Rachid Zarouf
email: rzarouf@cmi.univ-mrs.fr

EFFECTIVE H ∞ INTERPOLATION

Keywords: 2000 Mathematics Subject Classification. 30E05, 30H05, 32A35, 32A36, 46E20, 46J15 Complex interpolation, Nevanlinna-Pick interpolation, Caratheodory-Schur interpolation, Carleson interpolation, Hardy spaces, Bergman spaces

Given a finite subset σ of the unit disc D and a holomorphic function f in D belonging to a class X, we are looking for a function g in another class Y which satisfies g |σ = f |σ and is of minimal norm in Y . Then, we wish to compare g Y with f X . More precisely, we consider the interpolation constant

When Y = H ∞ , our interpolation problem includes those of Nevanlinna-Pick and Caratheodory-Schur. Moreover, Carleson's free interpolation problem can be interpreted in terms of the constant c (σ, X, H ∞ ). For Y = H ∞ , X = H p (the Hardy space) or X = L 2 a (the Bergman space), we obtain an upper bound for the constant c (σ, X, H ∞ ) in terms of n = card σ and r = max λ∈σ |λ|. Our upper estimates are shown to be sharp with respect to n and r.

1. Introduction 1.1. Statement and historical context of the problem. Let D = {z ∈ C : |z| < 1} be the unit disc of the complex plane and let Hol (D) be the space of holomorphic functions on D. The problem considered is the following: given two Banach spaces X and Y of holomorphic functions on D, X, Y ⊂ Hol (D) , and a finite subset σ ⊂ D, find the least norm interpolation by functions of the space Y for the traces f |σ of functions of the space X, in the worst case of f . The case X ⊂ Y is of no interest, and so one can suppose that either Y ⊂ X or X, Y are incomparable.

More precisely, our problem is to compute or estimate the following interpolation constant

c (σ, X, Y ) = sup f ∈X, f X ≤1 inf g Y : g |σ = f |σ .
If r ∈ [0, 1) and n ≥ 1, we also define C n, r (X, Y ) = sup {c(σ, X, Y ) : card σ ≤ n , |λ| ≤ r, ∀λ ∈ σ} .

Here and later on, H ∞ stands for the space (algebra) of bounded holomorphic functions on D endowed with the norm f ∞ = sup z∈D |f (z)| . The classical interpolation problems -those of Nevanlinna-Pick (1916) and Carathodory-Schur (1908) (see [START_REF] Nikolski | Operators, Function, and Systems: an easy reading[END_REF] p.231 for these two problems) on the one hand, and Carleson's free interpolation (1958) (see [START_REF] Nikolski | Treatise on the shift operator[END_REF] p.158) on the other hand-are of this nature and correspond to the case Y = H ∞ . Two first are "individual", in the sense that one looks simply to compute the norms f H ∞ |σ or f H ∞ /z n H ∞ for a given f . In the case of the third one, we consider infinite sets σ. Let l ∞ (σ) be the space of bounded functions (a λ ) λ∈σ on σ endowed with the norm a l ∞ (σ) = max λ∈σ |a λ |.

Carleson's free interpolation problem is to compare the norms a l ∞ (σ) and inf { g ∞ : g(λ) = a λ , λ ∈ σ} .

In other words, we want to estimate the interpolation constant defined as

c (σ, l ∞ (σ), H ∞ ) = sup a∈l ∞ (σ), a l ∞ ≤1
inf { g ∞ : g(λ) = a λ , λ ∈ σ} .

Let us now explain why our problem includes those of Nevanlinna-Pick and Carathodory-Schur.

(i) Nevannlinna-Pick interpolation problem. Given σ = {λ 1 , ..., λ n } a finite subset of D and W = {w 1 , ..., w n } a finite subset of C, find NP σ, W = inf { f ∞ : f (λ i ) = w i , i = 1, ..., n} .

The classical answer of Pick is the following:

NP σ, W = inf c > 0 : c 2 -w i w j 1 -λ i λ j 1≤i, j≤n ≫ 0 ,
where for any n × n matrix M , M ≫ 0 means that M is positive definite.

(ii) Carathodory-Schur interpolation problem. Given W = {w 0 , w 1 , ..., w n } a finite subset of C, find

CS W = inf { f ∞ : f (z) = w 0 + w 1 z + ... + w n z n + ...} .
The classical answer of Schur is the following:

CS W = (T ϕ ) n ,
where T ϕ is the Toeplitz operator associated with a symbol ϕ , (T ϕ ) n is the compression of T ϕ on P n , the space of analytic polynomials of degree less or equal than n, and ϕ is the polynomial n k=0 w k z k . From a modern point of view, these two interpolation problems (i) and (ii) are included in the following mixed problem: given σ = {λ 1 , ..., λ n } ⊂ D and f ∈ Hol(D), compute or estimate

f H ∞ /BσH ∞ = inf { g ∞ : f -g ∈ B σ Hol(D)} .
From now on, if σ = {λ 1 , ..., λ n } ⊂ D is a finite subset of the unit disc, then

B σ = n j=1 b λj
is the corresponding finite Blaschke product where b λ = λ-z 1-λz , λ ∈ D. The classical Nevanlinna-Pick problem corresponds to the case X = Hol(D), Y = H ∞ , and the one of Carathodory-Schur to the case λ 1 = λ 2 = ... = λ n = 0 and X = Hol(D), Y = H ∞ .

Looking at this problem in the form of computing or estimating the interpolation constant c (σ, X, Y ) which is nothing but the norm of the embedding operator X |σ , • X |σ → Y |σ , • Y |σ , one can think, of course, on passing (after) to the limit -in the case of an infinite sequence {λ j } and its finite sections {λ j } n j=1 -in order to obtain a Carleson type interpolation theorem X |σ = Y |σ , but not necessarily. In particular, even the classical Nevanlinna-Pick theorem (giving a necessary and sufficient condition on a function a for the existence of

f ∈ H ∞ such that f ∞ ≤ 1 and f (λ) = a λ , λ ∈ σ), does not lead immediately to Carleson's criterion for H ∞ |σ = l ∞ (σ).
(Finally, a direct deduction of Carleson's theorem from Pick's result was done by P. Koosis [START_REF] Koosis | Carleson's interpolation theorem deduced from a result of Pick, Complex analysis, operators, and related topics[END_REF] in 1999 only). Similarly, the problem stated for c (σ, X, Y ) is of interest in its own. It is a kind of "effective interpolation" because we are looking for sharp estimates or a computation of c (σ, X, Y ) for a variety of norms • X , • Y . b. The following partial case was especially stimulating (which is a part of a more complicated question arising in an applied situation in [START_REF] Baratchart | Rational and meromorphic approximation in Lp of the circle : systemtheoretic motivations, critical points and error rates[END_REF][START_REF] Baratchart | Rational approximation problem in the real Hardy space H 2 and Stieltjes integrals: a uniqueness theorem[END_REF]): given a set σ ⊂ D, how can one estimate c σ, H 2 , H ∞ in terms of n = card σ and max λ∈σ |λ| = r only? (Here, H 2 is the standard Hardy space of the disc D and is defined below in Subsection 1.3).

c. There is a direct link between the constant c (σ, X, Y ) and numerical analysis. For example, in matrix analysis, it is of interest to bound the norm of an

H ∞ -calculus f (A) ≤ c f ∞ , f ∈ H ∞ ,
for a contraction A on an n-dimensional arbitrary Banach space, with a given spectrum σ(A) ⊂ σ. The best possible con-

stant is c = c (σ, H ∞ , W ), so that c (σ, H ∞ , W ) = = sup f ∞ ≤1 sup { f (A) : A : (C n , | • |) → (C n , | • |) , A ≤ 1, σ(A) ⊂ σ} , where W = f = k≥0 f (k)z k :
k≥0 f (k) < ∞ stands for the Wiener algebra, and the interior sup is taken over all contractions on n-dimensional Banach spaces. An estimate for c (σ, H ∞ , W ) is given in [START_REF] Nikolski | Treatise on the shift operator[END_REF]. An interesting case occurs for f such that f |σ = 1 z | σ (estimates on condition numbers and the norm of inverses of n×n matrices) or f |σ = 1 λ-z |σ (for estimates on the norm of the resolvent of an n × n matrix). Notice that in the same spirit, the case Y = B 0 ∞,1 where B 0 ∞,1 is a Besov algebra presents an interesting case for the functional calculus of finite rank operators, in particular, those satisfying the so-called Ritt condition.

1.3. The spaces X and Y considered here. We systematically use the following conditions for the spaces X and Y , (P 1 )

Hol((1 + ǫ)D) is continuously embedded into Y for every ǫ > 0, (P 2 ) P ol + ⊂ X and P ol + is dense in X,

where P ol + stands for the set of all complex polynomials p, p = N k=0 a k z k ,

(P 3 ) [f ∈ X] ⇒ z n f ∈ X , ∀n ≥ 0 and lim z n f 1 n ≤ 1 , (P 4 ) [f ∈ X, λ ∈ D and f (λ) = 0] ⇒ f z -λ ∈ X .
Assuming X satisfies property (P 4 ) and Y ⊂ X, then the quantity c (σ, X, Y ) can be written as follows

c (σ, X, Y ) = sup f X ≤1 inf { g Y : g ∈ Y, g -f ∈ B σ X} .
General spaces X and Y satisfying (P i ) 1≤i≤4 are studied in Section 3. Then, we study special cases of such spaces: from Section 4 to the end of this paper, Y = H ∞ , but X may change from one section to another. In particular, in Sections 4 and 5, X = H p = H p (D), 1 ≤ p ≤ ∞ which are the standard Hardy spaces on the disc D (see [START_REF] Nikolski | Operators, Function, and Systems: an easy reading[END_REF] Chapter 2) of all f ∈ Hol(D) satisfying

sup 0≤r<1 T |f (rz)| p dm(z) 1/p < ∞,
m being the Lebesgue normalized measure on T. From now on, if f ∈ Hol(D) and k ∈ N, f (k) stands for the k th Taylor coefficient of f.

For p = 2, an equivalent description of H 2 is

H 2 =    f = k≥0 f (k)z k : k≥0 f (k) 2 < ∞    .
We also study (see Section 6) the case

X = l 2 a 1/ √ k + 1 , which is the Bergman space of all f = k≥0 f (k)z k satisfying k≥0 f (k) 2 1 k + 1 < ∞.
This space is also given by: X = L 2 a , the space of holomorphic functions f on D such that

D |f (z)| 2 dA < ∞,
where dA stands for the area measure.

Results

We start studying general Banach spaces X and Y and give some sufficient conditions under which C n, r (X, Y ) < ∞. In particular, we prove the following fact.

Theorem 2.1. Let X, Y be Banach spaces satisfying properties (P i ), i = 1, ..., 4.

Then C n, r (X, Y ) < ∞,
for every n ≥ 1 and r ∈ [0, 1).

Next, we add the condition that X is a Hilbert space, and give in this case a general upper bound for the quantity C n, r (X, Y ). Theorem 2.2. Let Y be a Banach space satisfying property (P 1 ) and X = (H, (•, •) H ) a Hilbert space satisfying properties (P i ) for i = 2, 3, 4. We moreover suppose that for every 0 < r < 1 there exists ǫ > 0 such that k λ ∈ Hol ((1 + ǫ)D) for all |λ| < r, where k λ stands for the reproducing kernel of X at point λ, and λ → k λ is holomorphic on |λ| < r as a Hol((1 + ǫ)D)-valued function. Let σ = {λ 1 , ..., λ 1 , λ 2 , ..., λ 2 , ..., λ t , ..., λ t } be a sequence in D, where λ s are repeated according to their multiplicity m s , t s=1 m s = n. Then we have, i)

c (σ, H, Y ) ≤ n k=1 E k 2 Y 1 2
, where (E k ) 1≤k≤n stands for the Gram-Schmidt orthogonalization (in the space H) of the sequence

k λ1,0 , k λ1,1 , ..., k λ1,m1-1 , k λ2,0 , k λ2,1 , ..., k λ2,m2-1 , ..., k λt,0 , k λt,1 , ..., k λt,mt-1 , and k λ, i = d dλ i k λ , i ∈ N. ii) For the case Y = H ∞ , we have c(σ, H, H ∞ ) ≤ sup ζ∈D P H Bσ k ζ H ,
where

P H Bσ = n k=1 (•, E k ) H E k stands for the orthogonal projection of H onto K Bσ (H), K Bσ (H) = span k λj , i : 1 ≤ i < m j , j = 1, ..., t .
After that, we deal with H ∞ interpolation (Y = H ∞ ). For general Banach spaces (of analytic functions in D) of moderate growth X, we formulate the following conjecture:

c 1 ϕ X 1 - 1 -r n ≤ C n,r (X, H ∞ ) ≤ c 2 ϕ X 1 - 1 -r n ,
where ϕ X (t), 0 ≤ t < 1 stands for the norm of the evaluation functional f → f (t) on the space X. We prove this conjecture for X = H p , L

1 p n 1 -|λ| 1 p ≤ c (σ n, λ , H p , H ∞ ) ≤ C n,r (H p , H ∞ ) ≤ A p n 1 -r 1 p
, where A p is a constant depending only on p and the left-hand side inequality is proved only for p ∈ 2Z + . For p = 2, we have ), and |λ| ≤ r. We have,

A 2 = √ 2. Theorem 2.4. Let 1 ≤ p ≤ ∞, n ≥ 1, r ∈ [0, 1
1 32 1 p n 1 -|λ| 1 p ≤ c (σ n, λ , H p , H ∞ ) ≤ C n,r (H p , H ∞ ) ≤ A p n 1 -r 1 p
, where A p is a constant depending only on p and the left-hand side inequality is proved only for p ∈ 2Z + . For p = 2, we have

A 2 = √ 2.
The above Theorems 2.1 , 2.3 and 2.4 were already announced in the note [START_REF] Zarouf | Interpolation avec contraintes sur des ensembles finis du disque[END_REF].

In order to prove (Theorem 2.1 , Theorem 2.2 and) the right-hand side inequality of Theorem 2.3 and Theorem 2.4, given f ∈ X and σ a finite subset of D, we first use a linear interpolation:

f → n k=1 f, e k e k ,
where •, • means the Cauchy sesquilinear form h, g = k≥0 ĥ(k)ĝ(k), and (e k ) 1≤k≤n is the Malmquist basis (effectively constructible) of the space K B = H 2 ⊖ BH 2 , whith B = B σ (see N. Nikolski, [START_REF] Nikolski | Treatise on the shift operator[END_REF] p. 117)). Next, we use the complex interpolation between Banach spaces, (see H. Triebel [START_REF] Triebel | Interpolation theory, functions spaces, differential operators[END_REF] Theorem 1.9.3-(a) p.59). Among the technical tools used in order to find an upper bound for

n k=1 f, e k e k ∞ (in terms of f X ), the most important one is a Bernstein- type inequality f ′ p ≤ c p B ′ ∞ f p for a (rational) function f in the star- invariant subspace K p B := H p ∩ BzH p , 1 ≤ p ≤ ∞ (for p = 2, K 2 B = K B )
, generated by a (finite) Blaschke product B, (K. Dyakonov [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF][START_REF] Dyakonov | Entire functions of exponential type and model subspaces in H p[END_REF]). For p = 2, we give an alternative proof of the Bernstein-type estimate we need and the constant c 2 we obtain is slightly better, see Section 6.

The lower bound problem of Theorems 2.3 and 2.4 is treated in Section 7 by using the "worst" interpolation n-tuple σ = σ n, λ , (the Carathodory-Schur type interpolation). The "worst" interpolation data comes from the Dirichlet kernels n-1 k=0 z k transplanted from the origin to λ. We notice that spaces X = H p , L 2 a , p ∈ [1, ∞) satisfy the condition X • b λ ⊂ X which makes the problem of upper and lower bound easier.

The paper is organized as follows. In Section 3 we prove Theorems 2.1 and 2.2. Sections 4 and 5 (resp. Section 6) are (resp. is) devoted to the proof of the upper estimate of Theorem 2.3 (resp. Theorem 2.4). In Section 7, we prove the lower bounds stated in Theorems 2.3 and 2.4. At the end of the paper, we shortly compare the method used in Sections 3, 4, 5 and 6 with those resulting from the Carleson free interpolation, see Section 8.

3.

Upper bounds for c(σ, X, Y ) 3.1. Banach spaces X, Y satisfying properties (P i ) 1≤i≤4 . In this subsection, X and Y are Banach spaces which satisfy properties (P i ) 1≤i≤4 . We prove Theorem 2.1 which shows that in this case our interpolation constant c(σ, X, Y ) is bounded by a quantity which depends only on n = card σ and r = max i |λ i | (and of course on X and Y ). In this generality, we cannot discuss the question of the sharpness of the bounds obtained. First, we prove the following lemma. Lemma 3.1. Under (P 2 ), (P 3 ) and (P 4 ), B σ X is a closed subspace of X and moreover if σ is a finite subset of D,

B σ X = {f ∈ X : f (λ) = 0, ∀λ ∈ σ (including multiplicities)} .
Proof. Since X ⊂ Hol(D) continuously, and evaluation functionals

f → f (λ) and f → f (k) (λ), k ∈ N ⋆ , are continuous on Hol(D), the subspace M = {f ∈ X : f (λ) = 0, ∀λ ∈ σ (including multiplicities)} , is closed in X.
On the other hand, B σ X ⊂ X, and hence B σ X ⊂ M. Indeed, properties (P 2 ) and (P 3 ) imply that h.X ⊂ X, for all h ∈ Hol((1 + ǫ)D) with ǫ > 0. We can write h = k≥0 h(k)z k with h(k) ≤ Cq n , C > 0 and q < 1. Then

n≥0 h(k)z k f X < ∞ for every f ∈ X. Since X is a Banach space we get hf = n≥0 h(k)z k f ∈ X.
In order to see that M ⊂ B σ X, it suffices to justify that

[f ∈ X and f (λ) = 0] =⇒ f /b λ = (1 -λz)f /(λ -z) ∈ X ,
but this is obvious from (P 4 ) and the previous arguments.

In Definition 1 below, σ = {λ 1 , ..., λ n } is a sequence in the unit disc D and B σ is the corresponding Blaschke product.

Definition 1. The model space K Bσ . We define K Bσ to be the n-dimensional space:

(3.1.1)

K Bσ = B σ H 2 ⊥ = H 2 ⊖ B σ H 2 . Malmquist basis. For k ∈ [1, n], we set f k (z) = 1 1-λ k z
, and define the family (e k ) 1≤k≤n , (which is known as Malmquist basis, see [START_REF] Nikolski | Treatise on the shift operator[END_REF] p.117), by

(3.1.2) e 1 = f 1 f 1 2 and e k =   k-1 j=1 b λj   f k f k 2 , for k ∈ [2, n], where f k 2 = 1 -|λ k | 2 -1/2 . The Malmquist family (e k ) 1≤k≤n
corresponding to σ is an orthonormal basis of K Bσ .

The orthogonal projection P Bσ on K Bσ . We define P Bσ to be the orthogonal projection of H 2 on its n-dimensional subspace K Bσ . In particular, (3.1.3)

P Bσ = n k=1 (•, e k ) H 2 e k ,
where (•, •) H 2 means the scalar product on H 2 .

Lemma 3.2. Let σ = {λ 1 , ..., λ n } be a sequence in the unit disc D , (e k ) 1≤k≤n be the Malmquist family corresponding to σ and •, • be the Cauchy sesquilinear form h, g = j≥0 ĥ(j)ĝ(j) for h, g ∈ Hol(D). For all f ∈ Hol(D) and 1 ≤ k ≤ n, the series f, e k are absolutely converging. Moreover, if Y is a Banach space satisfying (P 1 ) , the map Φ : Hol(D) → Y ⊂ Hol(D) given by

Φ : f → n k=1 f, e k e k ,
is well defined and has the following properties: (a) Φ |H 2 = P Bσ , (b) Φ is continuous on Hol(D) with the topology of the uniform convergence on compact sets of D, (c) if X satisfies (P 2 ), (P 3 ), (P 4 ) and Y ⊂ X, and if

Ψ = Id |X -Φ |X , then Im (Ψ) ⊂ B σ X, (d) if f ∈ Hol(D), then |Φ(f )(ζ)| = | f, P Bσ k ζ | = f, n k=1 e k (ζ)e k , for all ζ ∈ D, where P Bσ is defined in (3.1.2) and k ζ = 1 -ζz -1 .
Proof. First of all we set r = max λ∈σ |λ| . If f ∈ Hol(D) and ρ ∈]0, 1[, then

f (j) = (2π) -1 ρT f (w)w -j-1 dw,
for all j ≥ 0. For a subset A of C and for a bounded function h on A, we define

h A := sup z∈A |h(z)| . As a result, (3.1.4) | f, e k | ≤ j≥0 f (j) e k (j) ≤ (2πρ) -1 f ρT j≥0 | e k (j)| ρ -j .
Now if ρ is close enough to 1, it satisfies the inequality 1 ≤ ρ -1 < r -1 , which entails j≥0 | e k (j)| ρ -j < +∞ for each k = 1, ..., n, and the series f, e k are absolutely converging. Now, the the point (a) is a direct consequence of (3.1.3). In order to check point (b), let (f l ) l∈N be a sequence of Hol(D) converging to 0 uniformly on compact sets of D. We need to see that (Φ (f l )) l∈N converges to 0, for which it is sufficient to show that lim l | f l , e k | = 0, for every k = 1, 2, ..., n, which is clear applying (3.1.4) to f = f l .

We now prove point (c). Using point (a), since P ol

+ ⊂ H 2 , we get that Im Ψ |P ol+ ⊂ B σ H 2 . Now, since P ol + ⊂ Y and Im(Φ) ⊂ Y , we deduce that Im Ψ |P ol+ ⊂ B σ H 2 ∩ Y ⊂ B σ H 2 ∩ X,
since Y ⊂ X. Now Ψ (p) ∈ X and satisfies (Ψ (p)) |σ = 0 (that is to say (Ψ (p)) (λ) = 0, ∀λ ∈ σ (including multiplicities) for all p ∈ P ol + . Using Lemma 3.1, we get that Im Ψ |P ol+ ⊂ B σ X. Now, P ol + being dense in X (property (P 2 )), and Ψ being continuous on X (point (b)), we can conclude that Im (Ψ) ⊂ B σ X.

In order to prove (d), we simply need to write that

Φ(f )(ζ) = n k=1 f, e k e k (ζ) = f, n k=1 e k (ζ)e k , ∀f ∈ Hol(D), ∀ζ ∈ D and to notice that n k=1 e k (ζ)e k = n k=1 (k ζ , e k ) H 2 e k = P Bσ k ζ .
Proof of Theorem 2.1 . Let σ = {λ 1 , ..., λ n } be a sequence in the unit disc D and (e k ) 1≤k≤n the Malmquist family (3.1.2) associated to σ. Taking f ∈ X , we set g = Φ(f ) = n k=1 f, e k e k , where Φ is defined in Lemma 3.2. In the same spirit of that of (3.1.4), we notice that e k (j) = (2πi) -1 RT e k (w)w -j-1 dw, for all j ≥ 0 and for all R, 1 < R < 1 r . As a result,

| e k (j)| ≤ 2πR j+1 -1 e k RT and j≥0 f (j) e k (j) ≤ (2πR) -1 e k RT j≥0 f (j) R -j < ∞,
since R > 1 and f is holomorphic in D, (where • A is defined above in the proof of Lemma 3.2). We now suppose that f X ≤ 1. Since Hol r -1 D ⊂ Y , we have g ∈ Y and using Lemma 3.2 point (c) we get

f -g = Ψ(f ) ∈ B σ X,
where Ψ is defined in Lemma 3.2, as Φ. Moreover,

g Y ≤ n k=1 | f, e k | e k Y .
In order to bound the right-hand side, recall that for all j ≥ 0 and for

R = 2/(r + 1) ∈]1, 1/r[, j≥0 f (j) e k (j) ≤ (2π) -1 e k 2(r+1) -1 T j≥0 f(j) 2 -1 (r + 1) j .
Since the norm f → j≥0 f (j) 2 -1 (r + 1) j is continuous on Hol(D), and the inclusion X ⊂ Hol(D) is also continuous, there exists C r > 0 such that

j≥0 f (j) 2 -1 (r + 1) j ≤ C r f X ,
for every f ∈ X. On the other hand, Hol 2(r + 1) -1 D ⊂ Y (continuous inclusion again), and hence there exists K r > 0 such that

e k Y ≤ K r sup |z|<2(r+1) -1 |e k (z)| = K r e k 2(r+1) -1 T .
It is more or less clear that the right-hand side of the last inequality can be bounded in terms of r and n only. Let us give a proof to this fact. It is clear that it suffices to estimate sup

1<|z|<2(r+1) -1 |e k (z)| .
In order to bound this quantity, notice that

(3.1.5) |b λ (z)| 2 ≤ λ -z 1 -λz 2 = 1 + (|z| 2 -1)(1 -|λ| 2 ) |1 -λz| 2 ,
for all λ ∈ D and all z ∈ |λ| -1 D. Using the identity (3.1.5) for λ = λ j , 1 ≤ j ≤ n, and z = ρe it , ρ = 2(1 + r) -1 , we get

e k (ρe it ) 2 ≤   k-1 j=1 b λj (ρe it ) 2   1 1 -λ k ρe it 2 ≤ ≤   k-1 j=1 1 + (ρ 2 -1)(1 -|λ j | 2 ) 1 -|λ j | 2 ρ 2   1 1 -|λ k |ρ 2 ,
for all k = 2, ..., n. Expressing ρ in terms of r, we obtain

e k 2(r+1) -1 T ≤ 1 1 -2r r+1 2   n-1 j=1 1 + 2( 1 r 2 -1) 1 -r 2 4 (r+1) 2   =: C 1 (r, n),
and j≥0 f (j) êk (j) ≤ (2π) -1 C r e k 2(r+1) -1 T f X ≤ (2π) -1 C r C 1 (r, n) f X .
On the other hand, since

e k Y ≤ K r e k 2(r+1) -1 T ≤ K r C 1 (r, n), we get g Y ≤ n k=1 (2π) -1 C r C 1 (r, n) f X K r C 1 (r, n) = (2π) -1 nC r K r (C 1 (r, n)) 2 f X , which proves that c(σ, X, Y ) ≤ (2π) -1 nC r K r (C 1 (r, n)) 2 ,
and completes the proof of Theorem 2.1.

3.2.

The case where X is a Hilbert space. We suppose in this subsection that X is a Hilbert space and that X, Y satisfy properties (P i ) 1≤i≤4 . We prove Theorem 2.2 and obtain a better estimate for c (σ, X, Y ) than in Theorem 2.1 (see point (i) of Theorem 2.2 ). For the case Y = H ∞ , (point (ii) of Theorem 2.2 ), we can considerably improve this estimate. We omit an easy proof of the following lemma.

Lemma 3.3. Let σ = {λ 1 , ..., λ 1 , λ 2 , ..., λ 2 , ..., λ t , ..., λ t } be a finite sequence of D where every λ s is repeated according to its multiplicity m s , t s=1 m s = n. Let (H, (•, •) H ) be a Hilbert space continuously embedded into Hol(D) and satisfying properties (P i ) for i = 2, 3, 4. Then

K Bσ (H) =: H ⊖ B σ H = span k λj , i : 1 ≤ j ≤ t, 0 ≤ i ≤ m j -1 ,
where k λ, i = d dλ i k λ and k λ is the reproducing kernel of H at point λ for every

λ ∈ D, i.e. k λ ∈ H and f (λ) = (f, k λ ) H , ∀f ∈ H. Proof of Theorem 2.2 . i). Let f ∈ X, f X ≤ 1 . Lemma 3.3 shows that g = P H Bσ f = n k=1 (f, E k ) H E k
is the orthogonal projection of f onto subspace K Bσ . Function g belongs to Y because all k λj ,i are in Hol((1 + ǫ)D) for a convenient ǫ > 0, and Y satisfies (P 1 ).

On the other hand, g -f ∈ B σ H (again by Lemma 3.3). Moreover, using Cauchy-Schwarz inequality,

g Y ≤ n k=1 |(f, E k ) H | E k Y ≤ n k=1 |(f, E k ) H | 2 1/2 n k=1 E k 2 Y 1/2 ≤ ≤ f H n k=1 E k 2 Y 1/2 , which proves i). ii). If Y = H ∞ , then |g(ζ)| = P H Bσ f, k ζ H = f, P H Bσ k ζ H ≤ f H P H Bσ k ζ H ,
for all ζ ∈ D, which proves ii).

Upper bounds for

C n, r H 2 , H ∞
Here, we specialize the upper estimate obtained in point (ii) of Theorem 2.2 for the case X = H 2 , the Hardy space of the disc. Later on, we will see that this estimate is sharp at least for some special sequences σ (see Section 7). We also develop a slightly different approach to the interpolation constant c σ, H 2 , H ∞ giving more estimates for individual sequences σ = {λ 1 , ..., λ n } of D. We finally prove the right-hand side inequality of Theorem 2.3 for the particular case p = 2. Proposition 4.1. For every sequence σ = {λ 1 , ..., λ n } of D we have

(4.1.1) c σ, H 2 , H ∞ ≤ sup ζ∈D 1 -|B σ (ζ)| 2 1 -|ζ| 2 1/2
, and therefore

(4.1.2) c σ, H 2 , H ∞ ≤ √ 2 sup |ζ|=1 |B ′ (ζ)| 1 2 = √ 2 sup |ζ|=1 n i=1 1 -|λ i | 2 1 -λi ζ 2 B σ (ζ) b λi (ζ) 1/2 .
Proof. We prove (4.1.1). In order to simplify the notation, we set B = B σ . Applying point (ii) of Theorem 2.2 for X = H 2 and Y = H ∞ , and using

k ζ (z) = 1 1 -ζz and (P Bσ k ζ ) (z) = 1 -B σ (ζ)B σ (z) 1 -ζz ,
(see [START_REF] Nikolski | Treatise on the shift operator[END_REF] p.199), we obtain

P Bσ k ζ H 2 = 1 -|B σ (ζ)| 2 1 -|ζ| 2 1/2
, which gives the result. We now prove (4.1.2) using (4.1.1). The map

ζ → P B (k ζ ) = sup {|f (ζ)| : f ∈ K B , f ≤ 1} ,
and hence the map

ζ → 1 -|B(ζ)| 2 1 -|ζ| 2 1/2
, is a subharmonic function so

sup |ζ|<1 1 -|B(ζ)| 2 1 -|ζ| 2 1/2 ≤ sup |w|=1 lim r→1 1 -|B(rw)| 2 1 -|rw| 2 1/2
. Now applying Taylor's Formula of order 1 at points w ∈ T and u = rw, 0 < r < 1 (it is applicable because B is holomorphic at every point of T), we get

(B(u) -B(w)) (u -w) -1 = B ′ (w) + o(1),
and since |u -w| = 1 -|u|,

(B(u) -B(w)) (u -w) -1 = |B(u) -B(w)| (1 -|u|) -1 = |B ′ (w) + o(1)|.
Then we have

|B(u) -B(w)| ≥ |B(w)| -|B(u)| = 1 -|B(u)|, (1 -|B(u)|) (1 -|u|) -1 ≤ (1 -|u|) -1 |B(u) -B(w)| = |B ′ (w) + o(1)|,
and lim

r→1 (1 -|B(rw)|) (1 -|rw|) -1 1 2 ≤ |B ′ (w)|.
Moreover,

B ′ (w) = - n i=1 1 -|λ i | 2 (1 -λ i w) -2 n j=1, j =i b λj (w),
for all w ∈ T . This completes the proof since

1 -|B(rw)| 2 1 -|rw| 2 = (1 -|B(rw)|)(1 + |B(rw)|) (1 -|rw|)(1 + |rw|) ≤ 2 1 -|B(rw)| 1 -|rw| . Corollary 4.2. Let n ≥ 1 and r ∈ [0, 1[. Then, C n, r (H 2 , H ∞ ) ≤ 2 n(1 -r) -1 1 2 .
Proof. Indeed, applying Proposition 4.1 we obtain

|B ′ (w)| ≤ n i=1 1 -|λ i | 2 (1 -|λ i |) 2 ≤ n 1 + r 1 -r ≤ 2n 1 -r .
Now, we develop a slightly different approach to the interpolation constant c σ, H 2 , H ∞ .

Corollary 4.3. For every sequence

σ = {λ 1 , ..., λ n } of D, c σ, H 2 , H ∞ ≤ sup z∈T n k=1 1 -|λ k | 2 |z -λ k | 2 1/2 .
Proof. In order to simplify the notation, we set B = B σ . We consider K B (see Definition 1) and the Malmquist family (e k ) 1≤k≤n corresponding to σ (see Definition 1). Now, let f ∈ H 2 and

g = P B f = n k=1 (f, e k ) H 2 e k ,
(see (3.1.3)). The function g belongs to H ∞ (it is a finite sum of H ∞ functions) and satisfies g -f ∈ BH 2 . Applying Cauchy-Schwarz inequality we get

|g(ζ)| ≤ n k=1 |(f, e k ) H 2 | |e k (ζ)| ≤ n k=1 |(f, e k ) H 2 | 2 1/2 n k=1 1 -|λ k | 2 |1 -λ k ζ| 2 1/2
, for all ζ ∈ D. As a result, since f is an arbitrary H 2 function, we obtain

c(σ, H 2 , H ∞ ) ≤ sup ζ∈T n k=1 1 -|λ k | 2 |ζ -λ k | 2 1/2
, which completes the proof.

Corollary 4.4. For any sequence

σ = {λ 1 , ..., λ n } in D , c(σ, H 2 , H ∞ ) ≤   n j=1 1 + |λ j | 1 -|λ j |   1/2 . Proof. Indeed, n k=1 1 -|λ k | 2 |ζ -λ k | 2 ≤ n k=1 1 -|λ k | 2 (1 -|λ k |) 2 1/2
and the result follows using Theorem 4.3. Now we prove the right-hand side inequality of Theorem 2.3 for the special case p = 2.

Proof of Theorem 2.3 (p = 2, the right-hand side inequality only ). Since 1 + |λ j | ≤ 2 and 1 -|λ j | ≥ 1 -r for all j ∈ [1, n], applying Corollary 4.4 we get

C n,r (H 2 , H ∞ ) ≤ √ 2n 1/2 (1 -r) -1/2 .
Remark. As a result, we get once more the same estimate for C n,r (H 2 , H ∞ ) as in Corollary 4.2 , with the constant √ 2 instead of 2.

It is natural to wonder if it is possible to improve the bound √ 2n 1/2 (1 -r) -1/2 . We return to this question in Section 7 below.

Upper bounds for

C n, r (H p , H ∞ ) , p ≥ 1
In this section we extend Corollary 4.2 to all Hardy spaces H p : we prove the right-hand side inequality of Theorem 2.3 , p = 2. We first prove the following lemma.

Lemma 5.1. Let n ≥ 1 and 0 ≤ r < 1. Then,

C n,r (H 1 , H ∞ ) ≤ 2n(1 -r) -1 .
Proof. Let σ be a finite subset of D, (e k ) 1≤k≤n be the Malmquist basis corresponding to σ (see Definition 1), and f ∈ H 1 such that f H 1 ≤ 1. Let also g = Φ(f ) where Φ is defined in Lemma 3.2. Applying point (d) of Lemma 3.2, we get

|g(ζ)| ≤ f H 1 n k=1 e k e k (ζ) H ∞ ≤ n k=1 e k e k (ζ) H ∞ .
Since Blaschke factors have modulus 1 on the unit circle,

e k H ∞ ≤ (1 + |λ k |) 1/2 (1 -|λ k |) -1/2 .
As a consequence,

|g(ζ)| ≤ n k=1 e k H ∞ e k (ζ) ≤ n k=1 e k 2 H ∞ ≤ n k=1 (1 + |λ k |) (1 -|λ k |) -1 ≤ 2n(1 -r) -1 ,
for all ζ ∈ D, which completes the proof.

Before proving the upper bound in Theorem 2.3 , we give the following general observation which is a direct consequence of the classical complex interpolation between Banach spaces, see [START_REF] Bergh | Interpolation Spaces. An Introduction[END_REF][START_REF] Triebel | Interpolation theory, functions spaces, differential operators[END_REF]. In particular, we use the notation of [START_REF] Bergh | Interpolation Spaces. An Introduction[END_REF], Chapter 4.

Lemma 5.2. Let X 1 and X 2 be two Banach spaces of holomorphic functions in the unit disc D. Let also θ ∈ [0, 1] and (X 1 , X 2 ) [θ] be the corresponding intermediate Banach space resulting from the classical complex interpolation method applied between X 1 and X 2 . Then,

C n, r (X 1 , X 2 ) [θ] , H ∞ ≤ C n, r (X 1 , H ∞ ) 1-θ C n, r (X 2 , H ∞ ) θ , for all n ≥ 1, r ∈ [0, 1).
Proof. Let X be a Banach space of holomorphic functions in the unit disc D and let σ = {λ 1 , λ 2 , ..., λ n } ⊂ D be a finite subset of the disc. Let T : X -→ H ∞ /B σ H ∞ be the restriction map defined by

T f = {g ∈ H ∞ : f -g ∈ B σ X} ,
for every f ∈ X. Then,

T X→H ∞ /BσH ∞ = c (σ, X, H ∞ ) . Now, since (X 1 , X 2 ) [θ]
is an exact interpolation space of exponent θ (see [START_REF] Bergh | Interpolation Spaces. An Introduction[END_REF] or [START_REF] Triebel | Interpolation theory, functions spaces, differential operators[END_REF] Theorem 1.9.3-(a), p.59), we can complete the proof.

Now we prove the right-hand side inequality of Theorem 2.3 for the remaining case p = 2.

Proof of Theorem 2.3 (p = 2, the right-hand side inequality only ).

In the case X = H p , there exists 0 ≤ θ ≤ 1 such that 1/p = 1 -θ, and H 1 , H ∞ θ = H p (a topological identity: the spaces are the same and the norms are equivalent (up to constants depending on p only), see [START_REF] Jones | L ∞ estimates for the ∂ problem in the half plane[END_REF]). As a consequence, applying Lemma 5.2 with X 1 = H 1 and X 2 = H ∞ , we get

C n, r (H p , H ∞ ) ≤ γ p C n, r H 1 , H ∞ 1-θ (C n, r (H ∞ , H ∞ )) θ ≤ ≤ γ p 2n(1 -r) -1 1-θ (1) θ ,
where θ = 1 -1/p, and γ p is a constant depending only on p. Using Lemma 5.1 and the fact that

C n, r (H ∞ , H ∞ ) ≤ 1, we get C n, r (H p , H ∞ ) ≤ γ p 2n(1 -r) -1 1/p ,
which completes the proof.

6.

Upper bounds for C n, r L 2 a , H ∞ Our aim here is to generalize Corollary 4.2 to the case X = l 2 a ((k + 1) α ), α ∈ [-1, 0], the Hardy weighted spaces of all f = k≥0 f (k)z k satisfying

f 2 X := k≥0 f (k) 2 (k + 1) 2α < ∞.
Note that H 2 = l 2 a (1) and L 2 a = l 2 a (k + 1)

-1 2
. We prove the upper bound of Theorem 2.4. The main technical tool used here is a Bernstein-type inequality for rational functions.

6.1. Bernstein-type inequalities for rational functions. Bernstein-type inequalities for rational functions were the subject of a number of papers and monographs (see, for instance, [2-3, 7-8, 13]). Perhaps, the stronger and closer to ours (Proposition 6.1 ) of all known results are due to K.Dyakonov [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF][START_REF] Dyakonov | Entire functions of exponential type and model subspaces in H p[END_REF]. First, we prove Proposition 6.1 below, which tells that if σ = {λ 1 , ..., λ n } ⊂ D, r = max

j |λ j | , B = B σ and f ∈ K B , then (⋆) f ′ H 2 ≤ α n, r f H 2 ,
where α n, r is a constant (explicitly given in Proposition 6.1 ) depending on n and r only such that 0 < α n, r ≤ 3 n 1-r . Proposition 6.1 is in fact a special case (p = 2) of a K. Dyakonov's result [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF] for p = 2. It is important to recall that this result is in its turn, a generalization of M. Levin's inequality [START_REF] Levin | Estimation of the derivative of a meromorphic function on the boundary of the domain (Russian)[END_REF] corresponding to the case p = ∞. More precisely, it is proved in [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF] that the norm D K p B →H p of the differentiation operator Df = f ′ on the star-invariant subspace of the Hardy space H p , K p B := H p ∩ BzH p , (where the bar denotes complex conjugation) satisfies the following inequalities

c ′ p B ′ ∞ ≤ D K p B →H p ≤ c p B ′ ∞ ,
for every p, 1 ≤ p ≤ ∞ where c p and c ′ p are positives constants depending on p only, B is a finite Blaschke product and • ∞ means the norm in L ∞ (T). For the partial case considered in Proposition 6.1 below, our proof is different from [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF][START_REF] Dyakonov | Entire functions of exponential type and model subspaces in H p[END_REF]. It is based on an elementary Hilbert space construction for an orthonormal basis in K B and the constant c 2 obtained is slightly better. More precisely, it is proved in [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF] that c ′ 2 = 1 36c , c 2 = 36+c 2π and c = 2 √ 3π (as one can check easily (c is not evaluated in [START_REF] Dyakonov | Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness[END_REF])). It implies an inequality of type (⋆) (with a constant about 6.5 instead of 3).

In [START_REF] Zarouf | Asymptotic sharpness of a Bernstein-type inequality for rational functions in H 2[END_REF], we discuss the "asymptotic sharpness" of our constant α n, r . We find an inequality for

C n, r = sup D KB →H 2 ,
(sup is over all B with given n = deg B and r = max λ∈σ |λ|), which is asymptotically sharp as n → ∞. Our result in [START_REF] Zarouf | Asymptotic sharpness of a Bernstein-type inequality for rational functions in H 2[END_REF] is that lim n→∞ Cn, r n = 1+r 1-r for every r, 0 ≤ r < 1. Proposition 6.1. Let B = n j=1 b λj , be a finite Blaschke product (of order n), r = max j |λ j | , and f ∈ K B = H 2 ⊖ BH 2 . Then for every n ≥ 2 and r ∈ [0, 1),

f ′ H 2 ≤ α n, r f H 2 , where α n, r = 1 + (1 + r)(n -1) + √ n -2 (1 -r) -1
, and in particular

f ′ H 2 ≤ 3 n 1 -r f H 2 ,
for all n ≥ 1 and r ∈ [0, 1).

Proof. Using (3.1.3), f = P B f = n k=1 (f, e k ) H 2 e k , for all f ∈ K B . Noticing that, e ′ k e ′ k = k-1 i=1 b ′ λi b λi e k + λ k 1 1 -λ k z e k , for k ∈ [2, n], we get f ′ = n k=2 (f, e k ) H 2 k-1 i=1 b ′ λi b λi e k + n k=1 (f, e k ) H 2 λ k 1 1 -λ k z e k , = n i=1 b ′ λi b λi n-1 k=i+1 (f, e k ) H 2 e k + n k=1 (f, e k ) H 2 λ k 1 1 -λ k z e k .
Now using Cauchy-Schwarz inequality and the fact that e k is a vector of norm 1 in H 2 for k = 1, ..., n, we get

n k=2 λ k (f, e k ) H 2 1 1 -λ k z e k H 2 ≤ n k=1 |(f, e k ) H 2 | λ k 1 1 -λ k z ∞ e k H 2 ≤ ≤ 1 1 -r n k=1 |(f, e k ) H 2 | ≤ 1 1 -r n k=1 |(f, e k ) H 2 | 2 1 2 √ n ≤ 1 1 -r f H 2 √ n. Further, n-1 i=1 b ′ λi b λi n k=i+1 e k (f, e k ) H 2 H 2 ≤ n-1 i=1 b ′ λi b λi ∞ n k=i+1 (f, e k ) H 2 e k H 2 = = max 1≤i≤n-1 b ′ λi b λi ∞ n-1 i=1 n k=i+1 |(f, e k ) H 2 | 2 1 2 ≤ max i b ′ λi b λi ∞ n-1 i=1 f H 2 . Now, using b ′ λi b λi ∞ = |λ i | 2 -1 1 -λ i z (λ i -z) ∞ ≤ 1 + |λ i | 1 -|λ i | ≤ 1 + r 1 -r , we get n-1 i=1 b ′ λi b λi n k=i+1 (f, e k ) H 2 e k H 2 ≤ (1 + r) n -1 1 -r f H 2 .
Finally,

f ′ H 2 ≤ 1 + (1 + r)(n -1) + √ n -2 (1 -r) -1 f H 2 .
In particular,

f ′ H 2 ≤ 2n -2 + √ n (1 -r) -1 f H 2 ≤ 3n(1 -r) -1 f H 2 ,
for all n ≥ 1 and for every f ∈ K B .

6.2. An upper bound for c σ, L 2 a , H ∞ . Here, we apply Proposition 6.1 to prove the right-hand side inequality of Theorem 2.4. We first prove the following corollary. Corollary 6.2. Let σ be a sequence in D. Then,

c σ, l 2 a (k + 1) -1 , H ∞ ≤ 2 √ 10 n(1 -r) -1 3/2 .
Proof. Indeed, let X = l 2 a (-1), σ a finite subset of D and B = B σ . If f ∈ X, then using part (c) of Lemma 3.2, f |σ we get that Φ(f ) |σ = f |σ . Now, denoting X ⋆ the dual of X with respect to the Cauchy pairing •, • (defined in Lemma 3.2) and applying point (d) of the same lemma, we obtain X ⋆ = l 2 a (1) and

|Φ(f )(ζ)| ≤ f X P B k ζ X ⋆ ≤ f X K P B k ζ 2 H 2 + (P B k ζ ) ′ 2 H 2 1 2
, where

K = max 1, sup k≥1 (k + 1)k -1 = 2. Since P B k ζ ∈ K B , Proposition 6.1 implies |Φ(f )(ζ)| ≤ f X K P B k ζ 2 H 2 + 9 n(1 -r) -1 2 P B k ζ 2 H 2 1 2 ≤ ≤ A n(1 -r) -1 3/2 f X ,
where A = K (2/2 + 9)

1 2 = 2 √ 10, since P B k ζ 2 ≤ √ 2 n(1 -r) -1 1/2
, and since we can suppose n ≥ 2, (the case n = 1 being obvious). Now let us give the proof of the right-hand side inequality of Theorem 2.4.

Proof of Theorem 2.4 (the right-hand side inequality only). The case α = 0 corresponds to X = H 2 and has already been studied in Section 3 (we can choose A(0) = √ 2). We now suppose α ∈ [-1, 0) and we set θ = -α,

0 ≤ θ ≤ 1, so that α = (1 -θ)(1 -N ) + θ.(-N ). Since l 2 a (0) , l 2 a (-1) [θ] = l 2 a (α) ,
(see [START_REF] Bergh | Interpolation Spaces. An Introduction[END_REF][START_REF] Triebel | Interpolation theory, functions spaces, differential operators[END_REF]), this gives, using Lemma 5.2 with X 1 = l 2 a (0) = H 2 and X 2 = l 2 a (-1) , Corollary 6.2 , that

C n, r l 2 a (α) , H ∞ ≤ √ 2 n(1 -r) -1 1 2 1-θ 2 √ 10 n(1 -r) -1 3 2 θ = = A(0) 1-θ A(1) θ n(1 -r) -1 1-θ 2 + 3θ 2 .
It remains to use θ = -α, set A(α) = A(0) 1-θ A(1) θ and apply (5.1.2) with X = l 2 a ((k + 1) α ). In particular, for α = -1/2 we get (1 -θ)/2 + 3θ/2 = 1 and

A (-1/2) = A(0) (1-1/2) A(1) 1/2 = √ 2 1/2 (2 √ 10) 1/2 = √ 2.10
c(σ n, λ , H, H ∞ ) = sup f H ∞ /b n λ H ∞ : f ∈ H, f H ≤ 1 , where f H ∞ /b n λ H ∞ = inf { f + b n λ g ∞ : g ∈ H}.
Our goal in this subsection is to prove the sharpness of the upper estimate stated in Theorem 2.3 (p = 2) and in Theorem 2.4 for the quantities C n, r H 2 , H ∞ and C n, r L 2 a , H ∞ , that is to say, to get the lower bounds of Theorem 2.3 (p = 2) and Theorem 2.4.

In the proof, we use properties of reproducing kernel Hilbert space on the disc D, see for example [START_REF] Nikolski | Operators, Function, and Systems: an easy reading[END_REF]. Let us recall some of them adapting the general setting to special cases X = l 2 a ((k + 1) α ). The reproducing kernel of l 2 a ((k + 1) α ), by definition, is a l 2 a ((k + 1) α )-valued function λ -→ k α λ , λ ∈ D, such that (f, k w λ ) = f (λ) for every f ∈ l 2 a ((k + 1) -α ), where (•, •) means the scalar product (h, g) = k≥0 ĥ(k)ĝ(k)(k + 1) -2α . Since one has f (λ) = k≥0 f (k)λ k (k + 1) 2α (k + 1) -2α (λ ∈ D), it follows that

k α λ (z) = k≥0 (k + 1) 2α λ k z k , z ∈ D.
In particular, for the Hardy space H 2 = l 2 a (1) (α = 0), we get the Szeg kernel k λ (z) = (1-λz) -1 and for the Bergman space L 2 a = l 2 a (k + 1)

-1/2 (α = -1/2), the Bergman kernel k -1/2 λ (z) = (1 -λz) -2 .
We will use the previous observations for the following composed reproducing kernels (Aronszajn-deBranges, see [START_REF] Aronszajn | Theory of reproducing kernels[END_REF][START_REF] Nikolski | Operators, Function, and Systems: an easy reading[END_REF]): given the reproducing kernel k of H 2 and ϕ ∈ {z N : N = 1, 2}, the function ϕ • k is also positive definite and the corresponding Hilbert space is

H ϕ = ϕ(H 2 ) = l 2 a (k + 1) 1-N 2
.

It satisfies the following property: for every [START_REF] Nikolski | Operators, Function, and Systems: an easy reading[END_REF], page 320). We notice in particular that (7.1.1)

f ∈ H 2 , ϕ • f ∈ ϕ(H 2 ) and ϕ • f 2 ϕ(H 2 ) ≤ ϕ( f 2 H 2 ) (see
H z = H 2 and H z 2 = L 2 a .
The above relation between the spaces H 2 , L 2 a and the spaces ϕ(H 2 ) = H ϕ leads to establish the proof of the left-hand side inequalities stated in Theorem 2.3 (for p = 2 only) and in Theorem 2.4 .

Proof of Theorem 2.3 (p = 2) and Theorem 2.4. (left-hand side inequalities only) 0) For N = 1, 2 we set ϕ N (z) = z N .

1) We set

Q n = n-1 k=0 (1 -|λ| 2 ) 1/2 b k λ 1 -λz -1 , H n, N = ϕ N • Q n and Ψ = bH n, N , b > 0. Then Q n 2 2
= n, and hence by the above Aronszajn-deBranges inequality,

Ψ 2 Hϕ ≤ b 2 ϕ N Q n 2 2 = b 2 ϕ N (n). Let b > 0 such that b 2 ϕ N (n) = 1.
2) Since the spaces H ϕN and H ∞ are rotation invariant, we have

c (σ n, λ , H ϕN , H ∞ ) = c (σ µ,n , H ϕN , H ∞ )
for every λ, µ with |λ| = |µ| = r. Let λ = -r. To get a lower estimate for Ψ Hϕ

N /b n λ Hϕ N consider G such that Ψ -G ∈ b n λ Hol(D), i.e. such that bH n, N • b λ -G • b λ ∈ z n Hol(D).
3) First, we show that

ψ =: Ψ • b λ = bH n, N • b λ is a polynomial (of degree n if ϕ = z and 2n if ϕ = z 2 ) with positive coefficients. Note that Q n • b λ = n-1 k=0 z k (1 -|λ| 2 ) 1/2 1 -λb λ (z) = 1 -|λ| 2 -1 2 1 + (1 -λ) n-1 k=1 z k -λz n = = (1 -r 2 ) -1/2 1 + (1 + r) n-1 k=1 z k + rz n =: (1 -r 2 ) -1/2 ψ 1 . Hence, ψ = Ψ • b λ = bH n, N • b λ = bϕ N • 1 -r 2 -1 2 ψ 1 and ϕ N • ψ 1 = ψ N 1 (z), N = 1, 2. 4) Next, we show that m (ψ) =: m j=0 ψ(j) ≥ (2 √ 2) -1 n(1 -r) -1 if N = 1 16 -1 n(1 -r) -1 if N = 2 ,
where m = n/2 if n is even and m = (n + 1)/2 if n is odd.

Indeed, setting S n = n j=0 z j , we have both for N = 1 and N = 2

m ψ N 1 = m   1 + (1 + r) n-1 t=1 z t + rz n N   ≥ m S N n-1 . Next, we obtain m S N n-1 = m 1 -z n 1 -z N = = m (1 -z) -N = 1 (N -1)! m d N -1 dz N -1 1 1 -z = m j=0 C j N +j-1 . Now if N = 1, then m j=0 C j N +j-1 = m + 1 ≥ n 2 , whereas if N = 2 then m j=0 C j N +j-1 = (m + 1)(m + 2) 2 ≥ (n + 2)(n + 4) 8 ≥ n 2 8 . Finally, since m (ψ) = b m (ϕ N • ψ 1 ) = b 1 -r 2 -N/2 m (ψ N 1 ) we get m (ψ) ≥ (2(1 -r)) -1/2 nb/2 if N = 1 (2(1 -r)) -1 n 2 b/8 if N = 2 , with b = ϕ N (n) = n -1/2 if N = 1 n -1 if N = 2
. This gives the result claimed. 5)

Let F n = Φ m + z m Φ m , where Φ k stands for the k-th Fejer kernel. We have g ∞ F n L 1 ≥ g * F n ∞ for every g ∈ L ∞ (T) , and taking the infimum over all g ∈ H ∞ satisfying ĝ(k) = ψ(k), ∀k ∈ [0, n -1], we obtain

ψ H ∞ /z n H ∞ ≥ 1 2 ψ * F n ∞ ,
where * stands for the usual convolution product. Now using part 4),

Ψ H ∞ /b n λ H ∞ = ψ H ∞ /z n H ∞ ≥ 1 2 ψ * F n ∞ ≥ ≥ 1 2 |(ψ * F n ) (1)| ≥ 1 2 m j=0 ψ(j)≥ ≥ (4 √ 2) -1 n(1 -r) -1 if N = 1 32 -1 n(1 -r) -1 if N = 2 .
6) In order to conclude, it remains to use (7.1.1).

7.2. The case X = H p , 1 ≤ p ≤ +∞. Here we prove the sharpness (for even p) of the upper estimate found in Theorem 2.3. We first prove the following lemma.

Lemma 7.1. Let p, q such that p

q ∈ Z + , then c (σ, H p , H ∞ ) ≥ c (σ, H q , H ∞ ) q p
for every sequence σ of D.

Proof.

Step 1. Recalling that

c (σ, H p , H ∞ ) = sup f p ≤1 inf g ∞ : g ∈ Y, g |σ = f |σ , we first prove that c (σ, H p , H ∞ ) = sup f p ≤1, f outer inf g ∞ : g ∈ Y, g |σ = f |σ .
Indeed, we clearly have the inequality sup

f p≤1, f outer inf g ∞ : g ∈ Y, g |σ = f |σ ≤ c (σ, H p , H ∞ ) ,
and if the inequality were strict, that is to say sup

f p ≤1, f outer inf g ∞ : g ∈ Y, g |σ = f |σ < sup f p ≤1 inf g ∞ : g ∈ Y, g |σ = f |σ ,
then we could write that there exists ǫ > 0 such that for every f = f i .f o ∈ H p (where f i stands for the inner function corresponding to f and f o to the outer one) with f p ≤ 1 (which also implies that

f o p ≤ 1, since f o p = f p ),
there exists a function g ∈ H ∞ verifying both g ∞ ≤ (1 -ǫ)c (σ, H p , H ∞ ) and g |σ = f o|σ . This entails that f |σ = (f i g) |σ and since f i g ∞ = g ∞ ≤ (1 -ǫ)c (σ, H p , H ∞ ) , we get that c (σ, H p , H ∞ ) ≤ (1 -ǫ)c (σ, H p , H ∞ ), which is a contradiction and proves the equality of Step 1.

Step 2. Using the result of Step 1, we get that ∀ǫ > 0 there exists an outer function f o ∈ H q with f o q ≤ 1 and such that inf g ∞ : g ∈ Y, g |σ = f o|σ ≥ c (σ, H q , H ∞ ) -ǫ. Now let F = f q p o ∈ H p , then F p p = f o q q ≤ 1. We suppose that there exists g ∈ H ∞ such that g |σ = F |σ with g ∞ < (c (σ, H q , H ∞ ) -ǫ) p .

Then, since g (λ i ) = F (λ i ) = f o (λ i ) q p for all i = 1, . . . , n, we have g (λ i ) p q = f o (λ i ) and g p q ∈ H ∞ since p q ∈ Z + . We also have

g p q ∞ = g p q ∞ < (c (σ, H q , H ∞ ) -ǫ) q p ,
which is a contradiction. As a result, we have

g ∞ ≥ (c (σ, H q , H ∞ ) -ǫ) q p ,
for all g ∈ H ∞ such that g |σ = F |σ , which gives

c (σ, H p , H ∞ ) ≥ (c (σ, H q , H ∞ ) -ǫ) q p ,
and since that inequality is true for every ǫ > 0, we get the result.

Proof of Theorem 2.3 (the left-hand side inequality for p ∈ 2N, p > 2 only).

We first prove the lower estimate for c (σ n, λ , H p , H ∞ ) . Writing p = 2(p/2), we apply Lemma 7.1 with q = 2 and this gives c (σ n, λ , H p , H ∞ ) ≥ c σ n, λ , H 2 , H ∞ 2 p ≥ 32 -1 p n(1 -|λ|) -1 2 p for all integer n ≥ 1. The last inequality is a consequence of Theorem 2.3 (lefthand side inequality) for the particular case p = 2 which has been proved in Subsection 7.1.

Comparing our results with Carleson interpolation

Recall that given a (finite) set σ = {λ 1 , ..., λ n } ⊂ D, the Carleson interpolation constant C I (σ) is defined by

C I (σ) = sup a l ∞ ≤1 inf g ∞ : g ∈ H ∞ , g |σ = a .
We introduce the evaluation functionals ϕ λ for λ ∈ D, as well as the evaluation of the derivatives ϕ λ,s (s = 0, 1, ...) ϕ λ (f ) = f (λ), f ∈ X, and ϕ λ,s (f ) = f (s) (λ), f ∈ X. Theorem 8.1. Let X be a Banach space, X ⊂ Hol(D), and σ = {λ 1 , ..., λ n } be a sequence of distinct points in the unit disc D. We have, Theorem 8.1 (see [START_REF] Zarouf | Interpolation avec contraintes sur des ensembles finis du disque[END_REF] for its proof) tells us that, for σ with a "reasonable" interpolation constant C I (σ), the quantity c(σ, X, H ∞ ) behaves as max i ϕ λi . However, for "tight" sequences σ, the constant C I (σ) is so large that the estimate in question contains almost no information. On the other hand, an advantage of the estimate of Theorem 8.1 is that it does not contain card σ = n explicitly. Therefore, for well-separated sequences σ, Theorem 8.1 should give a better estimate than those of Theorem 2.3 and Theorem 2.4 . Now, how does the interpolation constant C I (σ) behave in terms of the characteristics r and n of σ? We answer this question in [START_REF] Zarouf | Interpolation avec contraintes sur des ensembles finis du disque[END_REF] for some particular sequences σ. More precisely, we compare these quantities for the cases X = H 2 , X = L 2 a and for three geometrically simple configurations: two-points sets σ, circular and radial sequences σ.

Let us recall that our specific upper bounds in Theorems C and D are sharp over all n elements sequences σ. However, we give in [START_REF] Zarouf | Interpolation avec contraintes sur des ensembles finis du disque[END_REF] some very special radial and circular sequences σ such that the estimate of c(σ, H 2 , H ∞ ) via the Carleson constant C I (σ) (using Theorem 8.1 ) is comparable with or better than the estimates of Theorem 2.3 (for X = H 2 ) and Theorem 2.4 (for X = L 2 a ) . We also give some examples of radial and circular sequences but also of two-points sets, such that it is worse (i.e. for which our estimate is better). More specific radial sequences are studied in [START_REF] Zarouf | Interpolation avec contraintes sur des ensembles finis du disque[END_REF]: sparse sequences, condensed sequences and long sequences.

1. 2 .

 2 Motivations. a. As it is mentioned in Subsection 1.1, one of the most interesting cases is Y = H ∞ . In this case, the quantity c (σ, X, H ∞ ) has a meaning of an intermediate interpolation between the Carleson one (when f X |σ ≍ sup 1≤i≤n |f (λ i )|) and the individual Nevanlinna-Pick interpolation (no conditions on f ).

1 4 . 7 . 7 . 1 .

 14771 Lower bounds for C n, r (X, H ∞ ) The cases X = H 2 and X = L 2 a . Here, we consider the standard Hardy and Begman spaces,X = H 2 = l 2 a (1) and X = L 2 a = l 2 a ((k + 1) -1/2), where the spaces l 2 a ((k + 1) α ) are defined in Section 6, and the problem of lower estimates for the one-point special case σ n, λ = {λ, λ, ..., λ} n , λ ∈ D. Recall the definition of our constrained interpolation constant for this case

ϕ

  λi ≤ c(σ, X, H ∞ ) ≤ C I (σ) max 1≤i≤n ϕ λi ,where C I (σ) stands for the Carleson interpolation constant.


Ismael Bouassida Rodriguez

Aymen Kamoun

Said Tazi

Thierry Villemur

Khalil Drira

Semantic Driven Dynamic Redeployment of Components in Pervasive systems

Keywords: pervasive computing, context, semantic, dynamic redeployment

The emergence of pervasive computing interweaving different devices and sensors introduces new challenges to ensure the continuity of the communication and collaboration between humans in distributed systems. The communication may be interrupted due to the change of context, such as a change of a users role or an energy level diminution, thus leading to lose the cooperation or shared data. This paper describes a framework that aims at ensuring the continuity of communication services even if changes happen. This framework is based on a multi-level modeling approach. Dynamic redeployment algorithms based on semantic description of collaboration with a scenario for validation are presented.

Introduction

Collaboration between human beings should gain from the development of pervasive computing environments. These environments should encourage people to communicate and cooperate everywhere at any time, because they are immersed, willingly or not, into environments containing large sets of devices, sensors, actuators and services that are available in a pervasive and automatic manner [START_REF] Lee | Ubiquitous device collaboration infrastructure: Celadon, in: Software Technologies for Future Embedded and Ubiquitous Systems[END_REF][START_REF] Perich | Collaborative joins in a pervasive computing environment[END_REF][START_REF] Touseau | How to guarantee service cooperation in dynamic environments?[END_REF]. Several issues of research arise from the use of pervasive computing for collaboration. For example the devices used in pervasive computing are heterogeneous in terms of communication and processing capabilities, mecanisms to ensure interoperability are needed [START_REF] Saha | Pervasive computing: A paradigm for the 21st century[END_REF][START_REF] Weiser | The computer for the 21st century[END_REF] .

Fluid communication between collaborators is needed when a change takes place, regardless of the level change occurs at, application level or otherwise. Indeed at the applicative level, the user's needs are dynamic because of complex situations of cooperation, such as where there is user mobility or the change of roles that necessitates new software components. Changes may also happen at a lower level, such as machine resources becoming limited due to a decreased level of energy, or a limitation of hardware resources such as main memory or CPU. In pervasive computing environments, the sessions of collaborative activities should be spontaneously initiated and managed (i.e. without user input), thus exploiting implicit collaboration situations. Adaptation such as dynamic deployment of components is needed for establishing multimedia sessions. Therefore, the main adaptation actions in a collaborative pervasive system involve component (re)deployments and/or flow (re)configurations. The goal of the research presented here is to enable continuous communication between collaborators no matter what may happen at different levels by providing adaptation mechanisms of architectures of components at runtime [START_REF] Mäntyjärvi | Adapting applications in handheld devices using fuzz context information[END_REF][START_REF] Cámara | Multiple concern adaptation for runtime composition in context-aware systems[END_REF]. In our approach, this is made possible through model driven architecture transformations, from abstractions of collaboration to concrete redeployment of software components applied on collaborative sessions. For instance, if a battery level drops during a session, the framework adapts the architecture by redeploying new components in the network to save the session and to ensure continuous communication and collaboration between the nodes (devices) of the session. The events that trigger adaptation actions are the changes in the context of the application. Context is divided into external context (e.g. user preferences, user presence and position, priority of communications, etc.) and execution resource context (e.g. battery level, CPU load and available memory of user devices, etc.) [START_REF] Baldauf | A survey on context-aware systems[END_REF][START_REF] Bellavista | A survey of context data distribution for mobile ubiquitous systems[END_REF]. Semantic computing technologies [START_REF] Ma | Soft Computing in Ontologies and Semantic Web[END_REF] and graph models [START_REF] Brandes | GraphML Progress Report[END_REF] have been used as the foundation for the algorithms we have implemented for adaptation concerns. We consider an intermediary level in which the semantics of collaboration are presented at a high conceptual level. At the application and collaboration levels, the semantics of the application and of the collaboration are represented as ontologies and as a set of rules. At the resource level, the set of components are represented as graphs where vertices represent software services and components, and edges represent the communication relationships between such components. Adaption algorithms use these models to detect the change of the context at runtime and to find the best component to deploy dynamically, in manner that ensure the continuity of communica-tion [START_REF] Maoz | Using model-based traces as runtime models[END_REF][START_REF] Morin | Models@ run.time to support dynamic adaptation[END_REF]. In this paper, we focus on architectural adaptation using the framework we have developed. The remainder of this paper is organized into seven sections. Related work is presented in Section 2. Section 3 introduces a scenario and an example of an application that illustrates several aspects of the identified problem. Section 4 presents the generic approach proposed in order to solve such a problem and a proposition of implementation. Section 5 details how our approach can be applied to the proposed scenario. Section 6 presents an evaluation of our approach being applied to the scenario. Finally, Section 7 provides conclusions and possible directions for future work.

Related work

We have classified related works into three subsections in accordance with the features highlighted in this section: model driven approaches, architecture oriented, and platform or existing frameworks.

Model driven approaches

Becker and Giese [START_REF] Becker | Modeling of correct self-adaptive systems: a graph transformation system based approach[END_REF] present an approach based on graph transformation techniques coupled with UML stereotypes in order to model self-adaptive systems. Adaptation, which is performed at run-time, is decomposed into three levels: goal management, change management and component control. However, in this approach, context and collaboration are not explicitly modeled. Edwards [START_REF] Edwards | Putting computing in context: An infrastructure to support extensible context-enhanced collaborative applications[END_REF] presents a system named Intermezzo that enables the construction of applications making use of rich, layered interpretations of context. It provides a context data store and an integrated notification service. This system is collaboration-oriented; contextual information is structured through activities, that represent the use of resources (e.g., documents). The underlying context model is very rich and deals with problems such as ambiguity, identity, evolution and equality of contextual data, providing solutions for each one of them. However, the case of low-level resources context is not considered.

Pad Ovitz et al. [START_REF] Padovitz | Multiple-agent perspectives in reasoning about situations for context-aware pervasive computing systems[END_REF] present a context model and reasoning approach developed with concepts from the state-space model, which describes context and situations as geometrical structures in a multidimensional space. A context algebra based on this model is also presented. This work shows how merging different points of view over context enhances the global context reasoning process. The authors provide a model (named Context Spaces) that unifies the context views of different entities into a single context representation. The cooperative aspect of this work focuses on migration, modeling and reasoning, partitioning, and merging context descriptions between agents for attaining optimal reasoning and context-awareness.

Architectural adaptation

Zhang et al. [START_REF] Zhang | An Adaptive Infrastructure Concept Model based on CORBA in Pervasive Computing[END_REF] propose an adaptive model infrastructure for pervasive computing environments. They propose three adaptive layers: adaptive collaboration layer, adaptive middleware layer and adaptive services layer. The adaptive collaboration layer provides a service cooperation platform in dynamic environments. The adaptive middleware layer is a self-reconfiguring layer that provides an optimized uniform high-level interface for implementation of distributed applications. The adaptive services layer provides an adaptive contents service and adaptive user interfaces. Using this model, the approach makes environmental changes invisible to collaboration among applications and users. From the architectural point of view, this work is similar to ours in that it defines three layers. However, collaboration in this work is limited to the collaboration between services to adapt contents to the user interface.

Platform oriented research

Ejigu et al. [START_REF] Ejigu | Coca: A collaborative context-aware service platform for pervasive computing[END_REF] propose a collaborative context-aware service platform named CoCA. This platform is data-independent and it may be used for context-aware application development in pervasive computing. It performs reasoning and decisions based on context data and domain-based policies using ontologies. The platform introduces a neighborhood collaboration mechanism to facilitate peer collaboration between pervasive devices in order to share their resources. The generic context management modeling deals with the way the context data are collected, organized, represented, stored and presented. In this work, collaboration is used between devices in order to better achieve context data acquisition.

Lee et al. [START_REF] Lee | Ubiquitous device collaboration infrastructure: Celadon, in: Software Technologies for Future Embedded and Ubiquitous Systems[END_REF] present the project Celadon in order to establish an infrastructure enabling on-demand collaboration between heterogeneous mobile devices and environmental devices. The Celadon project is a middleware architecture for ubiquitous device collaboration. Collaborative environments are organized into Celadon zones, which are public areas equipped with wireless access points for technologies. Collaboration in this work is limited to sharing hardware or software resources. Only the user context is considered in this work.

The Conami middleware [START_REF] Fawaz | Conami: Collaboration based content adaptation middleware for pervasive computing environment[END_REF] is a Collaboration-based Content Adaptation Middleware for dynamic pervasive computing environments. The authors consider the case of Mobile Ad hoc Networks (MANETs). The middleware allows devices in MANETs to collaborate with each other to perform content adaptation. Content adaptation is derived from the context of the user and the user environment and is done by composing available nearby services. In this work, collaboration is considered as a technique to adapt the content to the devices' capabilities. In this middleware, component deployment is not considered.

Perich et al. [START_REF] Perich | Collaborative joins in a pervasive computing environment[END_REF] present the design and implementation of a Collaborative Query Processing protocol that enables devices in pervasive computing environments to locate data sources and obtain data matching their queries. The features of the protocol allow devices to collaborate with other devices in order to obtain an answer for their queries regardless of their limited computing, memory, and battery resources. The presented approach deals mainly with low level context parameters.

Introductive Scenario (Collaborative warships game)

The scenario presented in this section illustrates the main features needed by collaborative applications in pervasive environments, such as contextawareness, spontaneous implicit sessions and automatic component deployment.

This scenario has been produced in the context of Itea2's A2Nets (Autonomic Services in M2M Networks) project1 . In our case, multi-media games are distributed over a fleet of buses and are available to passengers. Whenever a user boards a bus, his/her personal device notices the presence of the bus network and connects to it.

We consider a game called Collaborative Warships, which is a multi-player extension of the classic warships board game. Each player uses his/her mobile device to play and to interact with other players.

Ship Types

We distinguish three ship types: Battle Ships, Repair Ships and Spy Ships. Each ship has two main attributes: life points and speed.

Battle Ships (BS) can shoot at the ships of the opposite team. Any opposing ships that receive damage loose life points.

Repair Ships (RS) restores life points to the companion ships.

Spy Ships (SS) can detect the position of the enemy ships. One of the Battle Ships of a team is the leader of the BSs and SSs. This ship is called Battle Ship Leader (BSL). One of the Repair Ships of a team is the leader of the RSs. This ship is called Repair Ship Leader (RSL).

Game Rules

Two teams are required for the game to begin. Each team must have at least 1 BS. During the play, the positions of the enemy warships are unknown (except to SSs).

When a ship loses all its life points, it sinks, and leaves the game. The game ends when a team loses all its BS, and the team with ships remaining is declared the winner.

Two main requirements of pervasive collaborative applications are highlighted and instantiated in this example:

• Context awareness is necessary in order to detect changes in both external context (e.g. user presence) and execution context (e.g. battery level, available memory).

• Adaptation and especially adaptive deployment are necessary in order to respond to context changes. If, for instance, the device of a BSL has not enough energy to host several software components handling the exchanged flows, then a set of those components has to be moved and deployed on another mobile device in order to ensure a continuous communication between the team members.

Team Coordination

Players need to collaborate during the game. Therefore, data flows must be set up between the members of a team. Depending on the type of warship used by each player, different communication flows for cooperating with the other members of his/her team are needed. For example, when a BS reaches an enemy with its fire, it can inform the BSL of the position of the enemy. The BSL can then transmit that position to the other BSs of his/her team having more destruction points or having a more strategic position in order to shoot at the discovered enemy.

According to this description, several types of communication links are needed between the members of a team. Those links are represented in figure 1. Each RS is linked to the RSL of his/her team. BSs and SSs collaborate with the BSL of their team. The two leaders collaborate.

In the following sections, we propose a generic approach that may be used by developers in order to build collaborative pervasive applications.

Proposed approach

The main idea of our approach is to provide a generic framework which enables modeling, session management, multi-level adaptation, and automatic component deployment for pervasive applications. This enables code modularization and reutilization.

Multi-level Architecture Modeling

In order to clearly separate different concerns in our approach, a multilevel architecture is proposed.

A configuration is denoted as A n,i , where n is the considered abstraction level and i is the sequence number (i.e. an architecture A n,i evolves to A n,i+1 when it is reconfigured). For a given configuration A n,i at level n, multiple configurations (A n-1,1 , . . . , A n-1,p ) may be implemented at level n -1. Adapting the architecture to constraint changes at level n -1 by switching among these multiple configurations allows maintaining unchanged the n-level configuration.

Multi-level modeling of collaborative pervasive architectures

This subsection presents how we apply the multi-level approach presented above to collaborative pervasive systems in order to build a comprehensive generic framework for such systems.

Adaptation at the highest level should be guided by the evolution of activity requirements. Adaptation at the lowest levels should be driven by execution context constraint changes.

Application level

The application level represents applications needing collaboration among groups of users and/or devices. It contains software elements that are relevant to collaboration,and is represented as the abstraction level A 3,i .

Application designers also have to implement the refinement procedure which obtains the set A i 2 of collaboration level models that implement a given A 3,i model.

Collaboration level

The main issue addressed by the collaboration level is the determination of a high-level collaboration schema that responds to the application's collaboration needs. Hence, collaborative sessions may be managed from this level. The elements needed to implement such sessions are determined from this level. This model is inspired by classic graph-based session description formalisms such as dynamic coordination diagrams [START_REF] Baudin | E-Education applications: human factors and innovative approaches[END_REF].

Middleware level

The middleware level provides a communication model that masks lowlevel details in order to simplify the representation of communication channels. The architectural model produced by this level, A 1,i represents a detailed deployment descriptor containing the elements needed in order to implement the sessions defined by the collaboration level.

Context Capture and Representation

We target the adaptation of cooperative applications to different context changes. These changes may concern resource constraints (e.g. connectivity, energy level, available memory, etc.) or the evolution of the collaborative activity and environment (where participants can arrive and leave, change roles, active a "do not disturb" mode in their devices, etc.). We respectively call these two sets of parameters resources context and external context. Context parameters are captured by external modules.

Relevant events that are produced in the external context are taken into account by the application level (i.e. they are detected and translated into modifications on the A 3,i model, thus producing a new model A 3,i+1 ).

Relevant events that are produced related to run-time resource changes are taken into account by the middleware level. If it is possible, changes are handled by reconfiguring the A 1,i+1 model (i.e. by selecting a different model among the possible refinements of the collaboration model).

Generic Refinement and Selection Procedures

For a given architectural configuration A n,i in level n, a procedure, called Refine() computes the set A i n-1 . This set represents all possible architectural configurations in level n -1 that implement A n,i .

Given a set of possible architectures, it is necessary to choose an architecture to be effectively deployed. We present here a procedure, Select() (table 2), that allows for the choosing of an architecture depending on several parameters.

This procedure uses the resources context (e.g. variations of communication networks and processing resources) to eliminate the architectural configurations that cannot be deployed within the current resources levels. Among the set of selected architectures, the best configuration w.r.t architectural characteristics is selected.

The choice of an architecture must first take into account the resources context. The Context Adaptation() function (table 2, line 5) is a generic function that depends on the resources context. For example, it can express the availability level of a given resource (bandwidth, memory, energy level, etc.). This function is used for two purposes: it allows discarding architectures that cannot be deployed within the current resources context, and it allows for the selection of the architectures best adapted to that context.

The function Context Adaptation() associates a given architecture to a value that reflects its degree of adaptation to the current resources context. Our criterion for this function is as follows: well adapted architectures are those which have fewer nodes in a critical situation. A node is in a critical situation when its level for a certain resource is close to the threshold defined for that resource. This function is detailed in table 1. In this function, a set of resources Resource 1 . . . Resource R (e.g. Resource 1 =energy, Resource 2 =CPU load and Resource 3 =available RAM) is considered.

L i r Table 1: Context Aware Function Context Adaptation() { Let A 1,q be an architectural configuration at level 1 Let N = card(A 1,q ) Let Resource 1 . . . Resource R the set of considered resources Let R = card({Resource 1 . . . Resource R }) Let node q i be deployment node i of A 1,q Let L i
r the level of the resource r for node q i Let Tr be the threshold associated with the resource r Let α i r r ∈ [1..R] be weights associated with each resource r for node q i Let βr r ∈ [1..R] be weights associated with each resource r for A 1,q Let cadapt=0 for each i ∈ 1..N for each r ∈ 1..R

P i r = α i r L i r -Tr if Pr ≤ 0 then return -1 end for end for for each r ∈ 1..R cadapt=cadapt+βr min i (P i r ) end for return cadapt } Table 2: Generic selection procedure. 1 Select(Policy) 2 { 3 Let An,p ∈ An, p ∈ N 4 Let C denote the context attributes 5 Select S 1 = {A n-1,k ∈ A p n-1 , k ∈ N such that: Context Adaptation(A n-1,k , C) ≥ Context Adaptation(X, C), ∀X ∈ A p n-1 } 6 if card(S 1 ) = 1 7 if Policy == Dispersion 8 Select S 2 = {A n-1,k ∈ S 1 , k ∈ N such that: Dispersion(A n-1,k ) ≥ Dispersion(X), ∀X ∈ S 1 } 9 if Policy == Distance 10
Let An,p and An,q ∈ An, p, q ∈ N 11

Let A n-1,p the current mapping at level n -

1 of An,p 12
Select

S 2 = {A n-1,k ∈ A q n-1 , k ∈ N such that: Relative Cost(A n-1,p , A n-1,k ) ≤ Relative Cost(A n-1,p , X), ∀X ∈ S 1 } 13 if card(S 2 ) = 1 14
Select any configuration from S 2 15 } represents the available level of the resource Resource r for the node node i . L i r , expressed as a percentage, is calculated as the resource's level before deployment (given by the resources context module) minus the amount of resource consumed by each deployed component. The value T r is a threshold that indicates the critical percentage of the resource r, for a given node, under which the deployment is not possible. The coefficients α i r represent the importance assigned to each level L i r with respect to the characteristics of node i . For instance, the CPU load is more critical for a smartphone than for a laptop, because the smartphone needs CPU processing for other important functions (such as answering calls, etc.). Therefore, α i CP U is 1 for a laptop nodes and 0.5 for a smartphone (i.e. a smartphone will be considered as critical when its CPU level is lower than 2T CP U ). P i r is calculated for every node as the difference between the level of the Resource r and the threshold T r . If this difference is negative for a node, this means that the node is in a critical state with respect to Resource r , and hence the considered architecture cannot be deployed. Therefore, -1 is returned and the considered architecture will not be selected. If no node was found to be in a critical situation, then the degree of adaptation to the context (cadapt) of the considered architecture is calculated, as shown in table 1, line 20. First, for every resource, the minimum value of P i r found on any deployment node of the architecture is retained. Second, cadapt is calculated as an average of these minima (weighted by β r coefficients). The β r coefficients represent the global importance degree given to each resource ( R 1 β r = 1). Resources with higher β r are considered more important than other resources. These coefficients can be defined by the administrator or the business logic. This definition of the function Context Adaptation leads to the selection of the architectures having the highest values of resources for their more critical nodes.

When several architectures have the same value of Context Adaptation(), a policy (indicated by the parameter Policy) is used by the Select() procedure in order to retain the optimal configuration. If the chosen policy is Weight, the selection is based on minimizing the function Dispersion() (table 2, line 8). This generic function corresponds to the cost or the efficiency/performance of an architecture. For instance, it may be defined as the number of software components deployed per node. If the chosen policy is Distance, the selection minimizes the distance between two architectural configurations at level n -1, both implementing the corresponding n-level architectural configuration. This is performed using the function Relative Cost()

(

A Multi-level Collaborative Framework for pervasive Systems

The proposed multi-level modeling approach for collaborative pervasive systems remains generic with respect to implementation. As we have separated the approach and its implementation, this generic approach may be implemented in different ways by different designers. In this subsection we propose an implementation that can be used by application designers as a collaboration framework for pervasive systems, with our choices illustrated in figure 2. We first present the models used in each level, followed by the refinement and selection procedures that enable the transitions between levels.

Application level Model -A 3,i

We have chosen an ontology based model because it constitutes a standard knowledge representation system, allowing reasoning and inference. We have chosen to describe these models in OWL [START_REF] Smith | OWL Web ontology Langage Guide, W3C Recommendation[END_REF], the Semantic Web standard for metadata and ontologies.

In general, ontologies are separated in two levels: a generic ontology and a specific ontology. The former is a domain-wide ontology, but independent of applications. The latter ontology extends the generic one with terms specific to an application category. We have followed the same pattern in our implementation: there is a generic collaboration ontology (that describes sessions, users, roles, data flows, nodes, etc.) and an application ontology that extends the collaboration ontology with business-specific concepts and relations.

Session

Collaboration level Model -A 2,i

The collaboration level model is a graph, inspired by dynamic collaboration diagrams [START_REF] Baudin | E-Education applications: human factors and innovative approaches[END_REF]. This model is a generic collaboration ontology that is common to all applications, and therefore it is provided within the framework. This ontology2 is represented in figure 3. The main concept in this ontology is Session. A session contains one or more Flows, which have a source Node and a destination Node. Nodes are hosted on Devices. Each Node has one or more associated Roles. Flows are processed with Tools, which are composed of several Components (e.g. SenderComponents and ReceiverComponents). Related Flows, Tools and Components share the same DataType (e.g. Audio, Text or Video). This graph is expressed in the GraphML language (an XML dialect for representing graphs [START_REF] Brandes | GraphML Progress Report[END_REF]). Further explanations about this ontology and the associated choices can be found in [START_REF] Sancho | A Semantic-driven Auto-adaptive Architecture for Collaborative Ubiquitous Systems[END_REF].

Middleware level Model -A 1,i

For the middleware level, we have retained the Event Based Communication (EBC) paradigm [START_REF] Meier | Taxonomy of distributed event-based programming systems[END_REF]. The EBC model represents a well established paradigm for interconnecting loosely coupled components and it provides a one-to-many or a many-to-many communication pattern.

EBC entities are represented in the middleware level model. This model is a detailed graph containing a set of event producers (EP), event consumers (EC) and channel managers (CM) connected with push and push links. Multiple producers and consumers may be associated through the same CM. Since this model is also a graph, it is also expressed in the GraphML language.

Application-Collaboration Refinement and Selection

When a reconfiguration event related to the external context is detected, it is captured by the application level and translated into changes in the application level ontology, thus producing a new instance, A 3,1 . As the application level model is represented in OWL, we use SWRL rules [START_REF] Horrocks | SWRL: A Semantic Web Rule Language Combining OWL and RuleML, W3C Member Submission[END_REF] in order to implement its refinement to a collaboration architecture. Some rules have been included along with the generic collaboration ontology. For example, let us consider the rule shown in table 3.

Table 3: A SWRL Rule for AudioFlows

AudioFlow(?af) ∧ hasSource(?af,?src) ∧ hasDestination(?af,?dst) ∧ swrlx:createOWLThing(?asc,?src) ∧ swrlx:createOWLThing(?arc,?dst) → AudioSenderComponent(?asc) ∧ isDeployedOn(?asc,?src) ∧ AudioReceiverComponent(?arc) ∧ isDeployedOn(?arc,?dst)

This rule states that, whenever an instance of AudioFlow is found in the ontology, two components have to be instantiated3 : an AudioReceiverComponent having the flow's destination node as its deployment node, and an AudioSenderComponent having the source node as its deployment node. Similar rules are used for text and video flows, thus generating text and video sender and receiver components.

The rules implementing the transition from the application-specific ontology to the generic collaboration ontology are application-dependant, and therefore they have to be specified by the application designers along with the application ontology.

The processing4 of the SWRL rules along with the application ontology produces a new ontology instance (A 2,1 that describes the collaboration level graph in OWL language). This graph is translated into GraphML (by means of a simple XSLT transformation) in order to be shared with the middleware level. If the new model A 2,1 is equal to the previous A 2,0 , then the process ends, as no architecture reconfiguration is required in lower levels. Each application level model corresponds to a unique collaboration level model. Therefore the selection procedure at this level is straightforward.

Collaboration-Middleware Refinement and Selection

We use graph grammars [START_REF] Chomsky | Three models for the description of language, Information Theory[END_REF] for the Collaboration-Middleware Refinement.

In order to refine a given collaboration architecture into a set of EBC architectures, the graph grammar GG COLLAB→EBC ,is used 5 . In this graph grammar, non-terminal nodes are collaboration entities while terminals nodes are EBC entities. For clarity's sake, here only the case of audio sessions is considered. Therefore, the productions of this graph grammar refine AudioReceiverComponent and AudioSenderComponent (AR and AS) into EPs, ECs and CMs. Similar grammar productions have been developed for text and video components.

In order to select the optimal architecture among those built by the refinement process, the generic selection procedure presented in table 2 is used. We detail here our choices for the functions Dispersion() and Relative Cost().

The function Dispersion() is used to select architectures having fewer CMs deployed in the same device. The goal is to efficiently balance resources consumption and to be more robust. It associates an architecture A 1,q with the number of nodes containing at least one CM. This definition gives higher values to architectures having CMs dispersed in more nodes. The function Relative Cost() is used to select the closest architecture to the currently deployed architecture. We choose as criterion of selection the number of redeployments needed to switch from a given architecture to another.

is-a is-a is-a is-a is-a is-
If the selection process is not able to find a valid architectural configuration, then the high-level requirements cannot be implemented within the current resources context. Therefore, the middleware level sends a reconfiguration event to the application level. This level takes this into account in order to change the application level model into a model A 3,i that can be implemented.

Deployment Service

The A 1,i model produced by the middleware level is the detailed deployment descriptor that implements the low-level elements of the required architecture: producers, consumers, channel managers, and links. In order to effectively deploy such elements into real devices, a Deployment Service is needed. This service takes a deployment descriptor A 1,i as input and then it downloads, installs and starts the required components on each device. The implementation of this deployment service is based on the OSGi technology [START_REF]OSGi Service Platform Release[END_REF].

Application to the Scenario

Designers of collaborative pervasive applications only have to create i) the application level model, and ii) the SWRL rules that enable the refinement of application level models into collaboration level models. In this section, these design tasks are applied to the Collaborative Warships application.

Application Ontology for the Collaborative Warships Application

The domain ontology6 modeling the business concepts and relations of the Collaborative Warships application is illustrated in figure 4. The main concept of this ontology is Warship. The different types of warships (BS, RS, SS) are modeled as sub-concepts of Warship. Warships belong to a Team. This domain ontology is related to the generic collaboration ontology.

SWRL Rules for the Collaborative Warships Application

Application-dependent SWRL rules are needed in order to process instances of the application ontology presented in the previous subsection, and thus generate a collaboration level model.

In the case of the Collaborative Warships application, we consider, for instance, the rule presented in table 4. Here, we have only considered the case of audio flows. Text and video flows are handled in a similar way. BattleShip(?bs) ∧ BattleShipLeader(?bsl) ∧ belongsToSameTeam(?bs, ?bsl) ∧ differentFrom(?bs, ?bsl) ∧ swrlx:createOWLThing(?af1, ?bs) ∧ swrlx:createOWLThing(?af2, ?bs) → sessions:AudioFlow(?af1) ∧ sessions:hasFlow(sessionBS,?af1)∧ sessions:hasSource(?af1, ?bsl) ∧ sessions:hasDestination(?af1, ?bs) ∧ sessions:AudioFlow(?af2) ∧ sessions:hasFlow(sessionBS,?af2) ∧ sessions:hasSource(?af2, ?bs) ∧ sessions:hasDestination(?af2, ?bsl)

The BS BSL audio flows rule, presented in table 4, states that whenever a BS and a BSL belonging to the same team are found, two audio flows are created between them. This flow belongs to a session called sessionBS, which is an individual of the class sessions:session already existing in the ontology Both audio flows will be captured by the collaboration level rules and corresponding audio senders and receivers will be created on each node in the collaboration level graph.

Initial Refinement and Adaptation Process Examples

This subsection presents a top-down example showing the architecture models for the three considered levels as well as the refinement processes between levels. The adaptation process is also illustrated.

In this example, we focus only on one team in order to illustrate the initial refinement and the adaptation processes. The team has five players (player1 to player5). Each player represents a node in the application. Each node has an associated role: player1 is a BSL, player2 is a regular BS, player3 is a SS, player4 is a RSL and player5 is a regular RS. Each node has an associated device (m1 to m5).

This business level architectural configuration is captured by the game application and represented in the game ontology A 3,0 . The resulting ontology instances are represented in figure 5, application level. Concepts are drawn as ellipses, while relations are drawn as arrows going from one concept to another concept. As the figure shows, concepts and relations from the game-specific ontology (warships: prefix) and from the generic collaboration ontology (sessions: prefix) are instantiated together.

In order to refine this application model, SWRL rules are processed over these ontology instances. The BS BSL audio flows rule creates two audio flows between player1 and player2, the BSL RSL audio flows rule creates two audio flows between player1 and player4, etc. The rule presented in table 3 is processed for each instance of AudioFlow found, thus creating the corresponding AudioSenderComponent and AudioReceiverComponent at the endpoints of the audio flow.

The resulting collaboration graph is represented in figure 5, collaboration level. This graph contains 7 audio senders (AS) and 7 audio receivers (AR). The graph edges correspond to data flows, and are labeled by data type (audio) and by the session to which they belong. Each component has three attributes: the identifier, the type (sender or receiver) and the deploying machine identifier.

In order to refine this collaboration level graph, the GG COLLAB→EBC graph grammar, is used. This produces a set of valid configurations A 0 1 . The procedure Select() is used in order to find the optimal configuration. The retained configuration, A 1,0 , is presented in figure 5, middleware level. This configuration contains only terminal nodes (i.e. nodes belonging to the EBC level).This refinement creates a detailed deployment descriptor that is used by the deployment service in order to deploy the indicated components on each device, thus implementing the required application level session.

Evaluation

We conducted evaluation experiments using our engine GMTE for executing the graph grammar transformations, and the rule engine Jess for executing the SWRL rules.

In this example, we present results related to one team composed of 9 players (player1 to player9). Each node has an associated role: player1 is a BSL; player2; player6 and player7 are regular BSs; player3 is a SS; player4 is a RSL ;and player5 and player8 are regular RSs. Each node has an associated device (m1 to m9).

The scenario shown in the figure 6 consists of adding a new player every 2 minutes combined with important diminutions of the energy level of the players' devices as shown in Table 5. The underlined values in Table 5 represent degradations that trigger a new reconfiguration. Each time a player joins the game, a refinement and a selection process is triggered. Each time an energy level diminution of a player's device is detected, the selection process is triggered. In the case of this game we consider only three collaboration sessions as mentioned above and shown in figure 1. Since we deploy a channel manager for each session, 1, 2 or 3 channel managers can be redeployed after each selection process. In the worst case, 3 channel managers are redeployed.

Threshold calculation

Figure 7(a) illustrates the elapsed time for the generation of the collaboration ontology, the collaboration graph and the middleware graphs during each phase. Before starting the game, the framework needs to be initialized, at this time no player is connected to the game so no middleware graph is generated. Figure 7(a) (phase0) shows the time necessary for the initialization. The average initialization time is about 5 seconds. With respect to the initialization process that occurs only once at the beginning of the session, this duration is acceptable to the game.

This business level architectural configuration of phase 1 is captured by the game application and represented in the game ontology A 3,0 . The resulting ontology instance is represented in figure 5. We represent the time elapsed to generate the ontology instance and the corresponding collaboration and ECB graphs as shown in figure 7(a) (phase1) when player5 joins the game. The generation time (about 6 seconds) is still acceptable for this kind of application.

We now encounter an important diminution of the energy level of the device m1, and as a result the ECB level runs the Select() procedure. This procedure chooses a new configuration (e.g. A 1,5 ) more adapted to the new context parameters. Within this new configuration, m1 has fewer components deployed on it. The deployment service uses A 1,5 for effective redeployment. To study the time consumption of the Select() procedure, the scenario illustrates further diminutions of energy levels and the arrival of new players.

The figure 7(b) shows that the selection time is still acceptable within the range of 0.005 to 0.017 seconds. This duration indicates that the framework

Conclusion

This paper has presented a multi-level architecture modeling approach for collaborative pervasive systems. Architectural models for application, collaboration and middleware levels have been detailed. Ontologies, SWRL rules, graphs and graph grammars have been used for implementing a rule-based refinement process. These rules handle both transforming a given architecture within the same level and architectural mappings between different levels. This implementation constitutes a framework for building collaborative pervasive systems. The performances of the framework have being evaluated, especially the computing time of adaptation actions and the time of component (re)deployment. Our approach has been illustrated by a multi-media game distributed over a bus fleet. This game is an example of a collaborative pervasive application developed in the context of the European project A2Nets, on real field experiments using pervasive networks. A deployment service is being developed based on the OSGi technology. Also, the possibility of implementing a centralized and a P2P version of the deploy-ment service for different applications is under investigation. Future research actions include defining further adaptation rules that consider more context parameters (e.g. by extending one of the cited context ontologies), extending refinement procedures and extending graph grammars to handle middleware level abstractions other than Event Based Communications. Finally, the design and implementation of mechanisms enabling the spontaneous setup of implicit sessions will be considered.

Figure 1 :

 1 Figure 1: Team coordination description.

Figure 3 :

 3 Figure 3: Generic collaboration ontology.

Figure 4 :

 4 Figure 4: Collaborative Warships application ontology.

Figure 5 :

 5 Figure 5: Top-down refinement example.

Figure 6 :

 6 Figure 6: Experimentation scenario.

  Time elapsed for reconfiguration.

Figure 7 :

 7 Figure 7: Evaluation experiments

table 2 ,

 2 line 12).

	External Context Capture		Business Logic				Application Layer
				A	3,p	Ontology (OWL)
	SWRL		XSLT				Collaboration
	Processing	Engine	A	2,q	Graph (GraphML)	Layer
	Graph Transormation Engine	1,i A ... A 1,j	Selection	Graph (GraphML)	Communication (ECB) Layer
					1,k		Infrastucture
	Resources Context Capture					Layer

A Figure 2: Model-oriented leveled framework.

Table 4 :

 4 BS BSL audio flows Rule

Table 5 :

 5 Devices Energy Levels

	P P P P P P P P P Device Phase	1	2	3	4	5	6	7	8	9
	player1's device	95%	82%	82%	82%	81%	81%	81%	62%	62%
	player2's device	90%	88%	87%	87%	86%	85%	70%	70%	70%
	player3's device	90%	88%	87%	87%	86%	85%	85%	85%	84%
	player4's device	90%	88%	87%	80%	86%	50%	50%	49%	49%
	player5's device	91%	88%	87%	87%	86%	85%	85%	84%	82%
	player6's device	92%	88%	87%	87%	86%	85%	85%	83%	83%
	player7's device	92%	88%	88%	86%	86%	85%	85%	70%	70%
	player8's device	93%	89%	89%	87%	86%	85%	83%	50%	48%
	player9's device	90%	88%	87%	87%	86%	85%	85%	83%	82%

https://a2nets.erve.vtt.fi/

http://homepages.laas.fr/tazi/ontologies/sessions.owl

The SWRL built-in CreateOWLThing() creates new instances of existing concepts within a SWRL rule.

This processing is done with a rules engine such as Jess or a SWRL-enabled reasoning engine such as Pellet.

This implementation is done with a Graph Matching and Transformation Engine (GMTE), available at http://www.homepages.fr/khalil/GMTE.

http://homepages.laas.fr/tazi/warships.owl

Acknowledgement

This article was funded partially by the IST IP's IMAGINE project and the Itea2's A2Nets. We thank all the LAAS' members who have contributed to achieve earlier versions on top of which this work has been built. Our thanks are addressed in particular German SANCHO who has contributed to the development of FACUS and special thanks to David Allison for proofreading.