Anton Baranov
email: anton.d.baranov@gmail.com

Rachid Zarouf
email: rachid.zarouf@univ-amu.fr

A MODEL SPACE APPROACH TO SOME CLASSICAL INEQUALITIES FOR RATIONAL FUNCTIONS

We consider the set R n of rational functions of degree at most n ≥ 1 with no poles on the unit circle T and its subclass R n, r consisting of rational functions without poles in the annulus ξ : r ≤ |ξ| ≤ 1 r . We discuss an approach based on the model space theory which brings some integral representations for functions in R n and their derivatives. Using this approach we obtain L p -analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdélyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as n tends to infinity and the poles of the rational functions approach the unit circle T.

Introduction

The goal of this paper is to give a unified approach to several classical inequalities for rational functions. This approach is based on integral representations for rational functions and their derivatives. It makes possible to recover several known results and obtain their L p analogs where the estimate is given not only in terms of the degree of a rational function but also in terms of the distance from the poles to the boundary.

1.1. Notations. Let P n be the space of complex analytic polynomials of degree at most n ≥ 1 and let

R n = P Q : P, Q ∈ P n , Q(ξ) = 0 for |ξ| = 1 ,
be the set of rational functions of degree at most n (where deg P Q = max (deg P, deg Q)) without poles on the unit circle T = {ξ ∈ C : |ξ| = 1}. We denote by f L p , 1 ≤ p ≤ ∞, the standard norms of the spaces L p (T, m), where m stands for the normalized Lebesgue measure on T. Denote by D = {ξ ∈ C : |ξ| < 1} the unit disc of the complex plane and by D = {ξ ∈ C : |ξ| ≤ 1} its closure. For a given r ∈ (0, 1), we finally introduce the subset

R n, r = P Q : P, Q ∈ P n , Q(ξ) = 0 for r ≤ |ξ| ≤ 1 r
of R n , consisting of rational functions of degree at most n without poles in the annulus ξ : r ≤ |ξ| ≤ 1 r . We also introduce some notations specific to the theory of model subspaces of the Hardy space H p , 1 ≤ p ≤ ∞. Denote by Hol (D) the space of all holomorphic functions on D. The Hardy space H p = H p (D), 1 ≤ p < ∞, is defined as follows:

H p = f ∈ Hol (D) : f p H p = sup 0≤ρ<1 ˆT |f (ρξ)| p dm(ξ) < ∞ .
The first author is supported by the Chebyshev Laboratory (St. Petersburg State University) under RF Government grant 11.G34.31.0026, by JSC "Gazprom Neft" and by RFBR grant 12-01-00434.

As usual, we denote by H ∞ the space of all bounded analytic functions in D. For any σ = (λ 1 , . . . , λ n ) ∈ D n , we consider the finite Blaschke product

K Bσ = B σ H 2 ⊥ = H 2 ⊖ B σ H 2 .
The subspace K Bσ consists of rational functions of the form P/Q, where P ∈ P n-1 and Q is a polynomial of degree n with the zeros 1/λ 1 , . . . , 1/λ n of corresponding multiplicities.

For any Blaschke product B, the reproducing kernel of the model space K B corresponding to a point ξ ∈ D is of the form

k B ξ (z) = 1 -B(ξ)B(z) 1 -ξz .
1.2. Some classical inequalities for rational functions. In this subsection we give a brief review of several well-known inequalities for polynomials and rational functions.

1.2.1. Pointwise estimates for the derivatives of the functions in R n . Let us start with the following theorem.

Theorem. For any function f ∈ R n with the poles {a k } (counting multiplicities) we have

(1.1) |f ′ (ξ)| ≤ f L ∞ |a k |<1 1 -|a k | 2 |a k -ξ| 2 + |a k |>1 |a k | 2 -1 |a k -ξ| 2 , |ξ| = 1.
Inequality (1.1) has a long history. It was for the first time explicitly stated and proved (by two different methods) for the case when all poles are outside D in a monograph by V.N. Rusak [START_REF] Rusak | Rational Functions as Approximation Apparatus, Izdat. Beloruss[END_REF]Chapter III,Section 1] in 1979. Also, as Rusak mentions, this inequality is contained (only with a hint of a proof) in the book of V.I. Smirnov and N.A. Lebedev [SL,Chapter V,Section 3,Corollary 3].

At the same time this inequality (for poles both inside D and outside D) is a very special case of results of M.B. Levin [START_REF] Levin | An estimate of the derivative of a meromorphic function on the boundary of domain[END_REF][START_REF] Levin | Estimation of the derivative of a meromorphic function on the boundary of domain[END_REF] which were obtained already in 1974-1975, but remained unnoticed; these results apply to arbitrary functions admitting pseudocontinuation.

Further extensions of Levin-Rusak inequality were obtained in the 1990s independently by two groups of specialists in polynomial inequalities [START_REF] Borwein | Polynomials and Polynomial Inequalities[END_REF][START_REF] Li | Bernstein-type inequalities for rational functions with prescribed poles[END_REF]. In particular, in [START_REF] Borwein | Sharp extensions of Bernstein's inequality to rational spaces[END_REF] (see also [START_REF] Borwein | Polynomials and Polynomial Inequalities[END_REF]Theorem 7.1.7]) P. Borwein and T. Erdélyi obtained the following interesting improvement which shows that the sum in (1.1) may be replaced by the maximum.

Theorem. For any function f ∈ R n with the poles {a k } we have

(1.2) |f ′ (ξ)| ≤ f L ∞ max |a k |<1 1 -|a k | 2 |a k -ξ| 2 , |a k |>1 |a k | 2 -1 |a k -ξ| 2 , |ξ| = 1.
1.2.2. Spijker's Lemma. A well-known result by M.N. Spijker [S] (known as Spijker's Lemma) asserts that the image of T under a complex rational map f ∈ R n has length at most 2nπ, or, in other words,

(1.3) f ′ L 1 ≤ n f ∞ , f ∈ R n .
The inequality is apparently sharp (take f (z) = z n ). This result published by Spijker in 1991 ended a long search for the best bound in this inequality (e.g., in 1984 R.J. Leveque and L.N. Trefethen [LT] proved the above inequality with 2n in place of n). The importance of the sharp constants in this inequality is related to its role in the Kreiss Matrix Theorem [K]. However, it was recently noticed (see [N]) that inequality (1.3) was discovered already in 1978 by E.P. Dolzhenko [D] as a special case of more general results. Unfortunately, this paper (which appeared only in Russian) remained unknown to the specialists. Let us cite the following beautiful theorem from [D] where majorization on the whole circle is replaced by majorization on its subset.

Theorem. If E is a measurable subset of T or of a line in the complex plane and

f ∈ R n , |f (u)| ≤ 1, u ∈ E, then ´E |f ′ (u)| |du| ≤ 2πn; if f is
real valued, then the constant 2π can be replaced by 2. The latter estimate is sharp for n = 0, 1, 2, . . . and for each E with positive measure.

We refer to a recent paper by N.K. Nikolski [N] for a detailed account of the history of inequality (1.3) and its relations to the Kreiss Matrix Theorem, as well as for some new developments in the Kreiss theory.

Let us also mention that the Dolzhenko-Spijker inequality (1.3) follows easily from (1.2) (or from (1.1). This was mentioned by X. Li [Li] (see also [START_REF] Finis | On the spectral side of Arthur's trace formula -absolute convergence[END_REF]Remark 7]). Indeed, integrating (1.2) with respect to the normalized Lebesgue measure m, we obtain (1.3). 1.2.3. S.M. Nikolskii's inequalities. By the well-known results of S.M. Nikolskii [SMN] (1951), the essentially sharp inequality

(1.4) f L q ≤ c(p, q)n 1 p -1 q f L p
holds for all polynomials f of degree at most n and for all 1 ≤ p < q ≤ ∞ with a constant c(p, q) depending only on p and q. Analogous inequalities were proved in [SMN] for trigonometric polynomials of several variables. A few years later (1954) G. Szegö and A. Zygmund [SZ] among other more general results rediscovered (1.4) (for polynomials of one variable only) and extended it to the whole range 0 < p < q ≤ ∞.

Main results

In the present paper we use an approach based on the model space theory to obtain some extensions of inequalities of type (1.2), (1.3), and (1.4) for functions in R n or R n, r . The estimates we obtain will depend not only on the degree of a rational function n but also on the distance 1 -r from the poles to the boundary. We show the sharpness of the obtained inequalities as both n → ∞ and r → 1-.

Our main tools are integral representations for rational functions and their derivatives. Integral representations for a derivative of a rational function were introduced and successfully used by R. Jones, X. Li, R.N. Mohapatra and R.S. Rodriguez [START_REF] Jones | On the Bernstein inequality for rational functions with a prescribed zero[END_REF]Lemma 4.3] (for rational functions without poles in D) and by X. Li [START_REF] Li | Integral Formulas and inequalities for rational functions[END_REF]Lemma 3] in the general case. At the same time such representations are known in the setting of general model spaces where they were also used to produce some estimates for the derivatives [A,[START_REF] Baranov | Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings[END_REF][START_REF] Dyakonov | Differentiation in star-invariant subspaces I. Boundedness and compactness[END_REF].

From now on, for two positive functions U and V , we say that U is dominated by V , denoted by U V , if there is a constant c > 0 such that U ≤ cV ; we say that U and V are comparable, denoted by U ≍ V , if both U V and V U.

2.1. L p -version of the Borwein-Erdélyi inequality. We first give an analog of (1.1)-(1.2) in which the L ∞ -norm at the right is replaced by the L p -norm, p ≥ 1.

Theorem 2.1. Let 1 ≤ p ≤ ∞. Then (i) For any function f ∈ R n with the set of poles a = {a k } (repeated counting multiplicities) we have

(2.1) |f ′ (ξ)| ≤ D 1 p 1 (a) |a k |<1 1 -|a k | 2 |a k -ξ| 2 + D 1 p 2 (a) |a k |>1 |a k | 2 -1 |a k -ξ| 2 f L p for all |ξ| = 1, where (2.2) D 1 (a) = |a k |<1 1 + |a k | 1 -|a k | , D 2 (a) = |a k |>1 |a k | + 1 |a k | -1 .
(ii) Moreover, (2.1) is sharp in the following sense: for any p ∈ [1, ∞] there exists a constant c(p) > 0 such that for any n ≥ 2 and any r ∈ (0, 1) there exists f ∈ R n with the poles {a k } on the circle {|z| = 1 r } such that f ′ (-1)

f L p ≥ c(p)D 1 p 2 (a) |a k |>1 |a k | 2 -1 |a k + 1| 2 .
Clearly, inequality (1.1) is the limit case of (2.1) when p = ∞.

2.

2. An L p -version of the Dolzhenko-Spijker Lemma. Next we obtain a version of (1.3) where the L ∞ -norm at the right is replaced by the L p -norm, p ≥ 1. Note that this inequality does not follow from (2.1) by integration on the unit circle T, as it was the case for p = 1.

Theorem 2.2. For every rational function f ∈ R n having n 1 poles inside D and n 2 poles outside of D, we have

(2.3) f ′ L 1 ≤ n 1-1 p 1 D 1 p 1 (a) + n 1-1 p 2 D 1 p 2 (a) f L p where a = {a k } stand for the poles of f and D 1 (a) and D 2 (a) are defined in (2.2).
This inequality is asymptotically sharp when n = n 1 + n 2 tend to ∞ and dist ({a k }, T) → 0 (see Theorem 2.3 below).

2.3. L p -L q Bernstein-type inequality. Now we consider the following Bernstein-type problem, which could be interpreted as a generalization of (2.1) and (2.3): given n ≥ 1, r ∈ (0, 1) and 1 ≤ p, q ≤ ∞, let C n, r (L q , L p ) be the best possible constant in the inequality

f ′ L q ≤ C n, r (L q , L p ) f L p , f ∈ R n, r . One could also introduce C n (L q , L p ) = sup r∈(0,1) C n, r (L q , L p ) . The Dolzhenko-Spijker Lemma means exactly that C n (L 1 , L ∞ ) = n. It is however easy to see (take f (z) = (1 -rz) -1 as a test function) that C n (L q , L p ) = ∞ unless q = 1 and p = ∞.
Thus, the dependence on r (that is, on the distance from the poles to the boundary) appears naturally in the problem.

Theorem 2.3. Let n ≥ 1, r ∈ (0, 1), and 1 ≤ p, q ≤ ∞. We have

(2.4) C n, r (L q , L p ) ≍    n 1-r 1+ 1 p -1 q , q ≥ p, n (1-r) 1+ 1 p -1 q , q ≤ p,
with the constants depending only on p and q, but not on n and r.

Moreover, the constant in upper bound is in both of these two cases (1 + r) 1+ 1 p -1 q : we have

(2.5) C n, r (L q , L p ) ≤    (1 + r) 1+ 1 p -1 q n 1-r 1+ 1 p -1 q , q ≥ p, (1 + r) 1+ 1 p -1 q n (1-r) 1+ 1 p -1 q , q ≤ p.
The upper bound for C n, r (L q , L p ) is obviously sharp for the special case p = q = ∞, since it is reached by the Blaschke product b n r . Moreover, it is proved in [Z] that for any fixed r in (0, 1), there exists a limit

lim n→∞ C n, r (L 2 , L 2 ) n = 1 + r 1 -r ,
and thus, the bound

(1 + r) 1+ 1 p -1
q is again (asymptotically as n → ∞) sharp for p = q = 2. It is possible that this constant is sharp in the general case 1 ≤ p, q ≤ ∞.

In Subsection 4.3 we compare inequality (2.5) with a theorem by K.M. Dyakonov [START_REF] Dyakonov | Smooth functions in the range of a Hankel operator[END_REF]Theorem 11].

2.4. An extension of S.M. Nikolskii's inequality to rational functions. Direct analogs (with the dependence on n only) of (1.4) do not exist for functions in R n . As always, it is easy to check this fact considering the test function f (z) = (1 -rz) -1 as r tends to 1 -. A natural extension of (1.4) for functions in R n, r can be stated as follows:

Theorem 2.4. Let 1 ≤ p < q ≤ ∞, n ≥ 1 and r ∈ (0, 1).

(i) We have

(2.6) f L q ≤ 1 + r 1 -r 1 p -1 q n 1 p -1 q 1 + (n 2 + 1) 1 p -1 q f L p , f ∈ R n, r ,
where n 1 (respectively, n 2 ) is the number of poles of f inside D (respectively, outside D). In particular,

(2.7)

f L q n 1 -r 1 p -1 q f L p , f ∈ R n, r ,
with a constant depending only on p and q.

(ii) The inequality (2.7) is sharp: for 1 ≤ p < q ≤ ∞ there exists a constant c(p, q) > 0 such that for any r ∈ (0, 1) and n ≥ 2 there exists f ∈ R n, r with the property

f L q f L p ≥ c(p, q) n 1 -r 1 p -1 q .
Remark. It is possible that for any 1 ≤ p < q ≤ ∞ the upper bound (1+r)

1 p -1 q in (2.6
) is asymptotically sharp as n tends to infinity, for any fixed r ∈ (0, 1). We are able to provide a simple proof of this fact for the special case q = ∞, 2 ≤ p < ∞. Indeed, using the test function

f = 1 1+rz n-1 k=0 b k -r , r ∈ (0, 1), we clearly have f 2 L 2 = n 1-r 2 , since the family 1 1+rz b k -r is orthogonal in L 2 . Moreover, f L ∞ = f (-1) = n
1-r and thus

f p L p ≤ f p-2 L ∞ f 2 L 2 = 1 1 + r n 1 -r p-1 , 2 ≤ p < ∞. As a consequence, f L ∞ f L p ≥ n 1 + r 1 -r 1 p , 2 ≤ p < ∞,
which gives the result since the poles of f are all outside 1 r D, n 2 = n = deg f .

3.

Integral representations for rational functions and their derivatives

Preliminaries.

In what follows we may assume without loss of generality that the functions

f = P Q ∈ R n we consider are such that: (1) max (deg P, deg Q) = n, (otherwise f ∈ R m , m < n), (2) deg P ≤ deg Q : indeed, if n = deg P > l = deg Q and ξ ∈ T, then we define g (ξ) = f ξ and supposing that P (z) = n k=0 a k z k , a n = 0 and Q(z) = l k=0 b k z k , b l = 0, we obtain (multiplying by ξ n ) g (ξ) = n k=0 a k ξ -k l k=0 b k ξ -k = n k=0 a k ξ n-k l k=0 b k ξ n-k = n j=0 a n-j ξ j n j=n-l b n-j ξ j = P (ξ) Q (ξ)
,

where and (3) all the poles of f (i.e., the zeros of Q) are pairwise distinct: indeed, we can assume this perturbing slightly the poles of f and the result will follow by continuity. From now on, for every function f ∈ R n we will denote by σ 1 and σ 2 the sets of poles of f (repeated counting multiplicities) which are respectively inside D or outside D,

P (z) = n j=0 a n-j z j and Q(z) = n j=n-l b n-j z j are such that P , Q ∈ P n and deg P ≤ deg Q. Moreover, we clearly have |g ′ (ξ)| = f ′ ξ for all ξ ∈ T, g L p = f L p , 1 ≤ p ≤ ∞, and g ′ L q = f ′ L q , 1 ≤ q ≤ ∞, since g ′ (ξ) = -1 ξ 2 f ′ ξ , ξ ∈ T,
σ 1 = λ 1 , λ 2 , . . . , λ n 1 , σ 2 = {1/µ 1 , 1/µ 2 , . . . , 1/µ n 2 } .
Also, we will denote by

B 1 = n 1 j=1 b λ j , B 1 = n 1 j=1 b λ j and B 2 = n 2 j=1 b µ j ,
the corresponding finite Blaschke products and by

k B 1 ξ , k B 1 ξ , k B 2
ξ the reproducing kernels at the point ξ of the corresponding model spaces. Under the assumption (3), f can be written as

(3.1) f (ξ) = a + n 1 k=1 c k ξ -λ k + n 2 k=1 d k 1 -µ k ξ , c k , d k ∈ C, where n = n 1 + n 2 = deg f (a = 0 if and only if deg P < deg Q). We put g(ξ) = n 1 k=1 c k ξ-λ k and h(ξ) = n 2 k=1 d k 1-µ k ξ , so that f = a + g + h.
We denote by V σ 1 , σ 2 the vector space of all functions of the form (3.1).

3.2.

Integral representation for rational functions on the unit circle. We first obtain an integral representation for a function f ∈ R n .

Lemma 3.1. Keeping the notations and assumptions 1 and 2 of Subsection 3.1, for any function f ∈ R n we have

(3.2) f (ξ) = f, φ ξ , |ξ| = 1, where φ ξ (u) = k zB 2 ξ (u) + ξuk B 1 ξ (u), u, ξ ∈ T, and •, • = •, • L 2 stands for the scalar product in L 2 = L 2 (T, m).
Proof. Without loss of generality, we can suppose that f satisfies assumption 3 of Subsection 3.1 for the same reason of continuity. Thus, f satisfies the above formula (3.1):

(3.3) f (ξ) = a + h(ξ) + g(ξ)
.

Clearly, a + h ∈ K zB 2 . Thus, for a fixed ξ we have

a + h(ξ) = a + h, k zB 2 ξ .
Moreover, since g ∈ H 2 0 (where H 2 0 stands for the subspace of H 2 consisting of functions f such that f (0) = 0), we have

(3.4) a + h(ξ) = a + g + h, k zB 2 ξ = f, k zB 2 ξ , |ξ| ≤ 1,
Note that by the continuity of the kernel k zB 2 ξ in D × D, this formula extends to ξ ∈ T. To obtain an analogous formula for g, consider the function

(3.5) ϕ(ξ) = 1 ξ g 1 ξ ,
which belongs to K B 1 and as a consequence we can write, for |ξ| < 1,

(3.6) 1 ξ g 1 ξ = ϕ, k B 1 ξ = ˆT ϕ(u) 1 -B 1 (u)B 1 (ξ) 1 -uξ dm(u).
Now setting w = 1 ξ , |w| > 1, changing the variable v = ū and using the fact that ϕ(u) = ūg(ū) = vg(v), u ∈ T, we get

(3.7) g(w) = ˆT 1 u g 1 u 1 -B 1 (u)B 1 1 w w -u dm(u) = ˆT g(v)v 1 -B 1 (v)B 1 1 w w -v dm(v).
Note also that (3.7) holds for |w| > 1 and, by continuity, also for |w| = 1. Now, for |w| = 1,

(3.8) g(w) = w ˆT g(v)v 1 -B 1 (v) B 1 (w) 1 -vw dm(v) = w g, zk B 1 w = f, wzk B 1 w ,
where the finite Blaschke product B 1 is defined in Subsection 3.1 (we have

B 1 (v) = B 1 (v), |v| = 1),
and the last equality is due to the fact that the function zk B 1 w belongs to H 2 0 and thus, is orthogonal to a + h. Now, combining (3.4) and (3.8), we obtain for any ξ ∈ T :

f (ξ) = a + h(ξ) + g(ξ) = f, k zB 2 ξ + ξzk B 1 ξ ,
which completes the proof.

3.3.

Integral representation for the derivative of rational functions on the unit circle. The integral representation in this subsection essentially coincides with the representation due to X. Li [START_REF] Li | Integral Formulas and inequalities for rational functions[END_REF]Lemma 3] (for the case of rational functions without poles in D it was proved by R. Jones, X. Li, R.N. Mohapatra and R.S. Rodriguez [START_REF] Jones | On the Bernstein inequality for rational functions with a prescribed zero[END_REF]Lemma 4.3]). It is possible to reinterpret the proof of Li's Lemma using the theory of model spaces.

Lemma 3.2. Keeping the notations and assumptions 1 and 2 of Subsection 3.1, for any function

f ∈ R n , (3.9) f ′ (ξ) = f, ψ ξ , |ξ| = 1, where ψ ξ (u) = u k B 2 ξ (u) 2 -ξ 2 u k B 1 ξ (u) 2 , u, ξ ∈ T.
Proof. The scheme of the proof repeats the one of Lemma 3.1. We use again the fact that f can be written as in (3.1). This time, we notice first that h ∈ K B 2 . Then for a fixed ξ, we have

h ′ (ξ) = h, z (1 -ξz) 2 = h, z k B 2 ξ 2 .
Here the first equality is the standard Cauchy formula, while the second follows from the fact that

z(1 -ξz) -2 -z k B 2 ξ (z) 2 ∈ B 2 H 2 and h ⊥ B 2 H 2 .
Thus, for the case when all poles are outside the disc D, the formula is immediate. Moreover, since g ∈ H 2 0 (where H 2 0 is defined above in the proof of Lemma 3.1) we have

(3.10) h ′ (ξ) = g + h, z k B 2 ξ 2 = f, z k B 2 ξ 2 , |ξ| ≤ 1,
Again, by the continuity of the kernel in D × D, this formula extends to ξ ∈ T.

To obtain an analogous formula for g ′ , consider the function ϕ ∈ K B 1 defined by (3.5). Now setting in (3.6) w = 1 ξ , |w| > 1, we get

g(w) = ˆT ϕ(u) 1 -B 1 (u)B 1 1 w w -u dm(u).
Then, differentiating with respect to w, we obtain

g ′ (w) = ˆT ϕ(u) 1 -B 1 (u)B 1 1 w w -u ′ w dm(u) = -ˆT ϕ(u) 1 -B 1 (u)B 1 1 w w -u 2 dm(u),
since, by a direct computation,

1 -B 1 (u)B 1 1 w w -u ′ w + 1 -B 1 (u)B 1 1 w w -u 2 ∈ B 1 H 2
(as a function of u), while ϕ ⊥ B 1 H 2 . Changing the variable v = ū and using that ϕ(u) = ūg(ū) = vg(v), u ∈ T, we get

g ′ (w) = -ˆT g(v)v 1 -B 1 (v)B 1 1 w w -v 2 dm(v).
Recalling that the finite Blaschke product B 1 defined in Subsection 3.1 satisfies

B 1 (v) = B 1 (v), |v| = 1, we have (3.11) g ′ (w) = -ˆT g(v)v 1 -B 1 (v) B 1 (w) w -v 2 dm(v) = -ˆT f (v)v 1 -B 1 (v) B 1 (w) w -v 2 dm(v).
The last equality follows from the fact that h ∈ H 2 , while

v 1-B 1 (v) B 1 (w) w-v 2 ∈ H 2 0 .
Note also that (3.11) holds for |w| > 1 and, by continuity, also for |w| = 1. Now, applying formulas (3.10)-(3.11) to z = w ∈ T and recalling that f = a + g + h, a ∈ C, we conclude that:

f ′ (ξ) = ˆT(h + g)(u)ψ ζ (u)dm(u) = f, ψ ξ ,
since ψ ξ is orthogonal to 1.

Remark. The above integral representation immediately implies inequality (1.1) by Levin and Rusak.

Indeed, |f ′ (ξ)| ≤ ψ ξ L 1 f L ∞ = (|B ′ 1 (ξ)| + |B ′ 2 (ξ)|) f L ∞ (see (4.1) below).
It is, however, unclear, whether one can prove the Borwein-Erdélyi inequality (1.2) using the representation (3.9).

Proofs of the upper bounds

In this Section we prove the upper bounds from Section 2, that is, inequalities (2.1), (2.3), (2.5) and (2.6). All these proofs are based on

(1) integral representations (3.2) or (3.9);

(2) estimates of H p -norms of reproducing kernels k B ξ , where B is a finite Blaschke product. In particular, we will often use the fact that for any finite Blaschke product B = d j=1 b ν j , ν j ∈ D, and for any ξ ∈ T,

(4.1) k B ξ 2 L 2 = |B ′ (ξ)| = d j=1 1 -|ν j | 2 |ξ -ν j | 2 .
We will use here the assumptions and the notations of Subsection 3.1.

4.1. Proof of Theorem 2.1, inequality (2.1). We first assume 1 ≤ p < ∞ and denote by p ′ the conjugate exponent for p. Applying the Hölder inequality to the identity (3.9), we obtain

|f ′ (ξ)| ≤ f L p ψ ξ L p ′ ≤ f L p (k B 1 ξ ) 2 L p ′ + (k B 2 ξ ) 2 L p ′ = f L p k B 1 ξ 2 H 2p ′ + k B 2 ξ 2 H 2p ′ , |ξ| = 1.

Now for any finite

Blaschke product B = d j=1 b ν j , ν j ∈ D, we have for |ξ| = |u| = 1 (4.2) k B ξ 2p ′ H 2p ′ = ˆT k B ξ (u) 2(p ′ -1)+2 dm(u) ≤ k B ξ 2 H 2 max |u|=1 k B ξ (u) 2(p ′ -1) .
On one hand, by (4.1),

k B ξ 2 H 2 = d k=1 1 -|ν k | 2 |1 -ν k ξ| 2 , |ξ| = 1,
and on the other hand that for any u, ξ ∈ T,

(4.3) |k B ξ (u)| ≤ d j=1 1 -|ν j | 2 |1 -ν j ξ| |1 -ν j u| ,
which, by the Cauchy-Schwarz inequality, gives

(4.4) |k B ξ (u)| 2 ≤ d j=1 1 -|ν j | 2 |1 -ν j ξ| 2 d j=1 1 + |ν j | 1 -|ν j | , u ∈ T.
Thus combining (4.4) with (4.2), we obtain:

(4.5) k B ξ 2p ′ H 2p ′ ≤ d k=1 1 -|ν k | 2 |1 -ν k ξ| 2 p ′ d j=1 1 + |ν j | 1 -|ν j | p ′ -1
.

Applying the last inequality to B 1 and B 2 we get

k B 1 ξ 2 H 2p ′ + k B 2 ξ 2 H 2p ′ ≤ n 1 k=1 1 -|λ k | 2 |1 -λ k ξ| 2 n 1 k=1 1 + |λ k | 1 -|λ k | 1 p + n 2 k=1 1 -|µ k | 2 |1 -µ k ξ| 2 n 2 k=1 1 + |µ k | 1 -|µ k | 1 p ,
which, in view of the definition of D 1 (a) and D 2 (a), completes the proof.

4.2. Proof of Theorem 2.2. We use the integral representation (3.9) and Fubini-Tonelli's Theorem to get:

(4.6)

f ′ L 1 ≤ ˆT ˆT |f (τ )| k B 2 ξ (τ ) 2 + k B 1 ξ (τ ) 2 dm(τ ) dm(ξ) = ˆT |f (τ )| B ′ 2 (τ ) + B ′ 1 (τ ) dm(τ ),
where the last equality comes from (4.1). Now, for any finite Blaschke product B = B σ of degree d, corresponding to a set σ = {ν 1 , ν 2 , . . . , ν d } ⊂ rD, we have B ′ L 1 = d (this follows immediately if we integrate equality (4.1)).

On the other hand,

(4.7) B ′ L ∞ ≤ d k=1 1 -|ν k | 2 (1 -ν k z) 2 L ∞ ≤ d k=1 1 + |ν k | 1 -|ν k | .
This implies that for any 1 ≤ p ′ ≤ ∞,

(4.8) B ′ L p ′ ≤ B ′ 1-1 p ′ L ∞ B ′ 1 p ′ L 1 ≤ d 1 p ′ d k=1 1 + |ν k | 1 -|ν k | 1-1 p ′ .
Going back to (4.6) and applying the Hölder inequality, we get

f ′ L 1 = ˆT |f (τ )| |B ′ 2 (τ )| dm(τ ) + ˆT |f (τ )| | B ′ 1 (τ )|dm(τ ) ≤ f L p B ′ 2 L p ′ + B ′ 1 L p ′ ,
where p ′ is the conjugate of p,. Applying the last inequality combined with (4.8) to B 1 and B 2 we get

f ′ L 1 ≤ f L p n 1-1 p 1 n 1 k=1 1 + |λ k | 1 -|λ k | 1 p + n 1-1 p 2 n 2 k=1 1 + |µ k | 1 -|µ k | 1 p
, as required.

4.3. Proof of Theorem 2.3, inequality (2.5). The proof will consist of several steps.

Step 1. The case q = ∞, 1 ≤ p ≤ ∞. Clearly, for a function f ∈ R n, r having n 1 poles inside D and n 2 poles outside D we have

D 1 ≤ n 1 1 + r 1 -r , D 2 ≤ n 2 1 + r 1 -r .
Taking the supremum over all ξ ∈ T in (2.1) we obtain (4.9)

f ′ L ∞ ≤ 1 + r 1 -r 1+ 1 p (n 1 + n 2 ) f L p .
Step 2. The case q = 1, 1 ≤ p ≤ ∞. A direct consequence of the inequality (2.3) is that (4.10)

f ′ L 1 ≤ 1 + r 1 -r 1 p (n 1 + n 2 ) f L p ,
for any f ∈ R n, r having n 1 poles inside D and n 2 poles outside D.

Step 3. The case p = q. For any f ∈ R n, r (as always of the form (3.1)), we have by (3.9)

f ′ p L p = ˆT |f ′ (ξ)| p dm(ξ) = ˆT ˆT τ f (τ )u k B 2 ξ 2 -ξ 2 u k B 1 ξ 2 dm(τ ) p dm(ξ) ≤ ˆT ˆT |f (τ )| |k B 2 ξ (τ )| 2 + |k B 1 ξ (τ )| 2 dm(τ ) p dm(ξ).
Now applying the Hölder inequality (p ′ being the conjugate of p), we obtain

f ′ p L p ≤ ˆT k B 2 ξ (τ ) 2 + |k B 1 ξ (τ )| 2 dm(τ ) p p ′ ˆT |f (τ )| p k B 2 ξ (τ ) 2 + |k B 1 ξ (τ )| 2 dm(τ ) = |B ′ 2 (ξ)| + | B ′ 1 (ξ)| p p ′ ˆT |f (τ )| p k B 2 ξ (τ ) 2 + |k B 1 ξ (τ )| 2 dm(τ ),
where the last equality comes from (4.1). Now, integrating the last inequality on the unit circle with respect to ξ, we obtain

f ′ p L p ≤ ˆT |B ′ 2 (ξ)| + | B ′ 1 (ξ)| p p ′ ˆT |f (τ )| p k B 2 ξ (τ ) 2 + |k B 1 ξ (τ )| 2 dm(τ )dm(ξ) ≤ max |ξ|=1 |B ′ 2 (ξ)| + | B ′ 1 (ξ)| p p ′ ˆT ˆT |f (τ )| p k B 2 ξ (τ ) 2 + |k B 1 ξ (τ )| 2 dm(τ )dm(ξ) = max |ξ|=1 |B ′ 2 (ξ)| + | B ′ 1 (ξ)| p p ′ ˆT |f (τ )| p |B ′ 2 (τ )| + | B ′ 1 (τ )| dm(τ ).
Finally, using (4.7) we obtain

f ′ p L p ≤ max |ξ|=1 |B ′ 2 (ξ)| + | B ′ 1 (ξ)| p p ′ +1 f p L p ≤ n 1 k=1 1 + |λ k | 1 -|λ k | + n 2 k=1 1 + |µ k | 1 -|µ k | p p ′ +1 f p L p , whence (4.11) f ′ L p ≤ n 1 + r 1 -r f L p .
Step 4. The case 1 < q < p < ∞. This follows by interpolation between the cases q = 1 and q = p. For any f ∈ R n, r , we have, by the Hölder inequality with the exponents p-1 p-q and p-1 q-1 ,

f ′ q L q = ˆT |f ′ (τ )| p-q p-1 + p(q-1) p-1 dm(τ ) ≤ f ′ p-q p-1 L 1 f ′ p(q-1) p-1 L p
. Now, using both the inequalities (4.10) and (4.11) from Steps 2 and 3, we obtain the required estimate:

f ′ q L q ≤ n 1 + r 1 -r 1 p f L p p-q p-1 n 1 + r 1 -r f L p p(q-1) p-1 ≤ n q 1 + r 1 -r 1 p p-q p-1 + p(q-1) p-1 f q L p = n q 1 + r 1 -r q-1+ q p f q L p .
Step 5. The case 1 ≤ p ≤ q ≤ ∞. We now interpolate between q = p and q = ∞:

f ′ q L q ≤ f ′ q-p L ∞ f ′ p L p ≤ n 1 + r 1 -r 1+ 1 p -1 q f L p ,
where we used inequalities (4.9) and (4.11).

Remark. For the case when 1 ≤ q ≤ p ≤ ∞ and the function f ∈ R n has no poles in D, a different proof of the upper bound (but without explicit constants) could be given by an application of a result by K.M. Dyakonov [Dy1,Theorem 11]. Namely, applying inequality (11.2) from [START_REF] Dyakonov | Smooth functions in the range of a Hankel operator[END_REF] with s = 1 we obtain that for a rational function f of degree n with the poles 1/ν 1 , 1/ν 2 , . . . , 1/ν n / ∈ D we have (4.12)

f ′ L q ≤ C B ′ L γ f L p , where q ≤ p, 1 γ = 1 q -1 p ,
C is a constant depending on q and p which is not precised, and B = B σ is the finite Blaschke product corresponding to σ = (ν 1 , ν 2 , . . . , ν n ). It remains to apply inequality (4.8). 4.4. Proof of the upper bound in Theorem 2.4. Step 1. The special case q = ∞ and 1 ≤ p ≤ 2. Following the assumptions of Subsection 3.1 and applying Hölder inequality to (3.2), we obtain (with the notations of Lemma 3.1) that for any ξ ∈ T,

(4.13) |f (ξ)| ≤ f L p φ ξ L p ′ ,
where p ′ ≥ 2 stands for the conjugate of p. Moreover,

φ ξ L p ′ ≤ k zB 2 ξ L p ′ + k B 1 ξ L p ′
. Now, we prove that for any finite Blaschke product B = B σ of degree d, corresponding to a set σ = {ν 1 , ν 2 , . . . , ν d } ⊂ rD, we have

(4.14) k B ξ L p ′ ≤ 1 + r 1 -r d 1+ 1 p .
Indeed, as a direct consequence of (4.3), we get

(4.15) k B ξ L ∞ ≤ d j=1 1 + |ν j | 1 -|ν j | ≤ d 1 + r 1 -r .
Moreover, (4.1) clearly gives that for any ξ ∈ T,

(4.16) k B ξ 2 L 2 = d j=1 1 -|ν j | 2 |1 -ν j ξ| 2 ≤ d j=1 1 + |ν j | 1 -|ν j | ≤ d 1 + r 1 -r .
Thus, combining (4.15) and (4.16), we get that for any p ′ ≥ 2,

k B ξ L p ′ ≤ k B ξ 1-2 p ′ L ∞ k B ξ 2 p ′ L 2 ≤ d p ′ -1 1 + r 1 -r p ′ -1
.

Applying this to B = B 1 and to B = zB 2 , we get (4.17)

φ ξ L p ′ ≤ 1 + r 1 -r 1-1 p ′ n 1-1 p ′ 1 + (n 2 + 1) 1-1 p ′ ,
where n 1 (respectively, n 2 ) is the number of poles of f inside D (respectively, outside D). Thus, it follows from (4.13) and (4.17) that for any 1 ≤ p ≤ 2,

(4.18) f L ∞ ≤ 1 + r 1 -r 1 p n 1 p 1 + (n 2 + 1) 1 p f L p ,
as required.

Step 2. The case 1 ≤ p ≤ 2 and 1 ≤ p < q ≤ ∞. For any f ∈ R n, r , f q L q ≤ f q-p L ∞ f p L p , which gives using (4.18),

f L q ≤ f 1-p q L ∞ f p q L p ≤ 1 + r 1 -r 1 p -1 q n 1 p 1 + (n 2 + 1) 1 p 1-p q f L p ≤ 1 + r 1 -r 1 p -1 q n 1 p -1 q 1 + (n 2 + 1) 1 p -1 q f L p .
Step 3. The case 2 ≤ p ≤ ∞ and 1 ≤ p < q ≤ ∞. Let p ′ and q ′ be the conjugates to p and q, respectively. Then 1 ≤ q ′ < p ′ ≤ 2. By Step 2 we have for any V σ 1 , σ 2 ⊂ R n, r (see Subsection 3.1 for the definition)

Id (Vσ 1 , σ 2 , L q ′ )→(Vσ 1 , σ 2 , L p ′ ) ≤ 1 + r 1 -r 1 q ′ -1 p ′ n 1 q ′ -1 p ′ 1 + (n 2 + 1) 1 q ′ -1 p ′ ,
where Id is the identity operator. Now denoting by Id ⋆ the adjoint operator (for the usual Cauchy duality) of Id, we have Id ⋆ = Id. Therefore,

Id ( Vσ 1 , σ 2 , L p )→(Vσ 1 , σ 2 , L q ) = Id ⋆ ( Vσ 1 , σ 2 , L p )→(Vσ 1 , σ 2 , L q ) = Id (Vσ 1 , σ 2 , L q ′ )→(Vσ 1 , σ 2 , L p ′ ) ≤ 1 + r 1 -r 1 q ′ -1 p ′ n 1 q ′ -1 p ′ 1 + (n 2 + 1) 1 q ′ -1 p ′ = 1 + r 1 -r 1 p -1 q n 1 p -1 q 1 + (n 2 + 1) 1 p -1 q ,
which is the required estimate.

Proofs of the lower bounds

In this section we prove the asymptotic sharpness of the inequalities in Theorems 2.1, 2.3 and 2.4 as n tends to infinity and the poles of the rational functions approach the unit circle T. From now on, we denote by (5.1)

D n (z) = n-1 k=0 z k
the Dirichlet kernel of order n ≥ 1. The asymptotic behaviour of D n L q as n tends to ∞ is well known: for q > 1,

(5.2)

lim n→∞ D n L q n 1-1 q = 1 π ˆR sin x x q dx 1 q . Put I n = z ∈ T : z = e it , t ∈ π 4n , π 2n
. It is well known and easy to see that for q > 1 the integral over the arc I n gives a substantial contribution to the norms D n L q and D n D ′ n L q , namely (5.3)

ˆIn |D n (ξ)| q dm(ξ) n q-1 , ˆI2n |D n (ξ)D ′ n (ξ)| q dm(ξ) n 3q-1 .
Now we introduce the test functions from R n which will be used throughout this section to illustrate the sharpness of the inequalities in Theorems 2.1, 2.3 and 2.4. Put (5.4)

f (z) = b ′ -r (z) n-2 k=0 b k -r (z), b -r (z) = z + r 1 + rz .
Thus, f is essentially the Dirichlet kernel transplanted from the origin to the point λ = -r ∈ (0, 1) by the change of variable •b -r . Also, for n ≥ 4, we put

(5.5)

g(z) = b ′ -r (z) N k=0 b k -r (z) 2 ∈ R n ,
where N is the integer part of n-2 2 . It is interesting to note that other natural (and simpler) candidates such as h

(z) = (1 -rz) -n or h(z) = 1 1-rz b n-1 r (z)
will not give the right order of n in the inequality (2.5) for p < q.

We will need the following lemma.

Lemma 5.1. We have

(5.6) f L p n 1 -r 1-1 p , 1 < p ≤ ∞,

and

(5.7)

g L p n 2-1 p (1 -r) 1-1 p , 1 ≤ p ≤ ∞.
Proof. Let us compute the L p -norm of f . Making use of the change of variable ζ = b -r (ξ) we get

f p L p = ˆT b ′ -r (ξ) b ′ -r (ξ) p-1 n-2 k=0 b k -r (ξ) p dm(ξ) = ˆT b ′ -r (b -r (ζ)) p-1 |D n-1 (ζ)| p dm(ζ). (5.8) By a straightforward computation, b ′ -r • b -r = r 2 -1 (1+rb -r ) 2 = -(1+rz) 2 1-r 2 . Thus, f p L p = 1 (1 -r 2 ) p-1 ˆT |1 + rζ| 2(p-1) |D n-1 (ζ)| p dm(ζ) ≤ (1 + r) p-1 (1 -r) p-1 D n-1 p L p ,
and the statement follows from (5.2).

Analogously, changing the variable ζ = b -r (ξ), we obtain

g p L p = ˆT b ′ -r (b -r (ζ)) p-1 |D N +1 (ζ)| 2p dm(ζ) = 1 (1 -r 2 ) p-1 ˆT |1 + rζ| 2(p-1) |D N +1 (ζ)| 2p dm(ζ) ≤ (1 + r) p-1 (1 -r) p-1 D N +1 2p L 2p (1 + r) p-1 (1 -r) p-1 n 2p-1 .
5.1. Sharpness in Theorem 2.1. We prove here the statement (ii) of Theorem 2.1. Without loss of generality we assume that n ≥ 5 (for small n the statement is obvious, take the test function (1 -rz) -2 ). First we consider the case 1 < p ≤ ∞. For r ∈ (0, 1) let f be defined by (5.4). We have

f ′ = b ′′ -r n-2 k=0 b k -r + b ′ -r 2 n-3 k=0 (k + 1)b k -r . Moreover, b -r (-1) = 1, b ′ -r (-1) = -1+r 1-r , and b ′′ -r (-1) = 2r(1+r) (1-r) 2 . This gives f ′ (-1) = 2r(n -1)(1 + r) (1 -r) 2 + 1 + r 1 -r 2 n-3 k=0 (k + 1) > 1 + r 1 -r 2 n-3 k=0 (k + 1) n 1 -r 2 .
On the other hand, for a = {a k }, a k = 1 r , k = 1, . . . , n, we have

max |a k |>1 |a k | 2 -1 |a k + 1| 2 , |a k |<1 1 -|a k | 2 |a k + 1| 2 = n 1 + r 1 -r , D 2 (a) = |a k |>1 |a k | + 1 |a k | -1 = n 1 -r .
Thus, by (5.6),

max |a k |>1 |a k | 2 -1 |a k + 1| 2 , |a k |<1 1 -|a k | 2 |a k + 1| 2 D 1 p 2 (a) = n 1 + r 1 -r 1+ 1 p f ′ (-1) f L p .
In the case p = 1, we consider the test function g defined by (5.5). We have (5.9)

g ′ = b ′′ -r N k=0 b k -r 2 + 2 b ′ -r 2 N -1 k=0 (k + 1)b k -r N k=0 b k -r = - 2 1 + rz b ′ -r N k=0 b k -r r N k=0 b k -r + 1 -r 2 1 + rz N -1 k=0 (k + 1)b k -r .
As before, this gives g ′ (-1) = 2r(1 + r) (1 -r) 2 (N + 1) 2 + 2 1 + r 1 -r 2 (N + 1)

N -1 k=0 (k + 1) n 3 (1 -r) 2 ,
which gives the required estimate from below for g ′ (-1)/ g L p if we use (5.7). 5.2. Sharpness of (2.5). Here we show the asymptotic sharpness of the constants C n, r (L q , L p ) as n → ∞ and r → 1-. Clearly, we need to show the sharpness only for sufficiently large values of n (for small values of n one may use the test function (1 -rz) -2 ).

Step 1. The case 1 ≤ p ≤ q ≤ ∞. We consider the same test function g defined in (5.5). Let us estimate from below the norm g ′ L q using the representation (5.9) for g ′ . Taking b -r (ξ) as the new variable (as in the proof of Lemma 5.1), we get g ′ q L q = 2 q ˆT 1 1 + rb -r (ξ) q b ′ -r (b -r (ξ))

q-1 D N +1 (ξ) rD N +1 (ξ) + 1 -r 2 1 + rb -r (ξ) D ′ N +1 (ξ) q dm(ξ).

Since 1 + rb -r (z) = 1-r 2 1+rz and b ′ -r • b -r (z) = -(1+rz) 2 1-r 2 , we have

g ′ q L q = 2 q (1 -r 2 ) 2q-1 ˆT |1 + rξ| 3q-2 D N +1 (ξ) rD N +1 (ξ) + (1 + rξ)D ′ N +1 (ξ) q dm(ξ) ≥ 2 q (1 -r 2 ) 2q-1 ˆI2N D N +1 (ξ) rD N +1 (ξ) + (1 + rξ)D ′ N +1 (ξ) q dm(ξ),
where the arcs I N are defined at the beginning of the section. Now, by (5.2) and (5.3),

ˆI2N D N +1 (ξ) 2q dm(ξ) N 2-1 q and ˆI2N (1 + rξ)D N +1 (ξ)D ′ N +1 (ξ) q dm(ξ) ≥ (1 + r 2 ) q 2 ˆIN D N +1 (ξ)D ′ N +1 (ξ) q dm(ξ) N 3-1 q .
We conclude that (5.10)

g ′ L q 1 1 -r 2 2-1 q N 3-1 q .
As a result, combining (5.10) and (5.7) we obtain

g ′ L q g L p n 1 -r 1+ 1 p -1 q
for sufficiently large values of n.

Step 2. The case 1 ≤ q ≤ p ≤ ∞. We consider now a simpler test function h(z) = 1 1-rz b n-1 r (z). We have

h L p = 1 1 -rz L p 1 (1 -r) 1-
h ′ q L q = ˆT |b ′ r (ξ)| |b ′ r (ξ)| q-1 n -1 1 -rξ - r 1 -r 2 b r (ξ) dm(ξ) = ˆT |b ′ r (b r (ζ))| q-1 n -1 1 -rb r (ζ) - r 1 -r 2 ζ q dm(ζ) = 1 (1 -r 2 ) 2q-1

ˆT

(1 -rζ) 2 q-1 |(n -1)(1 -rζ) -rζ| q dm(ζ).

Now, integrating over the arc π 2 ≤ arg ζ ≤ 3π 2 , it is easily seen that for n ≥ 3, ˆT

(1 -rζ) 2 q-1 |(n -1)(1 -rζ) -rζ| q dm(ζ)

1/q ≥ cn,
where c > 0 is a numerical constant independent of q. Thus, h ′ L q n(1 -r) 1 q -2 , r ∈ (0, 1), and so

h ′ L q h L p n (1 -r) 1+ 1 p -1 q ,
r ∈ (0, 1), n ≥ 3.

5.3. Sharpness of (2.7). Let g be the test function defined in (5.5). Recall that, by (5.7),

g L p n 2-1 p (1 -r) 1-1 p , 1 ≤ p ≤ ∞.
On the other hand, we have (after the change of the variable) We conclude that for 1 ≤ p < q ≤ ∞,

g q L q = 1 (1 -r 2 ) q-1 ˆT |1 + rξ|
g L q g L p n 1 p -1 q (1 -r) 1 p -1 q .

  the elementary Blaschke factor associated to λ ∈ D. Define the model subspace K Bσ of the Hardy space H 2 by

  2 b r , and, by the change of variable ζ = b -r (ξ),

  2q-2 |D N +1 (ξ)| 2q dm(ξ) and ˆT |1 + rξ| 2q-2 |D N +1 (ξ)| 2q dm(ξ) ≥ ˆIN+1 |D N +1 (ξ)| 2q dm(ξ) n 2q-1 .


Bertrand Simon

Matthieu Debailleul

Vincent Georges

Olivier Haeberlé
email: olivier.haeberle@uha.fr

Vincent Lauer
email: lauer@waika9.com

Holographic microscopy and diffractive microtomography of transparent samples

Keywords: numbers: 42.30.Wb, 42.30.Kq, 42.40Kw Image reconstruction, tomography, Fourier optics, holographic interferometry, other holographic techniques

We present an optical tomographic diffractive microscope, a device able to image a complex refractive index distribution in three dimensions. Theoretical foundations are first recalled: diffraction under the first Born approximation explains the link between diffracted beam, object frequencies and physical properties of the object. We then describe our experimental setup, recording 2-D interferograms in the image space, and detail the image reconstruction process underlying our tomographic microscope, which involves 2-D transforms of the recorded interferograms, a peculiar 3-D mapping of the data, and a final 3-D Fourier reconstruction. We apply tomographic reconstruction to diatom skeletons, unicellular algae with cell walls made of silica, and compare it to holographic reconstruction. We further apply it to pollen grains and show differences between the real and imaginary parts of the measured complex refractive index. Finally, we also recall alternative tomographic methods.

Introduction

Optical tomographic microscopy allows for three-dimensional imaging of unstained samples by using successive illuminations in various conditions, recording of the diffracted waves, and finally reconstructing the sample from the diffracted waves. In this paper we use the approach developed in [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF] in which the sample is fixed and the direction of a coherent illuminating wave is varied. This method allows for high-resolution three-dimensional imaging and has a potential for high-speed, threedimensional imaging, as the imaging can be performed without moving the sample nor moving any heavy component: the direction of the illuminating beam can be varied using a lightweight galvanometer mirror or even an acousto-optic deflector. In our setup, we use a detector placed in an image plane, which simplifies the imaging device and improves speed.

In the present paper, we first describe the general principles underlying the obtainable resolution. We describe our optical setup and the method for obtaining three-dimensional images. We also discuss existing alternatives to do optical tomographic microscopy.

Theoretical background

We first give a short description, based on the approach given by Wolf [START_REF] Wolf | Three-dimensional structure determination of semi transparent objects from holographic data[END_REF], within the first Born approximation for weakly scattering objects. We recall the scalar propagation equation in an inhomogeneous medium, neglecting polarization effects:

∇ 2 + k(r) 2 ψ(r) = 0 with k(r) = 2π λ n(r) (1)
wherein n(r) = µ(r)ǫ(r) µ0ǫ0

is the complex refractive index and λ is the wavelength in vacuum of an incident plane wave.

Such an equation cannot be solved directly. Writing a solution in terms of Green's function

G(r) = exp(jk i |r|) 4π|r| (2)
wherein k i = 2π n0 λ is the modulus of the wave vector of the incident plane wave in the immersion medium having index n 0 , leads to [START_REF] Slaney | Limitations of Imaging with First-Order Diffraction Tomography IEEE transactions on microwaves theory and techniques[END_REF]:

ψ s (r) = G(r -r ′ )O(r ′ )ψ(r ′ )dr ′ (3)
where ψ(r) = ψ s (r) + ψ i (r) is the total field, ψ i (r) being the illumination field and ψ s (r) the scattered field, and

O(r) = k 2 i n 2 (r) -n 2 0 n 2 0 (4)
is the object function, n being the index of the specimen. The Born approximation assumes that objects are weakly scattering only. Then the diffracted field is small compared to the incident field and equation 3 is rewritten as:

ψ s (r) = G(r -r ′ )O(r ′ )ψ i (r ′ )dr ′ (5)
This equation can be interpreted as follows: Green's function represents an outgoing scalar field scattered by a point object. The field scattered by a point is thus the product of the Green function by the value at that specific point of the object function and the value at that specific point of the incident scalar field. The field scattered by the entire object is obtained by integration on all points of the object, i.e. it is obtained by convolution of Green's function by the product of the object function and the incident scalar field. Adding an ingoing scalar field to Green's function will not modify the detected field since in the far-field only the outgoing field is detected due to the orientation of detectors. It will modify the calculated field scattered by the entire object, but not the part of that field which we detect. This method, which is a scalar equivalent of the vector method used in [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF], allows us to write modified equations between the object function and the field as detected, which involve only homogeneous propagating waves. By adding a suitable ingoing field to the Green function (eq.2), we obtain :

G m (r) = exp(jk i |r|) 4π|r| - exp(-jk i |r|) 4π|r| (6)
the Fourier transform of G m (r) is:

Gm (k d ) = j 4πk i δ(|k d | -k i ), (7)
i.e. it represents a purely monochromatic propagative scalar field, unlike the Green function itself.

Thus in equation ( 5):

-we replace Green's function G(r) by the modified Green function G m (r).

-we consider an incident plane wave ψ i (r) = a i exp(jφ i )exp(jk i .r).

-we obtain a modified scattered field ψ m (r) which replaces the scattered field ψ s (r) and cannot be distinguished from the scattered field by detectors placed in the far-field and looking towards the object. This modified field can be viewed thought of as the detected field. It is also the field which would be reconstructed by a backscattering algorithm since such algorithms assume a monochromatic homogeneous scalar field.

In the Fourier space equation 5 then becomes:

ψm (k d ) = j 4πk i a i exp(jφ i )δ(|k d | -|k i |) Õ(k d -k i ) (8)
From 8, we then obtain the complex amplitude of the scalar field in the Fourier space:

A(k d ) = j 4πk i a i exp(jφ i ) Õ(k d -k i ) (9)
wherein this complex amplitude A(k d ) depends only on the direction of k d and is defined by : ψm

(k d ) = A(k d )δ(|k d | -|k i |) (10)
Equation 9 links the complex amplitude of the measured scattered field to the object function via their Fourier transforms. In the 3D tomographic microscope a detector placed in a Fourier plane detects the complex amplitude A(k d ). This complex amplitude is then corrected for variations of φ i and a i and mapped onto the object function Õ(k o ).

This scalar approximation has been shown to give good results and we will use it in the following. A more elaborate version can be obtained by replacing Green functions by the electric field vector radiated by a dipole [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF].

Also, note that under the first Born approximation, the reconstruction is strictly valid for a low phase change only, which is proportional to both the object index and size, while under the Rytov approximation, "the size of the object is not a factor" (from [START_REF] Slaney | Limitations of Imaging with First-Order Diffraction Tomography IEEE transactions on microwaves theory and techniques[END_REF]) and therefore the Rytov approximation is expected to give better results for larger objects.

We will use the spatial frequencies f d = k d /2π (spatial frequency of the diffracted wave) f i = k i /2π ( spatial frequency of the illumination wave) and f 0 = k 0 /2π (spatial frequency of the object). Using equation ( 9) we find that spatial frequencies are linked by: When considering a transmission geometry and only one direction of illumination, only one half of the Ewald sphere could be recorded at best. Moreover this half sphere is restricted to a cap of sphere only, because of the limited numerical aperture of the microscope objective (NA obj ) used in the detection system.

f d = f i + f o (11)
The wave diffracted by the weakly scattering object is recorded in both amplitude and phase using a holographic recording setup. As shown on figure 1(a), a very limited subset of the diffracted frequencies can be measured. For a given incident direction f i , and an observation direction f d , the corresponding spatial frequency of the object permittivity is f df i . The set of spatial frequencies f df i which are recorded is shown as a bold arc of circle on figure 1(b) (-f i is indicated by the dashed arrow on figure 1(b)). The resulting limited subset of recorded object frequencies will lead to low imaging quality, as will be shown in section 5.

The recorded spatial frequency space can be filled using successive variable directions of illumination, leading to recording a more complete subset of the threedimensional frequency representation of the object, in order to perform a more accurate reconstruction of the object.

In our approach, the detection system and the specimen are fixed, and tomographic acquisition is performed by changing the angle of incidence of the illumination wave. The incident direction is changed but the observation directions are the same. After proper shifting with respect to the incident illumination, other sets of the object frequencies are detected as shown in figure 1(c). Using a tilted illuminating wave permits to record higher frequencies from the Fourier transform of the specimen. It is therefore desirable to use high illumination angles. In our setup, this is performed using a high NA condenser, as in a classical incoherent transmission microscope.

The process depicted on figure 1(a,b,c) in the (f y , f z ) plane is also shown on figure 1(d,e,f) in the (f x , f y ) plane. The different parts of figure 1 give a complete explanation of how the spatial frequencies space is filled in a transmission setups. For example, the upper dashed disk in figure 1(f) corresponds to an illumination at 90 • azimuth angle. It shows up as the upper dashed arc of circle in figure 1(c).

When a large number of incidences is used, the support of the detected frequencies in the (x-y) plane becomes a disk as shown in figure 1(f).

In the (y-z) plane, the Fourier domain scanned by the diffractive tomographic microscope when a large number of incident angles is used takes the form of the well-known butterfly shaped support of the Optical Transfer Function (OTF) for a transmission microscope.

Experimental setup

Figure 2(a) depicts a sketch of the experimental setup of the tomographic diffractive microscope as originally described in [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. It is based on a Mach-Zehnder setup used to realise phase-shifting holography [START_REF] Zhang T Yamaguchi | Phase-shifting digital holography[END_REF] The light issued from a He-Ne laser is separated into a reference wave and an illumination wave by the beam splitter BS1. The reference wave is made plane using lenses L3 and L4. In order to perform phase-shifting holography, its phase is controlled by a mirror mounted on a piezoelectric element. Lenses L1 and L2 are used to illuminate the specimen with a plane illumination wave. The direction of this illumination wave is controlled via the motorized tilting mirror and the condenser: a slight tilt of the mirror generates a large angular variation of the plane wave illuminating the specimen. After diffraction by the specimen, the diffracted wave is collected by the objective. It then passes through lenses LT and L5 and reaches the CCD sensor which is placed at the focal point of the non-diffracted part of the illuminating wave, that is in a Fourier plane.

The reference wave is recombined with the diffracted wave using BS2. The interferences obtained are recorded on the CCD sensor. For each direction of the illumination wave, four images are recorded on the CCD with the phase of the reference wave being shifted by π/2 between each image. A complex image is computed using the formula A = I 0 -I π + j(I π/2 -I 3π/2 ) where A is the complex value, defined for each pixel of the CCD.

However, placing the CCD sensor in a Fourier plane has a major drawback: in such a plane, the non-diffracted part of the illuminating wave generates a very bright spot on the camera. If this bright spot is accurately detected, the diffracted wave itself is usually below the detection threshold of the sensor.

We solved this problem by placing the CCD sensor in an image plane as presented in 2(b). In this case, the non-diffracted part of the illuminating wave generates a continuous background, which does not tend to saturate the sensor. After the phaseshifting step, a numerical two dimensional Fourier transform is performed on the complex image, yielding the same spectrum which would have been directly obtained if the camera had been placed in a Fourier plane.

Note that changing the direction of illumination may be limited by total internal reflection. For living biological specimens, which index of refraction may be approximated to that of water (1.33), using an oil immersion condenser with numerical aperture N.A.=1.4 will lead to total internal reflection for illumination at the glass-toward-water interface. Similarly, the collection of the diffracted wave by the microscope objective would also be limited to N.A.=1.33 (at best). In order to avoid this phenomenon, and as the samples we studied here are non-living diatom skeletons and environment resistant pollens, we actually embedded these specimens into the same immersion oil as the objective or the condenser. Doing so, we could check that the illumination wave (without specimen) is indeed collected by the objective even at maximum illumination angle, which ensures that both the illumination and the dectection are performed at maximal numerical aperture. For 3-D imaging of living cells, using water immersion objectives (with N.A.=1.2) would probably be more appropriate.

Normalisation before mapping of the data

Equation 11 shows that one must properly determine f i in order to remap the Fourier space as described by figure 1. Because the object is weakly scattering, the location of this single frequency corresponds to the location of the maximum of intensity of the diffracted wave. We determine f i by extracting the coordinates f dmax = f i of the maximum of intensity of the diffracted wave (figure 3(a)). The complex amplitude A d (f dmax ) at that point is equal to the complex amplitude of the illumination wave a i exp(jφ i ). Because of mechanical limitations, the illumination wave may be affected by random phase shifts φ i (because mechanical rotation is inevitably accompanied with parasitic translations) as well as possible vibrations. Furthermore laser intensity fluctuations may happen and also the transmission of the illumination and detection optics may vary with the angle of incidence, changing a i . One must compensate for these possible fluctuations. This is simply performed by dividing the amplitude of the diffracted wave by the complex amplitude of the illumination wave (figure 3(b)):

A norm (f d ) = A d (f d )/A d (f dmax ) (12)
In application of equation 11 the normalised complex amplitude is then translated as shown on figure 3(c), which yields the complex amplitude of the object Fourier transform A obj : These normalisations must be repeated for the successive diffracted wavefronts obtained by varying the illumination direction. Note that this step constitutes a vital point in practice for a correct specimen reconstruction. Omitting this correction would lead to randomization of the acquired data that may result in a totally unexploitable image.

A obj (f o ) = A norm (f d -f dmax ) (13)
A d (f d ) A d (f dmax ) A norm (f d ) = A d (f d ) A d (f dmax ) A obj (f o ) = (a) (b) (c)
The successive complex amplitudes are accumulated in the Fourier space, yielding the frequency representation of the object. Some redundant frequencies may appear; in this case, we take the average of the complex amplitudes located at this redundant frequencies, which allows for an important reduction of the noise. Finally, performing a three-dimensional inverse Fourier transform yields the object three-dimensional representation. 4 for a (y-z) and two (x-y) planes. We imaged a Coscinusdiscus sp diatom whose skeleton (or frustule) is made of silica. These data have been acquired with NA obj =1.4 and in the tomography case, with a condenser numerical aperture corresponding to NA cond =1.4.

Results

A comparison between holographic (1 illumination angle) and tomographic reconstructions (1000 angles) is shown on figure

The vertical slicing capabilities of optical tomography are well illustrated by figure 4. In the left column of figure 4, the hologram is disturbed by out of focus parts of the object and thus does not allow for reasonable image quality. In the right column of figure 4, the thin slicing capabilities of optical tomography allows the obtention of a vertical slice (figure 4(b)) and of a middle (figure 4(d)) and top (figure 4(f)). As can be seen on figure (figure 4(f)), the top slice shows only the very top of the diatom and is undisturbed by out of focus information.

Diffractive tomographic microscope under the first Born approximation has the capability to image simultaneously the refractive index and the absorption of a semitransparent object. On figure [START_REF] Zhang T Yamaguchi | Phase-shifting digital holography[END_REF], both refractive index and absorptivity (imaginary part of the complex refractive index) of a snowdrop pollen are presented in a (x-y) plane at different z values. These two informations show different structures inside the pollen (figure 5(a) and(b)). The absorptive part of an organelle (white arrows) is well contrasted on figure 5(d) whereas almost invisible on figure 5(c). On the other hand, the refractive index on figure 5(e) is better contrasted than absorptivity on figure 5(f).

Our computation yields to the object function of equation 4 (or scattering potential). For small n(r) -n 0 this object function is simplified as:

O(r) = 2 k 2 i n 0 (n(r) -n 0 ) (14)
so that the object function is proportional to the complex refractive index, i.e. its real part is proportional to the real part of the refractive index and its imaginary part is proportional to the absorptivity. Nevertheless, it should be noticed that the actual experimental setup has some drawbacks : as seen on figure 4(d), horizontal walls are difficult to image, due to the missing cone along the optical longitudinal axis in the Fourier space.

Discussion

Tomographic microscopy has become the subject of intensive research and a number of alternative techniques have been developed. Different methods have been proposed and used for solving the problem of properly filling a reasonable part of the frequency space, mapping the data into the frequency space, detecting the diffracted wave, and improving the resolution.

A common feature to transmission microscopes is the presence of the so-called missing cone in the frequency space, which strongly limits the resolution along the optical axis. A popular alternative to obtain a more isotropic resolution is to rotate the sample instead of varying the illumination direction [START_REF] Woodford | The synthetic aperture microscope[END_REF][START_REF] Charrière | Living specimen tomography by digital holographic microscopy : morphometry of testae amoeba Optics[END_REF][START_REF] Gorski | Tomographic imaging of photonic crystal fibers Optics[END_REF]. In such a setup, the sample must be prepared within a rotating cylinder, which maybe more difficult. This setup can also be modified for using non-coherent illumination [START_REF] Fauver | Three-dimensional imaging of single isolated cell nuclei using optical projection tomography[END_REF].

Another alternative is to use simultaneous illumination by more than one parallel beam [START_REF] Fercher | Image formation by inversion of scattered field data : experiments and computational simulations[END_REF][START_REF] Mico | Synthetic aperture microscopy using off-axis illumination and polarization coding Optics[END_REF]. However, only a limited number of different illumination directions can be used in this method, the advantage being that they could in principle be used simultaneously, which would improve the acquisition speed.

Diffractive tomography Holography

Varying the wavelength [START_REF] Mico | Synthetic aperture microscopy using off-axis illumination and polarization coding Optics[END_REF] can also improve the imaging capabilities because it allows for the acquisition of different information, as the curvature radius of the recorded cap of spheres changes with the wavelength (a missing-cone does remain). For 2-D imaging only, the lateral resolution is anyway determined by the shortest wavelength, but for 3-D imaging, a more complete subset would be recorded, which would improve the imaging capabilities along the optical axis.

The possibility of simultaneously using reflexion and transmission to improve the frequency support was proposed in [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF][START_REF] Fukutake | Proposal of three-dimensional phase contrast holographic microscopy[END_REF] but has not yet been reduced to practice.

The method used in the present paper for filling the frequency space is to sequentially vary the directions of the illumination beam. It has the advantage of yielding an easy to use instrument without any unusual sample preparation, and yet allowing a high number of illumination to be used and a reasonably large part of the frequency space to be filled. However, each of the above alternatives has specific advantages in certain situations; especially, each method allows for a different filling of the frequency space.

We used phase stepping holography to measure the diffracted field. A possible alternative is to use in-line holography [START_REF] Devaney | The coherent optical tomographic microscope[END_REF] rather than using a separate illumination and reference beam. Doing so, only one hologram is to be recorded and processed in order to measure the diffracted components. This would also help to increase the acquisition speed. However the phase retrieval algorithms which must be used in this case cannot easily recover both the phase and amplitude of the detected diffracted beam. In-line holography can be modified for non-coherent light [START_REF] Noda | Three-dimensional phase-contrast imaging by a computedtomography microscope[END_REF]. In this case, the sample must be assumed to be a purely dephasing one, i.e. non-absorptive. This hypothesis may often be fulfilled in pratice, but must be handled with care: it is for example clearly taken at fault for pollens, as shown by figure [START_REF] Zhang T Yamaguchi | Phase-shifting digital holography[END_REF]. Alternatively, offaxis holography also permits to record the diffracted field with only one hologram [START_REF] Charrière | Living specimen tomography by digital holographic microscopy : morphometry of testae amoeba Optics[END_REF]. In that case, a camera with a larger CCD chip should be used in order to record the same field of view with the same resolution [START_REF] Zhang T Yamaguchi | Phase-shifting digital holography[END_REF].

In this work, the mapping of the frequency space is realised using the first Born approximation, and takes into account the diffraction in order to reach a subwavelength resolution. Such approximation has already been employed in [START_REF] Woodford | The synthetic aperture microscope[END_REF] and [START_REF] Lauer | New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope[END_REF]. In [START_REF] Charrière | Living specimen tomography by digital holographic microscopy : morphometry of testae amoeba Optics[END_REF][START_REF] Gorski | Tomographic imaging of photonic crystal fibers Optics[END_REF][START_REF] Fauver | Three-dimensional imaging of single isolated cell nuclei using optical projection tomography[END_REF]18], mapping is performed using a path integrated phase assumption and thus the reconstruction is realised using backprojection algorithm (Radon transform) and consequently may not be strictly considered as diffraction tomography. In [START_REF] Gorski | Tomographic imaging of photonic crystal fibers Optics[END_REF][START_REF] Vertu | Tomographic observation of transparent objects under coherent illumination and reconstruction by filtered backprojection and Fourier diffraction theorem[END_REF], a comparison between the Radon transform and exact mapping is detailed. In particular, the authors have highlighted possible reconstruction artefacts when using Radon reconstruction. When trying to get the maximum resolution, diffraction has to be taken into account in the reconstruction process.

In order to even further improve the resolution of the images, a priori informations about the observed object can be used for the reconstruction [START_REF] Kawata | Optical microscope tomography I Support constraints[END_REF][START_REF] Belkebir | Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography[END_REF]. Also, non-linear reconstruction methods can improve the quality of the images. These techniques could become a useful complement to the usual, linear version of optical tomography detailed in this paper, but their use still has to be investigated for such a set-up. Note also that the Fourier method we used requires that the Born approximation is satisfied, namely that the sample both is weakly absorbing and induces small phase changes only. The Rytov approximation, which is also linear, would be expected to provide better results for larger objects. Therefore, the method for microtomographic images reconstruction probably has to be tweeked with respect to the considered specimen and experimental goals, trying to detect the tiniest features in a smaller specimen, or trying to image a much larger specimen with a more modest resolution.

An important point in practice for biological application is also the speed of acquisition and reconstruction of the 3-D images. Our set-up is presently slow. For 1000 angles of illumination, 4000 holograms have to be recorded. The piezoelectric element has to physically translate a mirror, and the stepper motors driven-, tip-tilt mirror used for controlling the illumination must both ensure vibrationless motions. We therefore are restricted to slow movements, and have introduced temporisations to wait for vibration damping. As a consequence, 4000 holograms take 40 minutes to be recorded. This obviously is incompatible with live-cell imaging, but is tolerable for our specimens, or even for fixed cells.

Replacing these elements by high-speed systems may permit to overcome these limitations as demonstrated by Choi et al. [18], who recently presented a high speed version of our set-up. The reconstruction method they used being however based on an inverse Radon transform, the accuracy of the reconstruction is lower, as shown for example by Gorski and Olsten [START_REF] Gorski | Tomographic imaging of photonic crystal fibers Optics[END_REF]. We now plan to improve the speed of acquisition of our set-up, by replacing the speed limiting mechanical elements by faster systems, as for example galvanometric mirrors instead of a stepper motor driven mirror for illumination control, an electro-optic modulator instead of a piezo mirror for phase stepping, and a high-speed camera.

The reconstruction is fast: for a 32 bits 512 voxel complex final image, the 1000 2-D transforms, the 3-D mapping of the data and the final 3-D Fourier transform take less than 5 minutes on a 2.6Ghz dual-core PC using 64 bits Linux, and fast 3-D FFT transforms [START_REF] Frigo | The Design and Implementation of FFTW3[END_REF]. Note that most of the computational time is spent on reading the data on the hard-disk and not in the Fourier transforms themselves. Presently, we perform all the computations after completing the acquisitions. Alternatively, one could also perform all the 2-D computations and 3-D mapping progressively with the acquisitions, so that only the final 3-D Fourier transform would remain, which would also permit to save time.

Note that our method is comparable to the above described alternate methods in terms of reconstruction speed, as the involved mathematical operations are the same for all methods. However, rotating the specimen for tomography presents the drawback that inevitable parasitic mechanical translations of the sample oblige to a precise numerical compensation before mapping the information in the 3-D Fourier space [START_REF] Vertu | Tomographic observation of transparent objects under coherent illumination and reconstruction by filtered backprojection and Fourier diffraction theorem[END_REF]. While these parasitic movements also happens when tilting the mirror in our set-up, their corrections is very easy, as explained in section 4 (Fig. 4). This probably explains that results already presented by various authors using specimen rotation do not present optimal resolution, because then an inverse Radon transformation seems easier. Also, biologist often tend to prefer preparing their sample using more classical glass slide with cover glass method instead of having to insert it within a microcapillary.

Conclusions

Tomographic microscopy allows for quantitative three-dimensional imaging with optimized resolution. Whereas fluorescence microscopy has experienced a strong evolution with the generalized use of confocal fluorescence microscopy, there has been no corresponding evolution in transmission or reflexion microscopy. Use of confocal microscopes in transmission has always proven difficult and did not lend itself to a generalized use. We hope that tomographic microscopy will bring to transmission microscopy the improvement, which it needs to keep pace with confocal fluorescence imaging. The tomographic microscopy method presented in this paper should allow high-resolution and high-speed three-dimensional observation of unstained samples. Furthermore, it may allow confocal images to be superimposed to quantitative threedimensional images, instead of the more usual DIC contrast images.

Figure 1 .

 1 Figure 1. Principle of diffractive tomographic microscopy: construction of the detected frequency support in the case of a transmission microscope.

Figure 2 .

 2 Figure 2. Experimental setup of the transmission diffractive tomographic microscope. In 2(a) the CCD plane is localised in a Fourier plane, whereas in figure 2(b) it is localised in an image plane.

A

  norm (f df dmax )

Figure 3 .

 3 Figure 3. Normalisation for mapping of the Fourier frequencies

Figure 4 .Figure 5 .

 45 Figure 4. Holography vs tomography. On the left side, three slices extracted from the holographic reconstruction are shown : figure (4a) shows a (x-z) plane and figures 4(c,e) a (x-y) plane. On the right side, the same object is shown in the tomographic case (1000 angles): figure 4(b) shows a (x-z) plane and figure 4(d,f) show a (x-y) plane corresponding to figure 4(c,e). Scale bar represents 5 µm.

Acknowledgements

We gratefully acknowledge Dominique Voisin for the diatoms preparation and Joël Lambert for valuable help in the design and construction of the microscope.


Qihang Shu

Qiang Hu

Jun Zheng
email: junzheng@seu.edu.cn

Nathalie Mitton

A Dependable Slepian-Wolf Coding Based Clustering Algorithm for Data Aggregation in Wireless Sensor Networks

Keywords: Slepian-Wolf coding, dependable clustering, data aggregation, wireless sensor network

This paper considers the Slepian-Wolf coding based data aggregation problem and the corresponding dependable clustering problem in wireless sensor networks (WSNs). A dependable Slepian-Wolf coding based clustering (D-SWC) algorithm is proposed to provide dependable clustering against cluster-head failures. The proposed D-SWC algorithm attempts to elect a primary cluster head and a backup cluster head for each cluster member during clustering so that once a failure occurs to the primary cluster head the cluster members within the failed cluster can promptly switchover to the backup cluster head and thus recover the connectivity of the failed cluster to the data sink without waiting for the next-round clustering to be performed. Simulation results show that the D-SWC algorithm can effectively increase the amount of data transmitted to the data sink as compared with an existing nondependable clustering algorithm for Slepian-Wolf coding based data aggregation in WSNs.

I. INTRODUCTION

In a wireless sensor network (WSN), a number of sensor nodes are densely deployed in an area of interest with a data sink deployed within or near the area [START_REF] Akyildiz | Wireless sensor networks: A survey[END_REF] . The data observed by the sensor nodes usually have a certain spatial correlation and the degree of the spatial correlation increases as the distance between sensor nodes decreases [START_REF] Vuran | Spatio-temporal correlation: Theory and applications for wireless sensor networks[END_REF] . The spatial correlation can lead to considerable data redundancy in the network and transmitting redundant data would cause unnecessary energy consumption. To increase energy efficiency and prolong network lifetime, it is desirable to perform in-network data aggregation to remove the data redundancy in the network.

Slepian-Wolf coding [START_REF] Slepian | Noiseless coding of correlated information sources[END_REF][START_REF] Cover | Elements of Information Theory[END_REF] is a distributed source coding (DSC), which can remove data redundancy without communication between sensor nodes. To take advantages of the properties of Slepian-Wolf coding, sensor nodes must know the correlation structure of the entire network a priori and the correlation structure of the network depends on the characteristics of the observed phenomenon and the distances between the sensor nodes in the network [START_REF] Cover | Elements of Information Theory[END_REF] . However, applying Slepian-Wolf coding to a large network in a global manner is usually difficult because in that case every sensor node needs to know the correlation structure of the whole network, which would incur much additional control overhead.

An effective way to address the above issue is to introduce node clustering in a WSN to improve energy efficiency and prolong network lifetime [START_REF] Zheng | Wireless Sensor Networks: A Networking Perspective[END_REF] . This technique can help reduce the energy consumption of sensor nodes by allowing cluster members in a cluster to send data to their associated cluster head at a short distance and the cluster head to transmit locally aggregated data to the data sink at a long distance. Due to this fact, node clustering can be effectively combined with Slepian-Wolf coding to increase the energy efficiency in data aggregation. Using clustering, a network is configured into a certain number of clusters. Each cluster contains a smaller number of neighbor sensor nodes. In this case, each cluster node only needs to know the local correlation structure of the cluster when applying Slepian-Wolf coding to a large network, which is more feasible.

One concern that arises in node clustering is the clusterhead failure problem caused by physical damage or external attack, which would result in the loss of connectivity of all affected cluster members in a failed cluster, and disrupt the data communication within the cluster and even the normal operation of the entire network. To address this problem, it is desirable to find an efficient way to recover the connectivity of the affected cluster nodes in the event of a cluster-head failure. A traditional way to this end is to re-cluster the sensor nodes once a failure occurs [START_REF] Mhatre | Homogeneous vs heterogeneous clustered networks: A comparative study[END_REF][START_REF] Heinzelman | An application-specific protocol architecture for wireless microsensor networks[END_REF] , but such re-clustering is usually time-insensitive, and would disrupt the normal data communication of the affected sensor nodes during reclustering and lead to incomplete data transmission in the network.

In this paper, we consider Slepian-Wolf coding based data aggregation problem and the corresponding dependable clustering problem in WSNs. A dependable Slepian-Wolf coding based clustering (D-SWC) algorithm is proposed to provide survivable clustering against cluster-head failures, which elects a couple of cluster heads for each cluster member: a primary head and a backup head. Once a failure occurs to the primary cluster head, the cluster members within the failed cluster can promptly switchover to the backup cluster head and thus recover the connectivity of the failed cluster to the data sink without waiting for the next-round clustering to be performed. Simulation results show that the D-SWC algorithm can effectively increase the amount of data transmitted to the data sink as compared with an existing nondependable clustering algorithm for Slepian-Wolf coding based data aggregation in WSNs,

The rest of the paper is organized as follow. Section II describes the dependable clustering problem considered in this paper. Section III presents the proposed D-SWC algorithm. In Section IV evaluates the performance of the D-SWC algorithm through simulation results. Section V concludes this paper.

II. PROBLEM STATEMENT

This section first introduces the concept of Slepian-Wolf coding, and then describes the dependable clustering problem and review related work.

A. Slepian-Wolf Coding

Assume that a network has N sensor nodes uniformly distributed in an area of interest and each node i produces reading i X . All the readings forms a set of jointly ergodic sources, denoted by ) (

2 1 N X , , X , X … = X
. These readings follow a distribution of ) , , , (

2 1 N x x x p …
, which corresponds to the spatial correlation structure known by each node a priori. According to the Slepian-Wolf Theorem [START_REF] Slepian | Noiseless coding of correlated information sources[END_REF] , the sensor nodes can jointly encode their data without inter-node communication with an overall rate (in bits) lower-bounded by their joint entropy ) , , , (

X X X H … provided that their respective rates meet the following condition:

)] ( ) ( [ ) ( c | H R G X G X G ≥ for all } 2 1 { N , , , ∈ G , where } 2 1 { N , , , is a set of the indices of sensor nodes in the network, G is a set of sensor nodes, G c is the complementary set of G, ) ( X H
is the entropy of X, and

∑ ∈ = G G i i R R ) ( , } { ) ( G G X ∈ = j , X j .
According to chain theory [START_REF] Vuran | Spatio-temporal correlation: Theory and applications for wireless sensor networks[END_REF] , for any order of N nodes, there always exists a rate allocation

N i i R 1 } {
= which makes the total number of bits generated by all nodes equal to the value of their joint entropy, i.e., ) , , , (

2 1 1 N N i i X X X H R = ∑ = , where ) X ( H R 1 1 = ) X , , X , X / X ( H R i i i i 1 2 - 1 - = , N i ≤ ≤ 2
Therefore, all sensor nodes in a cluster, say cluster A, can encode their data with ) (

2 1 | | X , , X , X H
A bits using Slepian-Wolf coding without inter-node communication, and there always exists an optimal rate allocation which can achieve this local maximum compression gain.

B. Clustered Slepian-Wolf Coding

Assume that a network consists of a set of sensor nodes V. Each sensor node is initially a cluster-head candidate. The diameter of each candidate is equal, and all other nodes within the diameter can become its cluster members. The nodes within the cluster diameter of candidate v constitutes a finite point set v Ω with the cardinality of

| | v Ω , which is called the neighbor set of candidate v. ) ( v Ω P
denotes the power set of v Ω , which is a set whose elements are the subsets of v Ω , and it constitutes all possible combinations of nodes in v Ω .

Therefore, the cardinality of

) ( v Ω P is v Ω 2 .
Since a candidate v associated with each combination of nodes (cluster members) within its cluster diameter (e.g., a set of nodes v Δ , where

) ( v v Ω P Δ ∈
), can form a unique potential cluster (e.g., } {v

: v ∪ Δ A =
), it can generate up to v Ω 2 potential clusters [START_REF] Heinzelman | An application-specific protocol architecture for wireless microsensor networks[END_REF] . Further, since every node is initially a candidate, the network consists of Under the above assumptions, the clustered Slepian-Wolf coding problem is to select a set of disjoint potential clusters * C from the cluster set S to cover the whole network such that the global compression gain of Slepian-Wolf coding is maximized, or the total rate (bits) of the encoded data generated by all the clusters in the network is minimized [START_REF] Heinzelman | An application-specific protocol architecture for wireless microsensor networks[END_REF] , i.e.,

∑ ∈ ⊆ = C A S C A C )) ( ( min arg X H * , where , * V A C A = ∈ ∪ , * φ = ∈ A C A ∩ } { ) ( A A ∈ = j , X X j ,
and C denotes a set of disjoint clusters from the cluster set S to cover the whole network.

C. Dependable Clustering

We consider the dependable clustering problem, i.e., construct a cluster hierarchy with each cluster having two different cluster heads: a primary head and a backup head. This problem is similar to the domatic partition problem in graph theory [START_REF] Feige | Approximating the Domatic Number[END_REF] , where a domatic partition is a partition of vertices in which each part is a dominating set. With the domatic partition, each vertex in the graph is either in the dominating set or has a neighbor in the set. In a clustered network, each sensor node is either a cluster head or a cluster member. Therefore, all cluster heads in the network actually constructs a dominating set and the domatic partition can generate several different dominating sets. Correspondingly, a solution to the domatic partition problem can also partition the network such that each cluster member can be covered by several different sets of cluster heads. The domatic partition problem is a well-known NP-complete problem. A simple way to solve the problem is to greedily select small dominating sets and iteratively remove the selected dominating sets from the graph until the remainder is no longer dominating. However, the greedy algorithm aims to find multiple backup cluster heads, while in this work we only consider one primary cluster head and one backup cluster head for solving the dependable clustering problem.

D. Related Work

Slepian-Wolf coding based data aggregation has been studied in the literature [START_REF] Cristescu | Networked slepianwolf: theory, algorithms, and scaling laws[END_REF][START_REF] Cristescu | On network correlated data gathering[END_REF][START_REF] Marco | Reliability vs. efficiency in distributed source coding for field-gathering sensor networks[END_REF][START_REF] Wang | Distributed data aggregation using clustered Slepian-Wolf coding in wireless sensor networks[END_REF] . In [9-10], global Slepian-Wolf coding and shortest-path routing are studied with an aim to minimize the total cost for sending compressed data. The results show that applying Slepian-Wolf coding globally is difficult. However, this work does not consider distributed clustering with Slepian-Wolf coding. In [11], Marco and Neuhoff studied the application of Slepian-Wolf coding locally within each cluster and showed its advantage in overcoming the effect of node failures on the data reconstruction at the remote sink. However, this work does not propose any clustering protocol to construct a cluster hierarchy. In [12], the clustered Slepian-Wolf coding problem is studied and a distributed optimal compression clustering protocol (DOC 2 ) is proposed to solve the problem. On the other hand, most existing clustering protocols for WSNs do not consider correlation structure and thus cannot take advantage of Slepian-Wolf coding to increase the efficiency of data aggregation.

In the context of dependable clustering, there is not much related work reported in the literature. In [6-7], Mhatre and Heinzelman et al. introduced periodic re-clustering to address the problem. However, this would disrupt the normal communication in the failed cluster. In [13], Younis et al. proposed a robust energy-efficient distributed clustering (REED) for WSNs, which can construct a robust clustered hierarchy by selecting k independent sets of cluster heads. In [14], Gupta and Younis proposed a fault-tolerant clustering algorithm to perform re-clustering locally once most cluster heads reach a consensus about the occurrence of a clusterhead failure. In [15], Wang et al. proposed a dependable clustering protocol to solve the cluster-head failure problem in underwater sensor networks (UWSNs). However, these clustering algorithms do not consider Slepian-Wolf coding for data aggregation.

III. DEPENDABLE SLEPIAN-WOLF CODEING BASED CLUSTERING ALGORITHM

In this section, we present the proposed dependable Slepian-Wolf coding based clustering (D-SWC) algorithm for data aggregation in WSNs.

A. Overview

The D-SWC algorithm is based on the greedy algorithm proposed in [16] for solving the domatic partition problem and the distributed optimal-compression clustering protocol (DOC 2 ) for maximizing the amount of transmitted data in [12]. The objective of the D-SWC algorithm is to elect a primary cluster head and a backup cluster head for each cluster member during clustering, taking into account Slepian-Wolf coding. Compared with the greedy algorithm, the D-SWC algorithm has two major differences: 1) The greedy algorithm must ensure that multiple different sets of cluster heads be found so that each cluster member can be associated with multiple backup cluster heads. This makes it difficult to consider energy efficiency in clustering. For the D-SWC algorithm, however, it only needs to elect one primary cluster head and one backup cluster head, which makes it possible to consider energy efficiency in clustering.

2) The greedy algorithm must find a new dominating set or a set of backup cluster heads that cover not only the remaining cluster members but also the primary cluster heads. For the D-SWC algorithm, the elected backup cluster heads only need to cover the remaining cluster members because they become cluster heads only when the primary cluster heads fail.

B. Procedures

The D-SWC algorithm can be divided into three phases: initialization, clustering, and finalization.

a) Initialization phase

In the initialization stage, each node has three possible states: cluster head (or head), cluster member (or memb) and cluster-head candidate (or cand). Initially, each node performs neighbor discovery to discover its one-hop neighbors and maintain an uncovered neighbor set, which contains its onehop neighbors still in the cand state. A candidate can potentially generate a number of different clusters by combining different nodes in its uncovered neighbor set. If a candidate v has an uncovered neighbor set v U , the number of potential clusters the candidate can generate is

| | v U 2 .
Once the uncovered neighbor set of a candidate is determined, the candidate will select a cluster with the minimum average joint entropy as its qualified cluster from all the potential clusters that can be generated from the uncovered neighbor set.

For example, consider a cluster-head candidate v with a neighbor set v Ω . The power set of

v Ω , ) ( v Ω P
, constitutes all possible combinations of nodes in v Ω . Assume that v M is a cluster member set of candidate v. Our objective is to select a potential cluster with the minimum average entropy as the representative cluster of v, which is denoted by v A and satisfies the following conditions, i.e., ( )

( ({ } )) arg min { } |{ } | v v v v P v H X v v ⊆ ← ∪ ∪ M Ω M M M { } v v v ← ∪ A M * arg min ( ( )) v v H X ⊆ ∈ = ∑ C S A C C A where ) } ({ v v H M ∪
represents the joint entropy of the cluster, and

| } { | / ) } ({ v v v v H M M ∪ ∪
is considered as the average entropy of the cluster .

b) Clustering phase

In the clustering phase, each candidate v first performs the following procedures: 1) Construct its own potential clusters by comparing every combination of the neighbors in its one-hop uncovered neighbor set; 2) Calculate the average entropy of each potential cluster based on the distances between the nodes within the cluster; 3) Select the representative cluster with the maximum global compression gain among all the potential clusters; 4) Send the average entropy of its qualified cluster to all candidates within its 2-hop range.

After that, each candidate compares its own entropy with the average entropy received from other candidates within its 2-hop range. If the candidate itself has the minimum average entropy, it will become a cluster head. In this case, it will invite all the nodes in its representative cluster to join its cluster by an INVITE message. If a candidate receives an INVITE message destined to it, it will perform the following procedures: 1) Change its status from a cluster-head candidate to a cluster member; 2) Extract the cluster-head ID from the INVITE message; 3) Record the node sending the INVITE message as the primary cluster head and add the node's ID into its clusterhead (CH) list; 4) Broadcast a JOIN message to all the nodes within its cluster diameter to acknowledge the receipt of the INVITE message and to notify the other candidates within the cluster diameter that it has become a cluster member of some cluster head. If a candidate does not receive any INVITE message, it will stay in the candidate status and reselect its representative cluster because at this time some neighbors in its uncovered neighbor set may have become cluster heads or cluster members of other cluster heads. Even if a node has become a cluster member, it still needs to receive the INVITE messages and add the senders' IDs extracted from the messages into its cluster-head list. The above procedures are performed by all candidates until they all become either a cluster head or a cluster member. At the end of the clustering phase, there will be no candidate in the network and the cluster-head list of every cluster member will contain not only a primary cluster head, but also one or more backup cluster heads.

c) Finalization phase

In the clustering phase, the cluster-head list generated may contain one or more backup cluster heads except the primary cluster head. In this case, the one with the shortest distance to a cluster member is selected as the backup cluster head for that cluster member in the finalization phase. where K denotes the determinant of matrix K [START_REF] Cover | Elements of Information Theory[END_REF] . For matrix K, we use an exponential model of the covariance ( ) [10] to model its elements, where ij d represents the distance between the nodes i X and j X , and θ indicates the extent of correlation. For the sake of simplicity and without loss of generality, we apply the differential entropy to replace the discrete entropy because the differential entropy differs from the discrete entropy by a constant [10][15] . Figure 2 shows the total amount of data received by the data sink with D-SWC and DOC2 , respectively, within 10 simulation rounds. To investigate the impact of a cluster-head failure on the performance, four cluster-head failures are generated in each round, and the failed cluster heads are randomly selected from the primary cluster heads. It is seen that D-SWC transmits more data to the sink than DOC 2 during the 10 simulation rounds. This is because in the event of a cluster-head failure, D-SWC allows the cluster members within a failed cluster to transmit their data to a backup cluster head, while with DOC 2 the nodes have to wait until the next-round clustering is completed to transmit their data. Figure 3 shows the average residual energy of sensor nodes in the network with the D-SWC algorithm and the DOC 2 algorithm, respectively. It is seen that with the D-SWC algorithm the nodes consume a little more energy than that with the DOC 2 algorithm during the simulations. This is because with the DOC 2 algorithm the sensor nodes within a failed cluster do not to transmit data until the next-round reclustering is completed, while with the D-SWC algorithm the nodes can continue to transmit data to the backup cluster head, thus consuming more energy in the same simulation time. It should be pointed out that the consumed more energy is used for data transmission, not a waste.

2 exp ij ij k dθ = - in

V. CONCLUSIONS

In this paper, we proposed a dependable Slepian-Wolf coding based clustering (D-SWC) algorithm to solve the dependable clustering problem in Slepian-Wolf coding based data aggregation for WSNs. The D-SWC algorithm attempts to elect a primary cluster head and a backup cluster head for each cluster member during clustering so that once a failure occurs to the primary cluster head, the cluster members within the failed cluster can promptly switchover to the backup cluster head and thus recover the connectivity of the failed cluster to the data sink without waiting for the next-round clustering to be performed. The simulation results have shown that the D-SWC algorithm can effectively increase the amount of data transmitted to the data sink as compared with the DOC 2 algorithm. In future work, we will study the clusterhead switchover problem in the event of a cluster-head failure for supporting the D-SWC algorithm.

Fig. 1

 1 Fig.1The pseudo codes of the D-SWC algorithm.

Fig. 2

 2 Fig.2 Total amount of data received by the data sink.

Fig. 3

 3 Fig.3 Average residual energy of sensor nodes in the network.

This work was supported by the National Natural Science Foundation of China under Grant No. 61071115, the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20110092110007, and the Research Fund of National Mobile Communications Research Laboratory, Southeast University, China, under Grant No. 2012A02.

Figure 1 gives the pseudo codes of the D-SWC algorithm, the notations used in the pseudo codes are explained as follows:

v Ω : a finite point set within one hop of candidate v.

IV. SIMULATION RESULTS

In this section, we evaluate the performance of the proposed D-SWC algorithm through simulation results. For evaluation, we compare the D-SWC algorithm and the DOC 2 algorithm in [12] and the performance metrics include the amount of data transmitted to the data sink and the average residual energy of sensor nodes in the network.

In the simulation experiments, we used a network with 100 sensor nodes uniformly deployed in a region of 100m×100m. The cluster diameter is 15m. For the correlation structure, we use an approximation algorithm to calculate the amount of the joint entropy generated by a cluster with N nodes [4][10] . We assume that the observations 1 2 , , , N X X X … generated by N senor nodes can be modeled as an Ndimensional jointly Gaussian model with mean ( ) 0, 0, , 0 … and covariance matrix K. The density of X is ( ) ( )

and the differential entropy of ( )