P + (0 ∨ λg 0 ) is in H 2 (note that this is not trivial, since v occurred till now as an L 1 function). Now thanks to Lemma 7 we obtain, ∀u ∈ H 2,∞ < g 0f, u > I + < λg 0 , u > J = 0.

with h ∈ C ∞ c,R (I). We already now (as h is a valid variation) that Re < (fg 0 )g 0 , b > I = 0, which yields ∀h ∈ C ∞ c,R (I), < Im(λ), Im(b) > J = 0 by remarking that b is pure imaginary on J. Now using the same technique as in the proof of the constraint's saturation we obtain ∀u ∈ C R (J), < Im(λ), u > J = 0 which proves that λ takes real values. Finally using the fact that BEP 2,∞ is a convex problem we obtain by derivating one more time that:

Re < (g 0f ), b 2 > I ≥ 0 which leads to ∀u ∈ C R (J), < λ, u 2 > J ≥ 0.

Hence λ ≥ 0. Because (118) implies that (40) holds, the function (fg 0 ) ∨ λg 0 cannot vanish on a measurable set of positive measure unless it is the zero function. But this would imply f = g 0 a.e on I which contradicts the assumptions on f . This yields λ > 0 a.e on J.

  chap. 1, prop. 8]. Therefore (33) in fact holds for every continuous h supported on • I. Consequently (fg 0 ) g 0 must vanishes a.e. on

  and let g {n} 0 indicate the corresponding solution to problem (21). Then g {n} 0 |I converges to g 0| I in L 2 (I) and g {n} 0 |J converges weak-* to g 0 |J in L ∞ (J). If moreover ℓ(∂I) = 0 and f is not the trace on I of a H 2 -function less than 1 in modulus a.e. on J, then g {n} 0 converges to g 0 in L 2 (T). Proof. By definition g {n} 0 L ∞ (J) ≤ 1, and by Theorem 1

  {kn} 0 |I INRIA converges weakly to g ∞| I in L 2 (I). By weak (resp. weak-*) compactness of balls in L 2 (I) (resp. L ∞ (J)), we get |g ∞ | ≤ 1 a.e. on J and

•

  I, and b his Riesz-Herglotz transform. Since b has real part h on T, Lemma 3 gives us

  λ ≥ 0 a.e. on • J thus also on J since ℓ(∂J) = ℓ(∂I) = 0. As Re a = h and λ is real-valued, equation (47) gives us

1

 1 and vanishes at zero since it has zero mean on T. Clearly H has real part Imf ḡ0 |I ∨0 on T, so the Riesz-Herglotz representation (5) yields: i F (1/z) = 1 2π I e iθ + z e iθz Im f (e iθ ) g 0 (e iθ ) dθ , z ∈ D , and upon conjugating and changing z into 1/z we obtain (75) for z ∈ C \ D. As the right-hand side extends analytically to D across • J by reflection, (74) follows.

e

  it + z e itz log(|g 0 |)dt + bi+ǫn bi e it + z e itz log(|g 0 |)dt .

e|

  it + e ψ e ite iψ log(|g 0 |)dt ≤ ai+ǫn ai |e it + e ψ | |e ite iψ | | log(|g 0 |)|dt = log(|g 0 |)|dt.(110)The same is true with b i in place of a i . As the last term of (110) can be set arbitrarily small for n sufficiently large, the pointwise convergence of v n to g 0 is ensured. Finally remark that by construction |v n | ≤ |g 0 | + 1, which leads to the majoration |v ng 0 | 2 ≤ (2|g 0 | + 1) 2 . Hence Lebesgue's dominated convergence theorem applies and limn→∞ ||g 0v n || L 2 (I) = 0. Now let ǫ > 0 and 0 < α < 1 such that ||g 0αg 0 || L 2 (I) ≤ ǫ 4 . Let n 0 sufficiently large such that ||v n0g 0 || L 2 (I) ≤ ǫ 4 .For r < 1 we define u r belonging to disk algebra in the following way,∀θ ∈ [0, 2π], u r (e iθ ) = T P r (θt)v n0 (re it )dt,INRIAwhere P r is the Poisson kernel. Let e iφ ∈ J. Observe that by construction |v n | = 1 a.e on the sub-arc (e i(φ-ǫn 0 ) , e i(φ+ǫn 0 )). This is to the effect that|u r (e iφ )| ≤ T P r (φt)|v n0 (re it )|dt ≤ P r (ǫ n0 ) T |v n0 (re it )|dt + +ǫn 0 -ǫn 0 P r (t)dt ≤ P r (ǫ n0 )||v n0 || L 1 (T) + 1.Hence for r sufficiently close to 1, we have|u r | ≤ 1/α 2 on J and ||u r -v n0 || 2 L 2 (I) ≤ ǫ 4. Finally, we call q the truncated Taylor expansion of u r (which converges uniformly on T), where the truncation order has been chosen large enough so as to ensure that |q| ≤ 1/α on J and ||qu r || 2 L 2 (I) ≤ ǫ 4 . We have: ||αqg 0 || L 2 (I) ≤ α ||qu r || L 2 (I) + ||u rv n0 || L 2 (I) + ||v n0g 0 || L 2 (I) + ||g 0αg 0 || L 2 (I)≤ ǫ .

  lim n→∞ ||fk n || L 2 (I) = ||fg 0 || L 2 (I) .

  ∀m ∈ {0 . . . n}, < k nf, e imθ > I +

( 118 )

 118 In order to prove that λ ∈ R + , consider the valid variation h = g 0 b where b is defined as in[START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF], b(z) = 1 2 π I e it + z e itz h(e it ) dt = 1 2 π T e it + z e itz χ I (e it ) h(e it ) dt ,

  is well-defined rests on the fact that log |f | ∈ L 1 if f ∈ H 1 \{0}; this also entails that a H p function cannot vanish on a subset of strictly positive H p -function if, and only if ρ ∈ L p (T). A useful consequence is that, whenever g 1 ∈ H p1 and g 2 ∈ H p2 , we have g 1 g 2 ∈ H p3 if, and only if g 1 g 2 ∈ L p3 . In particularg 1 g 2 ∈ H p3 if 1/p 1 + 1/p 2 = 1/p 3 .It is a classical fact [19, ch. II, sec. 1] that a function f holomorphic in the unit disk belongs to H p if, and only |f | p , which is subharmonic in D, has a harmonic majorant there. This makes for a conformally invariant definition of Hardy spaces over general domains in C. In this connection, the Hardy space

	Lebesgue measure on T unless it is identically zero. For simplicity, we often say
	that a function is outer (resp. inner) if it is equal to its outer (resp. inner)
	factor.
	Intimately related to Hardy functions is the Nevanlinna class N + consisting of
	holomorphic functions in D that can be factored as jE, where j is an inner
	function and E an outer function of the form
	E(z) = exp	1 2π	0	2π	e iθ + z e iθ -z	log ρ(e iθ ) dθ ,	(14)
	with ρ a positive function such that log ρ ∈ L 1 (T) (although ρ itself need not be summable). Such a function again has nontangential limits of modulud ρ
	a.e. on T that serve as definition of its boundary values. The Nevanlinna class
	will be instrumental to us in that N + ∩ L p (T) = H p , see for example [17, thm 2.11] or [19, 5.8, ch.II]. Thus formula (14) defines a
							INRIA

  0∨(λ-1) is injective, because if g |I ∨ λg |J ∈ H2 0 for some g ∈ D we may multiply it by ḡ to obtain a H1 0 -function h which is real-valued on T and thus identically zero by Poisson representation of h(1/z) ∈ e iθ H 1 .

. By Beurling's theorem

[START_REF] Garnett | Bounded analytic functions[END_REF] chap. II, cor. 7.3

] φ 0∨(λ-1) is densely defined, for D contains w ρ H 2 where w ρ is the outer function with modulus 1 ∨ min(1, 1/λ). Note also that I + φ

  same is true of their decreasing rearrangements, thus by (6) and the inequality | log(1 + h/µ)| ≤ 2|h/µ| which is valid for |h/µ| ≤ 1/2, we obtain from (96)-(97) that

RR n 7087