L Baratchart

J Leblond

F Seyfert

Constrained extremal problems in H 2 and Carleman's formulas

we consider the extremal problem of best approximation to some function f in L 2 (I), with I a subset of the circle, by the trace of a Hardy function whose modulus is bounded pointwise by some gauge function on the complementary subset.

Introduction

If D is a simply 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. 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, recovery becomes then 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] Lavrentiev | Some improperly posed problems of mathematical physics[END_REF][START_REF] Aizenberg | Carleman's formulas in complex analysis[END_REF][START_REF] Partington | Interpolation, identification and sampling[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[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 2iπ 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. More precisely, for w an outer function (see definitions below), we call an expression of the form

g w = 1 2iπ I w(ξ) w(z) f (ξ) ξ -z dξ (2)
a Carleman formula for f , which produces an analytic function g w to approximate f in some way. Expression (2) may also be viewed as a (complex) normalized Cauchy transform.

On the unit disk D where E p (D) coincides with the Hardy class H p , Patil proved that if f ∈ H p with 1 < p < ∞, then the convergence in [START_REF] Ahern | On functions orthogonal to invariant subspaces[END_REF] actually holds in H p [START_REF] Patil | Representation of H p functions[END_REF]. 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 constant modulus 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 the restriction (f n ) | I 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. Also, the results of the present paper entail that if ϕ is any holomorphic function which is bounded on D together with its inverse, and if the boundary of I in T has linear measure 0, then (f n ) | I is closest to f in weighted L 2 (|ϕ | I | 2n , 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 introduced without reference to optimization. They are, however, implicit in that the constraint on T \ I depends on |f n | itself. To move on firmer ground, we make a slight twist and we rather investigate 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 that we raise is :

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

f -g 0 L 2 (I) = min g∈H 2 |g|≤M a.e. on J f -g L 2 (I) . (3)
This should be compared with the so-called bounded extremal problems studied 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][START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF] for 1 ≤ p ≤ ∞:

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

Note that in (3), we did not introduce a reference function ψ on J as in [START_REF] Ansari | Extremal vectors and invariant subspaces[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 holds further difficulties which left here for further research.

When I is of full measure, both problem (3) and BEP L p (I), L p (J) 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 L 2 (I), L 2 (J) is studied when 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 the general version of BEP L 2 (I), L 2 (J) 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 at. 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], and an extension to the case where additional interpolation conditions are imposed in D is carried out in [START_REF] Leblond | Recovery of harmonic functions from partial boundary data respecting internal pointwise values[END_REF]. Existence and uniqueness results for BEP L p (I), L p (J) in the range 1 ≤ p < ∞ are also presented in [START_REF] Baratchart | Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < ∞[END_REF]. Reformulations of BEP L p (I), L p (J) in an abstract setting involving Hilbert or smooth Banach spaces were carried out in [START_REF] Leblond | Constrained approximation and interpolation in hilbert function spaces[END_REF][START_REF] Chalendar | Constrained approximation and invariant subspaces[END_REF][START_REF] Smith | Constrained approximation in banach spaces[END_REF][START_REF] Chalendar | Approximation in reflexive banach spaces and applications to the invariant subspace problem[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 L 2 (I), L 2 (J) where the constraint bears on the imaginary part rather than the modulus, which is 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]. Problem BEP L ∞ (I), L ∞ (J) 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], together with its meromorphic generalization and related completion issues.

An initial incentive to study such problems lies with engineering issues, more precisely with linear system identification and design. This motivation is 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 further discussed 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][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][START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of toeplitz operators[END_REF], the results of which have been effective to identify hyperfrequency filters [START_REF] Baratchart | Identification d'un filtre hyperfréquences par approximation dans le domaine complexe[END_REF]. This connection is more transparent on the half plane, where f represents the so-called transfer-function of a linear dynamical system which is measured pointwise in a frequency band I of the imaginary axis, using harmonic identification techniques. Recall that a linear dynamical system is just a convolution operator on R (identified with the time axis), and that its transfer function is the Fourier-Laplace transform of its kernel [START_REF] Doyle | Feedback Control Theory[END_REF]. By the Paley-Wiener and the Hausdorff-Young theorems, the causality and the stability of the system from L r (R) to L s (R) imply that f belongs to the Hardy class H p of the right half plane with 1/p = 1/r -1/s, as soon as the latter is less than or equal to 1/2. Because f can only be estimated 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 (4) on the half-plane. This problem can be mapped to BEP L p (I), L p (J) 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] Partington | Interpolation, identification and sampling[END_REF][START_REF] Nikolskii | Operators, functions, and systems: an easy reading[END_REF]. Note that it is indeed essential here to bound the behaviour of g 0 on J, for traces of Hardy functions are dense in L p (I) (in C(I) if p = ∞) so that BEP L p (I), L p (J) 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 f -g L p (I) can be made arbitrarily small at the cost of g L p (J) becoming arbitrarily large, which is a version in this context of the classical trade-off between precision and robustess. Motivated by the fact that the transfer function sometimes has to meet uniform bounds for physical reasons (for instance it should be less than 1 in modulus when dealing with passive systems), BEP L p (I), L p (J) -like problems with a pointwise constraint on the modulus of the approximant were considered in [START_REF] Schneck | Constrained hardy space approximation[END_REF], when the approximated function f and the constraint M are assumed to be continuous on I and T, respectively. The present paper seems to be first to deal with a mixed situation, where an integral criterion is minimized on I under a pointwise constraint on J. Beyond the noteworthy connection with Carleman's formulas already mentioned, one motivation to study mixed norms stems again from system identification. Indeed, the L 2 norm on I has a probabilistic interpretation as the variance of the output when the input of the system is a noise whose spectrum is uniformly distributed in the bandwidth; more general spectra can also be handled by weighting the L 2 -norm in (3) with a boundedly invertible weight, which is but a small modification. If one requires in addition that the system to be identified is passive at higher frequencies, as is the case for instance with microwave circuits, one is led to consider problem (3) with M ≡ 1. A quantitative study of problems like (3) seems also relevant to estimate the growth of orthogonal polynomials, including for weights outside the Steklov class, see the discussion in [START_REF] Denisov | On the size of the polynomials orthonormal on the unit circle with respect to a measure which is a sum of the lebesgue measure and p point masses[END_REF]. Problem ( 3) is considerably more difficult to investigate than BEP L 2 (I), L 2 (J) , due to the fact that pointwise evaluation is not smooth -actually not even defined-in L 2 (J), and the analysis depends in a crucial manner on the multiplicative structure of Hardy functions. Beyond existence and uniqueness, our results 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 and conjugate functions. Section 3 deals with existence and uniqueness issues, along with saturation of the constraint. In section 4 we establish an analog, in the present nonsmooth and infinite-dimensional context, of the familiar critical point equation from convex analysis. It gives rise in Section 5 to a dual formulation of the problem which makes connection with Carleman's formulas and turns it into an unconstrained concave maximization issue. We express the derivative under mild assumptions which may be used to design an ascent algorithm.

Notations and preliminaries

Let T be the unit circle endowed with the normalized Lebesgue measure , and I a subset of T such that (I) > 0 with complementary subset J = T \ I. To avoid dealing with trivial instances of problem (3) we assume throughout that (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 on 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 [33, ch. 17, thm 11], namely if g ∈ H p then :

g(z) = 1 2 i π T g(ξ) ξ -z dξ , and also g(z) = 1 2 π T Re e iθ + z e iθ -z g(e iθ ) dθ , z ∈ D. (5)
Because of this Poisson representation, g(re iθ ) converges to g(e iθ ) in L p (T) as soon as 1 ≤ p < ∞. Moreover, (5) 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. (6)
The integral in the right-hand side of ( 6) is 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)
h L log + L ∆ = 1 0 h * (t) log(1/t) dt (7)
is finite [10, lem. 6.2.], which makes L log + L into a Banach function space. Then, it is a theorem of Zygmund [10, cor. 6.9.] that

h L 1 (T) ≤ C 0 h L log + L (8)
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 [10, 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 BM O ∆ = sup E 1 (E) E | h -h E | dθ < ∞, with h E ∆ = 1 (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 BM O ≤ C 1 h L ∞ (T) . (9)
The subspace of L

({t ∈ E : |ϕ(t) -ϕ E | > x}) (E) ≤ C exp -cx ϕ BM O . (10)
Conversely, if [START_REF] Bennett | Interpolation of operators[END_REF]

ω h (ρ) ≤ C 2 ρ 0 ω h (t) t dt + ρ π ρ ω h (t) t 2 dt ( 11
)
where C 2 is a constant independent of f . From [START_REF] Borwein | Convex Analysis and Nonlinear Optimization[END_REF] it follows easily that h is continuous if h is Dinicontinuous, and moreover that

h L ∞ (T) ≤ ω h (π) ≤ C 2 π 0 ω h (t) t dt, (12)
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 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 (13)
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 (14)
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) (15)
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 ( 14) 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 ( 13) 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 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 2π 0 e iθ + z e iθ -z log ρ(e iθ ) dθ , (16)
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 modulus ρ a.e. on T that serve as a definition of its boundary values. The Nevanlinna class will be instrumental to us in that

N + ∩ L p (T) = H p ,
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 particular g 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 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. ( 17
)
If E is a measurable subset of T, we set

< f, g > E = 1 2π E f (e iθ )g(e iθ ) dθ (18)
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 . (19)
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) (20)
and this pairing induces an isometric isomorphism between M (endowed with the norm of the total variation) and the dual of C(T) [33, 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] Goluzin | Generalized carleman formula and its application to analytic extension of functions[END_REF] to those measures whose Fourier coefficients of strictly negative index do vanish. Now, it is a fundamental theorem by F. and M. Riesz that such measures assume the form dν(θ) = g(e iθ ) dθ with g ∈ H 1 , so the Hahn-Banach theorem implies that H 1 is dual via (20) 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 , (21)
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, z ∈ D, P -(h)(s) = 1 2iπ T h(ξ) s -ξ dξ, s ∈ C \ D. (22)
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 the 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. 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 P + (0 ∨ v) ∈ L 2 (T) by hypothesis, we may define u ∈ L 1 (T) by the formula :

u = (0 ∨ v) -P + (0 ∨ v) ,
and by the very definition of u all its Fourier coefficients of non-negative index do vanish hence u ∈ H1 0 . Clearly 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 ( 23
)
where the next-to-last equality uses that the mean of the H1 0 -function ug is zero.

A bounded extremal problem and its well posedness

We first reduce problem (3) to a standard form 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

Normalized Problem 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) . (24)
We begin with a basic existence and uniqueness result :

Theorem 1 Problem (24) 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 (3) has a unique solution.

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

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 closed convex set in a Hilbert space (note that 0 ∈ C), we deduce that the solution g 0 to (24) uniquely exists, and is characterized by the variational inequality [12, thm V.2.]:

g 0| I ∈ C and Re < f -g 0 , φ -g 0 > I ≤ 0, ∀φ ∈ C. ( 25
)
Using φ = 0 in [START_REF] Yosida | Functional Analysis[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 approximant 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 < f -g 0 , h > I = 0 whence < f -g 0 , h > I = 0 upon replacing h by ih. Letting h = e ikθ for k ∈ N we see that (f -g 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. Theorem 1 entails that the constraint g L ∞ (J) ≤ 1 in Problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] is saturated (meaning it is an equality) unless f = g 0| I . If the boundary of I has measure zero, more in fact is true :

Theorem 2 Assume that (∂I) = 0 and let g 0 be the solution to Problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[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 (∂I) = 0 can be relaxed in the above statement. Reducing Problem (3) to Problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] as before, we obtain as a corollary :

Corollary 2 Assume that (∂I) = 0 and let g 0 be the solution to Problem (3). 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 later 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 ; c 0 , c k ∈ R , e i ψ k ∈ E , 1 ≤ k ≤ n, n ∈ N } , ( 26
) 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] Lavrentiev | Some improperly posed problems of mathematical physics[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 ,
where the empty product is 1. We claim that to each > 0 and p ∈ P R,n there exists r ∈ R R such that

||r -p|| L ∞ (K2) ≤ ,
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 ) ≤ 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) ≤ as claimed.
Lemma 3 Let f ∈ L 2 (I) and g 0 be the solution to problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[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, (27)
be the Riesz-Herglotz transform of h. Then b is continuous on D, and moreover

Re < (f -g 0 ) g 0 , b > I = 0 . ( 28
)
Proof.

It follows from ( 11) 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 (24) 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] Leblond | Recovery of harmonic functions from partial boundary data respecting internal pointwise values[END_REF].

Lemma 4 Let f ∈ L 2 (I) and g 0 be the solution to Problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF]. Then (f -g 0 ) g 0 has real mean on I :

Re < (f -g 0 ) g 0 , i > I = 0. ( 29
)
Proof.

For each α ∈ [-π, π], the function g 0 e iα belongs to H 2 and is a candidate approximant in [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] since it has the same modulus as g 0 . Hence the function α → f -g 0 e iα L 2 (I) reaches a minimum at α = 0, and differentiating under the integral sign yields [START_REF] Nikolskii | Operators, functions, and systems: an easy reading[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 (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 (K) > 0, and since K ⊂ • J it is at distance η > 0 from I. 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 ( 30
)
is the Riesz-Herglotz transform of χ F . For λ < log(1/(1 -ε)) 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 → f -g 0 w λ,F L 2 (I) reaches a minimum at λ = 0. From (30), we see that A F is uniformly bounded on I because |e it -e iθ | ≥ η > 0 whenever e it ∈ F and e iθ ∈ I. Therefore we may differentiate under the integral sign to compute the derivative of f -g 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 . ( 31
)
Let e it0 be a density point of K and I l denote the arc centered at e it0 of length l, so that (I l ∩ K)/l → 1 as l → 0. In particular (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,

and observing that (f -g 0 )g 0 ∈ L 1 (I), we get from ( 31)-(32) that Re < (f -g 0 )g 0 , e it0 + e iθ e it0 -e iθ > I = lim

l→0 Re < (f -g 0 ) g 0 , 2π (I l ∩ K) A I l ∩K > I = 0 . (33)
Thus, if we let D K denote the set of density points of K, we may recap ( 33) and ( 29) by saying that (f -g 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 (∂I) = 0 we can replace I by Ī in this product :

Re < (f -g 0 ) g 0 , r >Ī = 0 , ∀ r ∈ S K . (34)
As (K) > 0 and almost every point of K is a density point by Lebesgue's theorem [33, 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 (34) that Re < (f -g 0 ) g 0 , iφ >Ī = 0 , ∀φ ∈ C R ( Ī), (35)
which entails that (f -g 0 ) g 0 is real-valued a.e. on Ī. In particular, if h is a Dini-continuous real function • I thus also on I. Hence, 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 consider the continuity of the solution to problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] with respect to the data. Theorem 3 Let f ∈ L 2 (I) and g 0 be the solution to problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF]. Assume that f {n} converges to f in L 2 (I) as n → ∞, and let 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 inequalities :

f -g 0 L 2 (I) ≤ f -g ∞ L 2 (I) ≤ lim inf n→∞ f {kn} -g {kn} 0 L 2 (I) ≤ lim sup n→∞ f {kn} -g {kn} 0 L 2 (I) . ( 37
)
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) (38)
for infinitely many n. But f -f {kn} L 2 (I) < ε/2 for large n, thus (38) 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 (37), whence g ∞ = g 0 by the uniqueness part of Theorem 1. Equality in (37) 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 (∂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), again because 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 show 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] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] and if (∂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 (39) Using Hölder's inequality in (39) 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 (I) < 1, we obtain

< |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) , 1/p + 1/2 + 1/s 0 = 1.
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) . (40)
Now, Dini-continuous functions supported on

• I are dense in L s0 ( • I)
, hence also in L s0 (I) as (∂I) = 0. Therefore (40) 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. ( 41
)
Hölder's inequality in (39), using this time (41) instead of g 0 L 2 (I) ≤ f L p (I) , strengthens (40) 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 . ( 42
)
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 (42) becomes arbitrarily close to 1 + K p/2 , and the result now follows on letting k → +∞.

In problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF], it would be interesting to know whether g 0 ∈ BM OA when f ∈ L ∞ (I) and (∂I) = 0.

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 half-space 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 gradient of the objective function in the ambient space is a vector lying orthogonal to the constraint; this vector is an implicit parameter of the critical point equation, known as a Lagrange parameter.

In problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] the variational inequality is [START_REF] Yosida | Functional Analysis[END_REF], and 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 (∂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 (∂I) = 0. Then, g 0 ∈ H 2 is the solution to problem (24) 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 . (43)
Moreover, if f ∈ L p (I) for some p such that 2 < p < ∞, then λ ∈ L p (J).

Remark: note that (43) is equivalent to saying that (g 0| I -f ) ∨ λ g 0| J ∈ L 1 (T) and

P + (g 0| I -f ) ∨ λ g 0| J = 0 (44)
which is the critical point equation proper, with Lagrange parameter λ. Observe that log λ ∈ L 1 (J), otherwise the H1 0 -function (g 0| I -f ) ∨ (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] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[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, (45)
denotes its Riesz-Herglotz transform, then a is continuous on I and we have that

Re < (f -g 0 ) g 0 , a > I ≥ 0 . ( 46
)
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 ∞ with modulus exp{t h}. As h ≥ 0, the function g 0 w t is a candidate approximant in problem (24) when t ≤ 0. Since t → f -g 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 < (f -g 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] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF]. If f = g 0| I and (∂I) = 0, then there exists a unique λ ∈ L 1 R (J) such that

(g 0| I -f ) g 0| I ∨ λ ∈ H1 0 . ( 47
)
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] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF], and if we write by (47)

g 0 {n} | I -f {n} g 0 {n} | I ∨ λ {n} ∈ H1 0 , with λ {n} ∈ L 1 R (J), (48)
then the sequence of concatenated functions in (48) converges weak-* in H1 0 to the function (47).

Proof. The uniqueness of λ is clear because if λ 1 ∈ L 1 R (J) satisfies (47), then 0 ∨ (λ -λ 1 ) ∈ H1 0 so that λ = λ 1 . 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 0 -f ) 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 (45) and put

L(h) = Re < (f -g 0 ) g 0 , a > I . ( 49
)
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 Hlder'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).

(50) By Lemma 5, L is a positive functional on bounded functions supported on • J. Hence λ ≥ 0 a.e. on

• J thus also on J since (∂J) = (∂I) = 0. As Re a = h and λ is real-valued, equation (50) gives us

L(h) = Re < λ , a > J , h ∈ L q (J), (51)
and therefore, substracting (49) from (51), we get

Re < (g 0| I -f ) g 0| I ∨ λ , a > T = 0 ( 52
)
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 (∂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 . ( 53
)
whenever b is the Riesz-Herglotz transform of some φ ∈ L q R (I). As λ is real-valued and Re b = 0 a.e. on J, we may rewrite (53) in the form

Re < (g 0| I -f ) g 0| I ∨ λ , b > T = 0. ( 54
)
Now, by [START_REF] Baratchart | Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of toeplitz operators[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 (54), ( 52), [START_REF] Nikolskii | Operators, functions, and systems: an easy reading[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 . ( 55
)
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 (55), since f {n} ∈ L ∞ (I), that there is a unique λ {n} meeting (48). 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. ( 56
)
Besides, since λ {n} ≥ 0 and the mean on T of a H1 0 -function is zero, (48) 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 (56) 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} (57)
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 (56) 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 meet (47). 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| I -f ) g 0| I on I, there is a unique weak-* accumulation point of the bounded sequence (57) 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 (44) 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 = u -g 0 , we obtain

< λg 0 , u -g 0 > J = -< P + (f -g 0| I ) ∨ 0 , u -g 0 > T = -< f -g 0 , u -g 0 > I . ( 58
)
Since Re < λg 0 , u -g 0 > J = Re < λ , uḡ 0 -1 > J is non-negative because λ ≥ 0 and Re(uḡ 0 ) ≤ |u| ≤ 1, we see from (58) that ( 25) is met. Proving necessity is a little harder. For this, let g 0 solve problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] 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 (43). 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 (47) 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 . ( 59
)
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 (59) by ḡ0 then yields

(g 0| I -f ) ∨ λ g 0| j ∈ w-1 H2 0 (60)
which means that the concatenated function in (60) 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 (60) (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 ( 14) and S µ the singular inner factor defined by [START_REF] Denisov | On the size of the polynomials orthonormal on the unit circle with respect to a measure which is a sum of the lebesgue measure and p point masses[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 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}.
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 (24) since |g| = |g 0 | on T, thus the map Q → f -γ 0 b Q 2 L 2 (I) (61)
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 (61) 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 , ∀ x j , y j ∈ R
for every j ∈ {0, . . . , n -1}. After a short computation, this gives us Re < (f -g 0 ) γ 0 , (x j + iy j )e ijθ q R (e iθ ) -(x j -iy j )e i(n-j)θ q(e iθ ) (q R (e iθ ))

2

> I = 0 , ∀ x j , y j ∈ R ,
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 < (f -g 0 ) g 0 , (x j + iy j )e ijθ q(e iθ ) -(x j -iy j )e i(n-j)θ q R (e iθ ) > I = 0 , ∀ x j , y j ∈ R .

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 (62) is pure imaginary on T, and since λ is real a.e. on J Re < λ , (x j + iy j )e ijθ q(e iθ ) -

(x j -iy j )e i(n-j)θ q R (e iθ ) > J = 0 , ∀ x j , y j ∈ R . (63)
Therefore, substracting (62) from (63), we obtain from (i) and (59) that Re < F , (x j + iy j )e ijθ q(e iθ ) -

(x j -iy j )e i(n-j)θ q R (e iθ ) > T = 0 , ∀ x j , y j ∈ R . (64)
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 ( 19) that (64) simplifies to Re < F , (x j + iy j )e ijθ q(e iθ ) > T = 0 , ∀ x j , y j ∈ R .

As x j + iy 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 , (65)
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 , (66)
and consequently from (65) and (66), 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 µ -µ 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 [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] because

|β 0 j a | = |g 0 | a.e. on T. Therefore the map a → f -β 0 j a 2 L 2 (I) (67)
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 (67) 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 < (f -g 0 ) g 0 , (x + iy) S µ0 -(x -iy)S µ0 > I = 0 , ∀x , y ∈ R .

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 -(x -iy)S µ0 > J = 0 , ∀x , y ∈ R . (69)
Substracting (68) from (69), we deduce from (i) and (59) that Re < F , (x + iy) S µ0 -(x -iy)S µ0 > T = 0 , ∀x , y ∈ R.

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. ( 70
)
Put F (e iθ ) = e -iθ g(e iθ ) with g ∈ H 2 , and take conjugates in (70) after multiplying both arguments by e iθ : < g , e -iθ S µ0 > T = 0.

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 (71) 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. (72)
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 . (73)
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] Duren | Theory of H p spaces[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] Duren | Theory of H p spaces[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 ( 17) once more we get that ḡ{n} 0 (z) → ḡ0 (z) for fixed z ∈ C \ D. Altogether, in view of (73), we obtain:

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 (72) of F , this yields (43) and achieves the proof.

Using Theorem 4 it is easy to characterize the solution to problem [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[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 (∂I) = 0.

Then, for g 0 ∈ H 2 to be the solution to problem (3), 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 , (74)
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 (74) amounts to :

(g 0| I -f ) ∨ λ g 0| J ∈ H1 0 . (75)
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 (74) 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 (3) to problem [START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF] given in section 3. If λM ∈ L 1 (J) so does λg 0| J by (i), and the function (74) lies in e -ıθ N + ∩ L 1 (T) = H1 0 . Relation (75) 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 L 2 (I), L 2 (J) . There, λ is a constant and the operators involved are continuous. In our case we consider the Toeplitz operator φ 0∨(λ-1) having symbol 0 ∨ (λ -1), with values in H 2 and domain D = {g ∈ H 2 ; λg

| J ∈ L 1 (J), P + (0 ∨ λg | J ) ∈ H 2 }: φ 0∨(λ-1) (g) = P + 0 ∨ (λ -1)g | J .
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 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 .

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 (∂I) = 0. If g 0 is the solution to problem (3) and λ is as in (74), then

g 0 = I + φ 0∨(λ-1) -1 P + (f ∨ 0) . (76)
Proof. From (75) we see that λg 0| J ∈ L 1 (J) and that P + (0 ∨ λg 0| J ) = P + ((f -g 0| I ) ∨ 0) ∈ H 2 , hence g 0 ∈ D. Using that g 0 = P + (g 0 ), we now obtain (76) on rewriting (75) 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 (∂I) = 0. If g 0 denotes the solution to problem (3) and λ ∈ L 1 (M 2 dθ, J) is the non-negative function such that (74) holds, then λM 2 extends across

• J to a holomorphic function F on C \ I satisfying F (1/z) = F (z), z ∈ C \ I. (77)
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. (78)
Proof. By (i) of Corollary 3 we know that |g 0 | = M a.e. on J, hence multiplying (74) by ḡ0 we get

|g 0| I | 2 -f ḡ0| I ∨ λ M 2 ∈ e -iθ N + ∩ L 1 (T) = H1 0 . (79)
Call F the concatenated function on the left of (79), so that H(z) = i F (1/z) lies in H 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 (6) yields : 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 non-negative functions in L 1 (M 2 dθ, J) whose logarithm lies in L 1 (J).

i F (1/z) = H(z) = - 1
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 (∂I) = 0, and let g 0 ∈ H 2 be the solution to Problem 3 with λ as in (74). Then, it holds that

f -g 0 2 L 2 (I) = sup µ∈L 1 + (M 2 dθ,J) inf g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ = inf g∈H 2 sup µ∈L 1 + (M 2 dθ,J) f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ . (80)
Moreover, the sup inf and the inf sup in (80) are attained for g = g 0 and µ = λ.

Proof. Let A, B respectively denote the sup inf and the inf sup in (80). Setting g = g 0 for each µ, we get f -g 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(g -g 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 = 2Re < (g 0| I -f ) ∨ λg 0| J (g -g 0 ) , 1 > T . (81)
Now (g 0 -f ) ∨ λg 0 ∈ e -iθ N + by (74), and since g -g 0 ∈ H 2 it also holds that g -g 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 (81) that Ψ (0) = 0, hence Ψ meets a minimum at 0 by convexity. Expressing that f -g 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 (80) and we have also shown it is attained for g = g 0 and µ = λ.

To establish that f -g 0 2 L 2 (I) = B, observe first that sup 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. Therefore we may restrict the minimization in the second line of (80) to those g such that |g| ≤ M a.e. on J. For such g the supremum is f -g 2 L 2 (I) , for the integral term is nonpositive and we can pick µ to be a positive but arbitrary small function. As g 0 minimizes g → f -g 2 L 2 (I) by definition, and since the integral term is always 0 if we put g = g 0 by Corollary 3-(i), we may set g = g 0 and µ = λ to attain the inf sup. 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 (80) 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 of problem (3) which acts on L 1 + (M 2 dθ, J) as follows (compare [11, sec. 4.3]):

Φ M (µ) = inf g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ , µ ∈ L 1 + (M 2 dθ, J). (82)
As an infimum of affine functions, Φ M is concave and upper semi-continuous with respect to µ ∈ L 1 + (M 2 dθ, J), when endowed with the natural norm (i.e. the L 1 -norm on J with weight M 2 ). Note that the extra-condition log µ ∈ L 1 (J) makes L 1 + (M 2 dθ, J) non-complete. In view of (80), solving problem (3) amounts to maximize Φ M over the convex set L 1 + (M 2 dθ, J). As we shall see momentarily, the true nature of Carleman-type formulas in this context is that they solve for g 0 in (74) as a function of f and λ, and more generally for the optimal g in (82), whenever the inf is attained (cf. Proposition 4).

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 (∂I) = 0, and and since µ is kept fixed g µ is clearly also an infimizer of inf

g∈H 2 f -g 2 L 2 (I) + J µ (|g| 2 -M 2 ) dθ
which is but the right-hand side of (82). This proves the second assertion of the proposition. By (86)) and ( 22), 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 (85) 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. (87)
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 In other words, w µ 1/2 n converges pointwise to w µ 1/2 on • I and therefore almost everywhere on I since (∂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, (88)
and from (87), (88), and (85) which is known to hold with µ replaced by µ n , we see that

lim n→∞ Φ M (µ n ) = P -f w µ 1/2 ∨ 0 2 L 2 (T) -µ 1/2 M 2 L 2 (J) . (89)
In another connection, it is plain that

lim sup n→∞ Φ M (µ n ) ≤ Φ M (µ) ≤ lim inf n→∞ Φ M (µ n ), (90)
where the first inequality comes from (87) and the upper semi-continuity of Φ M in L 1 + (M 2 dθ, J) while the second inequality is obvious from (82), (87), and the fact that µ ≤ µ n . Now (85) follows from (89) and (90).

We mentioned early in Section 3 that problem (3) reduces to the case where M ≡ 1. If moreover f ∈ L ∞ (I) and we let A f ∨0 : H 2 → H2 0 denote the Hankel operator with symbol f ∨ 0 defined by A f ∨0 (u) = P -((f ∨ 0)u), Proposition 4 yields a formula for the value of the criterion which may be compared with the Nehari theorem (see e.g. [29, Thm. 1.3.2]):

Corollary 5 When f ∈ L ∞ (I) \ H 2
| I and (∂I) = 0, the squared value of problem (24) is:

sup u∈H 2 , |u | I |≡1 A f ∨0 (u) 2 L 2 (T) -u 2 L 2 (T) + (I) . (91)
Proof. This is straightforward from (85) and the first half of (80), except that the maximization bears on outer u only. However, since for every inner function Θ it holds that

P -((f ∨ 0)Θu) 2 = P -(Θ(P -((f ∨ 0)u)) 2 ≤ Θ(P -((f ∨ 0)u) 2 = P -((f ∨ 0)u) 2 ,
maximizing over all u ∈ H 2 having modulus 1 a.e. on I does not increase the value of the problem.

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] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF]. It also sheds light on the role of λ as a "pointwise" Lagrange multiplier. The next proposition does such a computation, 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 5 Assumptions and notations being as in Proposition 4, suppose in addition that |f | ∈ L p (I) for some p > 2. 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 (86), it holds that h|g µ | 2 ∈ L 1 (J) and that

Φ M (µ + h) -Φ M (µ) - J h(|g µ | 2 -M 2 ) dθ = o h/µ L ∞ (J) , (92)
where the function oK is a little "o" of its argument near 0, uniformly with respect to µ.

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). Using [START_REF] Jacob | A constrained approximation problem arising in parameter identification[END_REF], we may rewrite (86) in the following form: Note that ∆ h ∈ BM OA and e ∆ h ∈ H ∞ since log(1 + h/µ) ∈ L ∞ (J) for h/µ L ∞ (J) < 1. Now, 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) = 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 = o h/µ L ∞ (J) . (97)
For this, since P + + P -= id (the identity operator), we observe from ( 19) 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 (93) 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 (97) is therefore equal to < P -f w µ 1/2 ∨ 0 2 , Re 2(e ∆ h -1) -(0 ∨ h/µ) > T .

By (95), it holds on T that 2∆ h = 0 ∨ log(1 + h/µ) + iϕ where ϕ denotes the conjugate function of 0 ∨ log(1 + h/µ). Thus, the quantity above can be rewritten as

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 both Q 1 and Q 2 are o h/µ L ∞ (J) ; hereafter, we use the same symbol o for different functions as this causes no confusion. On the one hand, 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, therefore

|Q 2 | ≤ 2C f 2 L 2 (T) h/µ 2 L ∞ (J) (98)
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 . (99
< P -f w µ 1/2 ∨ 0 2 , 1 -cos(ϕ/2) > T ≤ 2 B(ϕ) P -f w µ 1/2 ∨ 0 2 L log + L h/µ L ∞ (J)

  35) holds with φ = h | Ī . Hence by Lemma 3 (where I may be replaced by Ī), we get that < (f -g 0 ) g 0 , h >Ī = 0. (36) However, by regularization, Dini-continuous functions are uniformly dense in the space of continuous functions with compact support on • I [25, chap. 1, prop. 8]. Therefore (36) in fact holds for every continuous h supported on • I. Consequently (f -g 0 ) g 0 must vanish a.e. on

  solution to problem[START_REF] Krein | Approximation of L 2 (ω 1 , ω 2 ) functions by minimum-energy transfer functions of linear systems[END_REF]. 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 (∂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 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 {kn} 0 | I 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 < |g 0 | 2 , h > I = Re < f g 0 , b > I .

0 whenever b 1

 1 is a finite Blaschke product dividing b, i.e. such that b = b 1 b 2 with b 2 a Blaschke product.

  2π I e iθ + z e iθ -z Im f (e iθ ) g 0 (e iθ ) dθ , z ∈ D , and upon conjugating and changing z into 1/z we obtain (78) for z ∈ C \ D. As the right-hand side extends analytically to D across • J by reflection, (77) follows.

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 .

w µ 1 2 ∨

 12 /2 g µ = P + (f w µ 1/2 ∨ 0),(93) and since |w µ 1/2 | = 1 on I we get that w µ 1/2 g µ ∈ H 2 with norm at most f L 2 (I) because P + is a contraction in L 2 (T). As |w µ 1/2 | 2 = µ on J, we thus have that h|gµ| J | 2 = (h/µ)|w µ 1/2 g µ | 2 | J ∈ L 1 (J) with norm bounded by f 2 L 2 (I) when h/µ L ∞ (J) < 1.In particular, the integral in the left-hand side of (92) is well-defined. Next, multiplying the H2 0 -functionw µ 1/2 g µ -(w µ 1/2 f ∨ 0) = -P -(w µ 1/2 f ∨ 0) by the H2 -function w µ 1/2 g µ (compare (84)) yields that (|g µ| I | 2 -f ḡµ| I ) ∨ µ|g µ| J | 2 ∈ H1 0 , with norm at most f 2 L 2 (I) . 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 is nonnegative and lies in L p/2 (T) by assumption, thus a fortiori in H 1 because p > 2. Adding up yields|g µ | 2 | I + |f | 2 2 + |g µ| I -f | 2 µ|g µ | 2 | J ∈ L 1 (T),with norm bounded by some constant C(f ) depending only on f , and since the function under brace is positive it lies in L log + L with norm bounded by some constant C (f ), thanks to the M. Riesz theorem (cf.[START_REF] Baratchart | Hardy approximation to L ∞ functions on subsets of the circle[END_REF] and the remark thereafter). 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 = |g µ| I -f | 2 ∨ µ|g µ | 2 | J ∈ L log + L(94) with norm bounded by C (f ), where we used (93) again. 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. (95)

)= ( 1 -

 1 Put for simplicity F ∆ = P -f w µ 1/2 ∨ 0 2 and B(ϕ)∆ cos(ϕ/2))/ϕ, noting that |B(ϕ)|, hence also |B(ϕ)| log + |B(ϕ)|, is bounded above independently of ϕ. Then, it holds that< P -f w µ 1/2 ∨ 0 2 , 1 -cos(ϕ/2) > T = < B(ϕ)F , ϕ > T .(100)Now, by[START_REF] Baratchart | Problems of Adamjan-Arov-Krein type on subsets of the circle and minimal norm extensions[END_REF], we have thatϕ BM O ≤ C 1 log(1 + h/µ) L ∞ (J) ,and by (8) the function B(ϕ)F + i B(ϕ)F lies in H 1 with norm at most C 0 B(ϕ)F L log + L . Therefore, by Fefferman's duality [19, Ch. VI, Thm. 4.4], we get that |< B(ϕ)F , ϕ > T | ≤ C 4 B(ϕ)F L log + L log(1 + h/µ) L ∞ (J) (101) for some absolute constant C 4 . Hence, by the inequality | log(1 + h/µ)| ≤ 2|h/µ| which is valid for |h/µ| ≤ 1/2, we obtain from (100)-(101) that

  by a theorem of M. Riesz [19, chap. III, thm 2.3]; in this range of exponents, we will denote its norm by K p . It follows easily from Parseval's relation that 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 < ∞[START_REF] Pichorides | On the best values of the constants in the theorems of m. riesz, zygmund and kolmogorov[END_REF]. A sufficient condition for h to be in L 1 (T) is that h belongs to the the so-called 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 ) = ({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

  holds for some finite A > 0 in place of ϕ BM O , every arc E and any x > 0, then ϕ ∈ BM O and A ∼ ϕ BM O . The John-Nirenberg theorem easily implies that BM O ⊂ L p for all p < ∞. The space of H 1 -functions whose boundary values lie in BM O will be denoted by BM OA, and BM OA/C is a Banach space equipped with the BM O-norm. Clearly BM OA ⊂ H p for 1 ≤ p < ∞,

	and h + i h ∈ BM OA whenever h ∈ L ∞ (T). A sufficient condition for the boundedness of h is that h be
	Dini-continuous; recall that a function h defined on T is said to be Dini-continuous if ω h (t)/t ∈ L 1 ([0, π]),
	where
	ω h (t) = sup	h e iθ1 -h e iθ2 , t ∈ [0, π],
	|θ1-θ2|≤t
	is the modulus of continuity of h. Specifically [19, chap. III, thm 1.3], it holds that

let g 0 be the solution to problem (3) with λ ∈ L 1 (M 2 dθ, J) the non-negative function such that (74) 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

Conversely, if λ is a positive function on J such that log λ ∈ L 1 (J) and if g 0 defined by (83) lies in H 2 , then g 0 is the solution to problem [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF] where M = |g 0 | | J . In this case λ is the function appearing in (74).

Proof. Assume g 0 is the solution to problem (3) so that (i) and (ii) of Corollary 3 hold. Dividing (74) by wλ 1/2 and using that |w λ 1/2 | 2 = 1 ∨ λ, we deduce

w-1

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

But w λ 1/2 g 0 ∈ H 2 because it clearly belongs to N + ∩ L 2 (T), so that (84) implies:

Now, (83) follows from this and [START_REF] Jacob | A constrained approximation problem arising in parameter identification[END_REF]. Conversely, assume that g 0 defined by (83) lies in H 2 and set

, we see from (83) and ( 22) that g 0 w λ 1/2 ∈ H 2 and that

which implies (84). Thus w λ 1/2 (g 0 -(f ∨ 0)) ∈ H2 0 and multiplying by wM wλ 1/2 ∈ H2 yields

from which (74) follows. As (i) of Corollary 3 is met by definition, g 0 indeed solves for problem [START_REF] Alpay | Some extremal problems linked with identification from partial frequency data[END_REF].

Theorem 5 justifies one assertion made in the introduction. Namely if w is outer, we may write it as w I w J where w I (resp. w J ) has modulus 1 on I (resp. J). Then, expression (2) coincides with formula (83) if f gets replaced by f w I , g 0 by g w w I , and w λ 1/2 by w J . So, if w is invertible in H ∞ , it follows from the theorem that g w is a best approximant to f in L 2 (|w| 2 I dθ, I) among those H 2 functions not exceeding |g w | in modulus, pointwise on J. Next, we compute the dual functional Φ M (µ) introduced in (82).

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 (∂I) = 0 and let µ ∈ L 1 + (M 2 dθ, J). Write w µ 1/2 for the outer function with modulus µ 1/2 a.e. on J and modulus 1 a.e. on I. Then, the function Φ M (µ) defined by (82) can be expressed as

Moreover, if we set

then the infimum in the right-hand side of (82) is attained at g = g µ , whenever the latter belongs to H 2 .

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] Jacob | A constrained approximation problem arising in parameter identification[END_REF] shows that the integral in (86) lies in H 2 . From Theorem 5 it follows that g µ is the solution to problem (3) where M gets replaced by |g µ |, and µ plays the role of λ in (74). Hence Proposition 3 implies that g µ is an infimizer of inf

as soon as h/µ L ∞ (J) < 1/2. Therefore, to prove that (99

But F = P -f w µ 1/2 ∨ 0 2 is bounded in L p/2 (T)-norm, independently of µ, by K p f 2 p in view of the M. Riesz theorem and because w µ 1/2 has modulus 1 on I, hence F log + F is bounded in L α (T) for every

the limiting relation (102) follows from Hölder's inequality and the fact that the BM O norm dominates the L q (T) norm for all 1 < q < ∞.

In the same vein we show that

Indeed, since P -is a contraction in L 2 (T) and |w µ 1/2 | ≡ 1 on I, we have that

which can be treated like the right-hand side of (99) to obtain (103). In view of (85), ( 96), ( 97) and ( 103), the proof is complete once we have observed that

Remark: it is unclear whether Proposition 5 holds true as soon as |f | 2 ∈ L log + L. In this case, the difficulty is of course to prove (102). When f merely lies in L 2 (I), it is easy to check using ( 12), (98), (99), and (104) that

which is a weak substitute to (92) under the (much stronger) assumption that 0 ∨ h/µ is Dini-continuous. When f ∈ L p (I) with p > 2, Proposition 5 may be used to constructively approach problem (3) from the point of view of convex optimization using an ascent algorithm, thanks to the uniform character of the function o with respect to µ in (92).