AND Rafik Aguech

Wissem Jedidi

COMPLETE MONOTONICITY AND BERNSTEIN PROPERTIES OF FUNCTIONS ARE CHARACTERIZED BY THEIR RESTRICTION ON N 0

Keywords: Completely monotone functions, completely monotone sequences, Bernstein functions, completely alternating functions, completely alternating sequences, Hausdorff moment problem, Hausdorff moment sequences, selfdecomposability

We give several new characterizations of completely monotone functions (CM) and Bernstein functions (BF) via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on N 0 . We give a complete answer to the following question: Can we affirm that a function f is completely monotone (resp. a Bernstein function) if we know that the sequence (f (k)) k is completely monotone (resp. alternating)? This approach constitutes a kind of converse of Hausdorff's moment characterization theorem in the context of completely monotone sequences.

INTRODUCTION

Traditionally, completely monotone functions (CM) are recognized as Laplace transforms of positive measures and Bernstein functions (BF) are their positive antiderivatives. The literature devoted to these two classes of functions is impressive since they have remarkable applications in various branches, for instance, they play a role in potential theory, probability theory, physics, numerical and asymptotic analysis, and combinatorics. A detailed collection of the most important properties of completely monotone functions can be found in the monograph of Widder [START_REF] Widder | The Laplace Transform[END_REF] and for Bernstein functions, the reader is referred to the elegant manuscript of Schilling, Song and Vondraček [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF]. Hausdorff's moment characterization theorem [START_REF] Hausdorff | Moment probleme für ein endliches Intervall[END_REF] is explained in details, and also in the context of measures on commutative semigroup in the Book of Berg, Christensen and Ressel [START_REF] Berg | Harmonic Analysis on Semigroups, Theory of Positive Definite and Related Functions[END_REF]. The references [START_REF] Berg | Harmonic Analysis on Semigroups, Theory of Positive Definite and Related Functions[END_REF] and [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] were a major support in the elaboration of this paper and constitute for us a real source of inspiration.

Theorem 2 below, is borrowed from [START_REF] Berg | Harmonic Analysis on Semigroups, Theory of Positive Definite and Related Functions[END_REF] and gives the complete characterization of completely monotone (respectively alternating) sequences: a sequence (a k ) k is interpolated by a function f in CM (respectively BF) if and only if (a k ) k completely monotone (respectively alternating) sequence and minimal (see Definition 2 for minimality). Completely monotone sequences are also known as the Hausdorff moment sequences. In this spirit, a natural question prevailed, what about the converse? i.e:

Can we affirm that a function f belongs to CM (respectively BF) if we know that the sequence (f (k)) k is completely monotone (respectively alternating)? In other terms, could a completely monotone (respectively alternating) and minimal sequence (a k ) k be interpolated by a regular enough function f , which is not in CM (respectively BF)?

Date: November 8, 2015. [START_REF] Alili | On exponential functionals, harmonic potential measures and undershoots of subordinators[END_REF] We prove that under natural regularity assumptions on f , the answer is affirmative for the first question (and then infirmative for the second) and this constitutes a kind of converse of Hausdorff's moment characterization theorem [START_REF] Hausdorff | Moment probleme für ein endliches Intervall[END_REF]. Mai, Schenk and Scherer [START_REF] Mai | Two novel characterizations of self-decomposability on the half-line[END_REF] adapted a Widder's result [START_REF] Widder | The Laplace Transform[END_REF] and used a specific technique from Copula theory in order to state, in their Lemma 3.1 and Theorem 1.1, that:

(i) a continuous function f with f (0) = 1 belongs to CM if and only if the sequence (f (xk)) k is completely monotone for every x ∈ Q ∩ [0, ∞);

(ii) a continuous function f with f (0) = 0 belongs to BF and is self-decomposable if and only if the sequence (f (xk) -f (yk)) k is completely alternating for every x > y > 0. (See Section 8 below for the definition of self-decomposable Bernstein functions).

The idea of this paper was born when we wanted to remove the dependence on x in characterizations (i) and (ii) and to study general non bounded completely monotone functions and general Bernstein functions. Our answer to the question is given in Theorems 4 and 5 below that say:

(iii) a bounded function f belongs to CM if and only if it has an holomorphic extension on Re(z) > 0 which remains bounded there and the sequence f (xk) k≥0 is completely monotone and minimal for some (and hence for all) x > 0. If f is unbounded, then a shifting condition is necessary;

(iv) a bounded function f is a Bernstein function if and only if it has an holomorphic extension on Re(z) > 0, and the sequence f (xk) k≥0 is completely alternating and minimal for some (and hence for all) x > 0. If f is unbounded, then a boundedness condition on the increments is necessary.

For each of Theorems 4 and 5 we shall give two proofs based on two different approaches, the first one uses Blaschke's theorem on the zeros of a function on the open unit disk and the second one is based on a Greogory-Newton expansion of holomorphic functions (see Section 6 below for the last two concepts). We emphasize that these two approaches require some boundedness (especially in the completely monotone case). In Corollary 4.2 of Gnedin and Pitman [START_REF] Gnedin | Moments of convex distribution functions and completely alternating sequences[END_REF] the necessity part of (iv) above is stated without the holomorphy and minimality condition, their formulation is equivalent to Theorem 2 below. We discovered the idea of our second proof (for the Bernstein property context) hidden in the remark right after their corollary. The authors surmise that the sufficiency part of (iv) could be proved by Gregory-Newton expansion of Bernstein functions and we will show that their idea works. Since we are studying general, non necessarily bounded functions in CM and in BF, there was a price to pay in order to avoid these kind of restrictive conditions. For this purpose, we develop in Section 3 and 4 there several algebraic tools, based on the scale, shift and difference operators, giving new characterizations for the CM and BF classes. We did our best to remove redundant assumptions of regularity (such as continuity or differentiability or boundedness or global dependence on parameters) in the our sufficiency conditions. This kind of redundancy often appears, because the classes CM and BF are very rich in information. These tools, that we find intrinsically useful, can also be considered as a major contribution in this work. They were also crucial in the proofs of the results given in Section 5. Throughout this paper, we give different proofs, whenever it is possible, and when the approaches were clearly distinct.

The paper is organized as follows. Section 2 gives the basic setting and definitions. In Section 3 and 4, we recall classical characterizations of complete monotonicity and alternation for functions and sequences, we develop several other characterizations and we discuss the concept of minimal sequences. Section 6 is devoted to specific pre-requisite for the proofs of the main results. We recall there and adapt some results around functional iterative equations and asymptotic of differences of functions. We also adapt some results stemming from complex analysis and from interpolation theory. Section 7 is devoted to the proofs and Section 8 gives an alternative characterization for self-decomposable Bernstein functions to point point (ii) above, in the spirit of point (iv) above.

BASIC NOTATIONS AND DEFINITIONS

Throughout this paper, N 0 denotes the set of non-negative integers and N = N 0 {0}. A sequence (a k ) k∈N 0 is seen as a function a : N 0 → R so that a(k) = a k . The symbols ∧ and ∨ denote respectively the min and the max. All the considered functions are measurable, the measures are positive, Radon with support contained in [0, ∞). For functions f : D ⊂ C → C, the scaling, the shift and the difference operators acting on them are respectively denoted, whenever these are well defined, by

σ c f (x) := f (cx), σ = σ 1 = Identity, τ c f (x) := f (x + c), τ = τ 1 , ∆ c f (x) := f (x + c) -f (x), ∆ = ∆ 1 , θ c f (x) := f (c) -f (0) + f (x) -f (x + c), θ = θ 1 ,
and their iterates are given by σ

0 c f = τ 0 c f = ∆ 0 c f = θ 0 c f = f and for every n ∈ N, σ n c = τ c n , τ n c = τ cn , ∆ n c f = ∆ c (∆ n-1 c f ), θ n c f = (-1) n ∆ n c f -∆ n c f (0) , so that for every n ∈ N 0 , ∆ n c f (x) = n i=0 n i (-1) n-i f (x + ic) (1)
θ n c f = n i=0 n i (-1) i f (x + ic) -f (ic) . Definition 1 (Berg [3] p. 130). Let D = (0, ∞) or [0, ∞) or N 0 . A function f : D → f (D) is called completely monotone on D, and we denote f ∈ CM(D), if f (D) ⊂ [0, ∞) (respectively completely alternating if f (D) ⊂ R and we denote f ∈ CA(D) ), if for all finite sets {c 1 , • • • , c n } ⊂ D and x ∈ D , we have (-1) n ∆ c 1 • • • ∆ cn f (x) ≥ 0 (respectively ≤ 0). Remark 1. (i) Every function f in CM(D) (respectively CA(D)
) is non-increasing (respectively nondecreasing). We will see later on that f is necessarily decreasing (respectively increasing) when it is not a constant. (iv) By linearity of the difference operators, the classes CM(D) and CA(D) are convex cones.

CLASSICAL CHARACTERIZATIONS OF COMPLETELY MONOTONE AND ALTERNATING

Ψ(λ) = [0,∞) e -λx ν(dx ), λ > 0 (respectively λ ≥ 0); (2)
(c) Ψ is infinitely differentiable on (0, ∞) (respectively continuous on [0, ∞), infinitely differentiable on (0, ∞)) and satisfies (-1) n Ψ (n) ≥ 0 for every n ∈ N 0 .

The measure ν in (2) will be referred in the sequel as the representative measure of Ψ.

Remark 2. (i) Every function Ψ ∈ CM(0, ∞) such that Ψ(0+) exists, naturally extends to a continuous bounded function in CM[0, ∞), this is the reason why we identify, throughout this paper, such functions Ψ with their extension on [0, ∞).

(ii) By Corollary 1.6 p. 5 in [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF], the closure of CM[0, ∞) (with respect to pointwise convergence) is CM[0, ∞). This insures that Ψ ∈ CM(0, ∞) if and only if τ cn Ψ ∈ CM[0, ∞) for some positive sequence c n tending to zero or equivalently τ c f ∈ CM(0, ∞) for every c > 0. It is also immediate that Ψ ∈ CM(0, ∞) if and only if σ c f ∈ CM(0, ∞) for some (and hence for all) c > 0.

(iii) It is not clear at all to see that functions in CM(0, ∞) are actually infinitely differentiable just using Definition 1. The latter is clarified by point (b) of Theorem 1. Furthermore, Dubourdieu [START_REF] Dubourdieu | Sur un theoreme de M. S. Bernstein relatif ala transformation de Laplace-Stieltjes[END_REF] pointed out that strict inequality prevails in point (c) of for all non-constant completely monotone functions, for these and their derivatives are then strictly monotone.

We start with a taste of what we can obtain as algebraic characterization. The following proposition has to be compared with the Remark 1 (ii): Proposition 1. (a) A function Ψ : (0, ∞) → [0, ∞) belongs to CM(0, ∞) if and only if for some (and hence for all) c > 0 the function -∆ c Ψ belongs to CM(0, ∞) and the Laplace representative measure in (2) of -∆ c Ψ gives no mass to zero.

(b) In this case, the sequence of functions (-∆ nc )Ψ converges pointwise, locally uniformly, to a function in CM(0, ∞) that does not depend on c, more precisely

Ψ(λ) = lim x→∞ Ψ(x) + lim n→∞ (-∆ nc )Ψ(λ), λ > 0.
The same holds for the successive derivatives of (-∆ nc )Ψ.

3.2.

Completely alternating functions and Bernstein functions. The well known class BF of Bernstein functions consists of those functions Φ : (0, ∞) → [0, ∞), infinitely differentiable on (0, ∞) and satisfy (-1) n-1 Φ (n) (λ) ≥ 0, for every λ > 0 and n ∈ N. In other terms, Φ is a Bernstein function if it is non-negative, infinitely differentiable and Φ ∈ CM(0, ∞). It is also known (see Theorem 3.2 p. 21 [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] for instance) that any function Φ ∈ BF admits a continuous extension on [0, ∞), still denoted Φ, and represented by

Φ(λ) = q + dλ + (0,∞) (1 -e -λx )µ(dx ), λ ≥ 0 , (3)
where q, d ≥ 0 and the so-called Lévy measure µ satisfies the integrability condition

(0,∞) (1 -e -x ) µ(dx ) < ∞ which is equivalent to (0,∞) (1 ∧ x) µ(dx ) < ∞.
An integration by parts gives

(0,∞) e -λx µ (x, ∞) dx = Φ(λ) -q λ -d, λ > 0, so that q = Φ(0), d = lim x→∞ Φ(x)
x and the relation between Φ and the triplet (q, d, µ) becomes one-to-one.

The following proposition unveils the link between completely alternating functions and Bernstein functions:

Proposition 2. 1) The class of Bernstein functions coincides with the class of non-negative and completely alternating functions on [0, ∞).

2) The class of completely alternating functions on (0, ∞) is given by

CA(0, ∞) = {f : (0, ∞) → R, differentiable, s.t. f ∈ CM(0, ∞)}.
In particular, if g ∈ CM(0, ∞), then -g ∈ CA(0, ∞).

It is clear that the subclass BF b of bounded Bernstein function is given by

BF b = {Φ ∈ BF, s.t. lim λ→∞ Φ(λ) < ∞} = {Φ ∈ BF, s.t. Φ(λ) = q + (0,∞) (1 -e -xλ ) µ(dx ), with q ≥ 0, µ (0, ∞) < ∞} and that Φ ∈ BF b if and only if Φ ≥ 0 and Φ(∞) -Φ ∈ CM[0, ∞). (4) We denote BF 0 b = {Φ ∈ BF b , s.t. Φ(0) = 0} = {Φ ∈ BF, s.t. Φ(λ) = (0,∞) (1 -e -xλ ) µ(dx ), with µ (0, ∞) < ∞}.
We also have the following equivalences Φ ∈ BF ⇐⇒ Φ ≥ 0 and σ c Φ ∈ BF for some (and hence for all) c > 0 (5) ⇐⇒ λ → Φ(λ + c) -Φ(c) ∈ BF, for every c > 0.

Equivalence ( 5) immediate and ( 6) is justified as follows: by differentiation get Φ (.+c) ∈ CM[0, ∞), for all c > 0 and closure of the class CM(0, ∞) (Corollary 1.6 p.5 [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF]) insures that Φ ∈ CM(0, ∞).

A natural question is to ask whether (6) remains true if expressed with a single fixed c > 0. The answer is negative because for every Φ 0 ∈ BF, the function Φ(λ) = Φ 0 (|λ -c|), λ ≥ 0, is not in BF despite that λ → Φ(λ + c) -Φ(c) ∈ BF. A closed transformation is studied in Corollary 3.8 (vii) p. 28 in [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF] which says that Φ ∈ BF yields θ c Φ ∈ BF for every c > 0. We propose the following improvement:

Proposition 3. (a) A function Φ : [0, ∞) -→ [0, ∞) belongs to BF if and only if for some (and hence for all) c > 0,

λ → θ c Φ(λ) = Φ(c) -Φ(0) + Φ(λ) -Φ(λ + c) ∈ BF 0 b . (b)
In this case, the sequence of functions θ nc Φ converges pointwise, locally uniformly, to a function in BF, null in zero, that does not depend on c. More precisely

Φ(λ) = Φ(0) + λ lim x→∞ Φ(x) x + lim n→∞ θ nc Φ(λ), λ ≥ 0.
The same holds for the successive derivatives of θ nc Φ.

Remark 3. By (4), point (a) is also equivalent to λ → Φ(λ + c) -Φ(λ) ∈ CM[0, ∞), for some (and hence for all) c > 0.

CLASSICAL CHARACTERIZATION OF COMPLETELY MONOTONE AND ALTERNATING SEQUENCES AND ADDITIONAL RESULTS

A characterization of completely monotone (respectively alternating) sequences, closely related to Hausdorff moment characterization theorem [START_REF] Hausdorff | Moment probleme für ein endliches Intervall[END_REF], could be found in the monograph of Berg et al. [START_REF] Berg | Harmonic Analysis on Semigroups, Theory of Positive Definite and Related Functions[END_REF]: Theorem 2. [3, Propositions 6.11 and 6.12 p. 134] Let a = (a k ) k≥0 a positive sequence. Then, the following conditions are equivalent: (a) the sequence a is completely monotone (respectively alternating); (b) for all k ∈ N 0 , n ∈ N 0 (respectively n ≥ 1), we have

(-1) n ∆ n a(k) ≥ 0 (respectively ≤ 0); (7)
(c) there exists a positive Radon measure ν on [0, 1] (respectively q ∈ R, d ≥ 0 and a positive Radon measure µ on [0, 1)) such that we have the representation

a 0 = ν([0, 1]), a k = (0,1] u k ν(du), k ≥ 1 ( 8
)
respectively a 0 = q, a k = q + d k + [0,1) (1 -u k ) µ(du), k ≥ 1 . (9)
4.1. Comments on CM(N 0 ) and CA(N 0 ).

Comment 1:

In the completely monotone case, the measure ν in ( 8) is not only Radon but also finite because of the convention a 0 = ν ([0, 1]). In the completely alternating case, we have that a 0 = q and the measure µ in ( 9) is only Radon, satisfying the integrability condition [0,1) (1 -u) µ(du) < ∞. By the dominated convergence theorem, we retrieve d = lim k→∞ (a k /k) . Furthermore, in both cases, ν (respectively (q, d, µ)) uniquely determine the sequence (a k ) k≥0 , which is justified as follows:

1-In the completely monotone case: use Fubini argument, get that the exponential generating function of the sequence (a k ) k≥0 is the Laplace transform of ν,

k≥0 a k (-t) k k! = [0,1]
e -tu ν(du), t ≥ 0, and finally conclude with the injectivity of the Laplace transform.

2-In the completely alternating case: making an integration by parts, write

a k -q -dk -µ ({0}) = k 1 0 u k-1 µ (0, u] du, k ≥ 1,
then by a Fubini argument, get that the exponential generating function of the sequence (a k ) k≥0 leads to a Bernstein function build with the triplet (q, d, µ):

h(t) : = k≥0 a k t k k! , t ≥ 0 = (q + d t)e t + µ ({0}) (e t -1) + t 1 0 e tu µ (0, u] du = (q + d t)e t + µ ({0}) (e t -1) + (0,1) (e t -e tv ) µ(dv ) e -t h(t) = q + d t + µ ({0}) (1 -e -t ) + (0,1) (1 -e -tw ) µ(dw ) ( 10
)
where µ is the image of the measure µ obtained by the change of variable w = 1 -v, and finally conclude with the unicity through the Bernstein representation in equality [START_REF] Lorch | On the composition of completely monotonic functions and completely monotonic sequences and related questions[END_REF].

Comment 2: Completely monotone sequences are always positive, whereas a completely alternating sequence is non-negative if and only if the corresponding q-value in ( 9) is non-negative (see [START_REF] Athavale | Bernstein functions, complete hyperexpansivity and subnormality -I[END_REF]).

4.2.

The classes CM * (N 0 ) and CA * (N 0 ) of minimal completely monotone and alternating sequences. A lot of care is required if one modifies some terms of a completely monotone or alternating sequence. We clarify, with our own approach, the following fact we have found in [START_REF] Lorch | A remark on completely monotonic sequences, with an application to summability[END_REF] and [START_REF] Lorch | On the composition of completely monotonic functions and completely monotonic sequences and related questions[END_REF], and extend it to completely alternating sequences: strict inequality prevails throughout [START_REF] Flajolet | Mellin transforms and asymptotics: finite differences and rice's integrals[END_REF] for a completely monotone sequence unless

a 1 = a 2 = • • • = a n = • • • , that is,
unless all terms except possibly its first are identical. We can state that A sequence a = (a k ) k≥0 in CM(N 0 ) (respectively in CA(N 0 )) ceases to strictly alternate, in differences, at a certain rank if and only if the sequence a is constant (respectively if and only if if the sequence a is affine).

Our argument uses the explicit computation (1) of the quantities (-1) n ∆ n a(k), n ∈ N, k ∈ N 0 , which does not seem to be fully exploited in the literature we encountered:

(-1) n ∆ n a(k) = µ({0}) 1 l k=0 + (0,1) u k (1 -u) n ν(du) if a ∈ CM(N 0 ) -µ({0}) 1 l k=0 -(0,1) u k (1 -u) n µ(du) if a ∈ CA(N 0 ).
Let α = ν or µ. Based on the fact that (0,1) u k (1 -u) n α(du) = 0, for some n ∈ N and k ∈ N 0 , if and only if α (0, 1) = 0, then an elementary reasoning shows that

(-1) n ∆ n a(k) = 0 for some n ∈ N, k ∈ N 0 ⇐⇒ a m = µ({1}), ∀m ≥ 1 if a ∈ CM(N 0 ) a m = q + µ({0}) + d m, ∀m ≥ 1 if a ∈ CA(N 0 ).
As an example, fix > 0 and consider the completely monotone (respectively alternating) sequence

b 0 = , b k = 0, k ≥ 1 (respectively b 0 = 0, b k = , k ≥ 1). It satisfies: (-1) n ∆ n b(k) = 1 l k=0 (respectively -1 l k=0 ), n ∈ N, k ∈ N 0 .
By linearity of the operators (-1) n ∆ n , we obviously have

(-1) n ∆ n (a -b)(k) = (µ({0}) -) 1 l k=0 + (0,1) u k (1 -u) n ν(du) if a ∈ CM(N 0 ) -µ({0}) 1 l k=0 -(0,1) u k (1 -u) n µ(du) if a ∈ CA(N 0 ).
Since ν is finite (respectively µ integrates 1 -u), then the dominated convergence theorem ensures that (i) a is called minimal and we denote a ∈ CM * (N 0 ) (respectively a ∈ CA * (N 0 )) if the sequence

lim n→∞ (0,1) (1 -u) n ν(du) = lim n→∞ (0,1) (1 -u) n µ(du) = 0, so that the quantities (-1) n ∆ n (a -b)(0)
{a 0 -, a 1 , • • • , a k , • • • } (respectively {a 0 , a 1 -, • • • , a k -, • • • })
is not completely monotone (respectively alternating) for any positive . (ii) Equivalently, a is minimal if and only if the measure ν in (8) (respectively µ in ( 9)) has no point mass at zero.

Example 1. The sequence a = ((k + 1) -1 ) k≥0 ceases to be completely monotone if a 0 = 1 is replaced by a 0 = 1 -, since

(-1) n ∆ n a(0) = 1 n + 1 -, n ∈ N 0 .
The analogous constatation holds for the completely alternating sequence (1 -(k + 1) -1 ) k≥0 accordingly to Definition 2.

After the above comments and considerations on minimal sequences, Theorem 2 could be specified as follows: taking ν and µ obtained as the image of the measures ν and µ on (0, ∞) in ( 8) and ( 9 (b) Equivalently, a sequence (a k ) k≥0 belongs to CM * (N 0 ) (respectively belongs to CA * (N 0 ) and positive) if and only if there exist a unique finite measure ν on (0, ∞) (respectively a unique triplet (q, d, µ) where q, d ≥ 0 and the measure µ satisfying (0,∞) (1 ∧ u) µ(du) < ∞), such that:

a k = [0,∞) e -ku ν(du) resp. a k = q + dk + (0,∞) (1 -e -ku ) µ(du) , k ≥ 0. (11)
It is clear that the subclass CM * (N 0 ) and the subclass of positive sequences in CA * (N 0 ) are convex cones.

LINKING FUNCTIONS AND SEQUENCES OF THE COMPLETELY AND ALTERNATING TYPE

In the spirit of Theorems 2 and 3, a natural question is to ask whether the completely monotone (respectively Bernstein) character of function f is entirely recognized via its associated sequence (f (k)) k . This constitutes a kind converse of Hausdorff's moment characterization theorem [START_REF] Hausdorff | Moment probleme für ein endliches Intervall[END_REF] which is formulated in Theorem 2 or 3. A complete answer is given in the following two subsections. (ii) the sequence τ n Ψ(k) k≥0 = Ψ( n + k) k≥0 completely monotone and minimal.

Corollary 2. Two completely monotone functions on (0, ∞) coincide on the set of positive integers starting from a certain rank if and only if they are equal. If one of them extends to [0, ∞), then so does the other and they coincide on [0, ∞).

5.2.

Complete monotonicity property of functions is recognized by their restriction on a lattices of the form α n N 0 , where α n → 0. The following two results characterize complete monotonicity of functions only in terms of minimal completely monotone sequences, i.e. condition (a) in Theorem 4 and Corollary 1 would be self contained.

Proposition 4. A function Ψ : [0, ∞) → [0, ∞) belongs to CM[0, ∞)
if and only if it is continuous and for some (and hence for all) sequence (α n ) n≥0 of positive numbers tending to zero, there corresponds a sequence (Ψ n ) n≥0 in CM[0, ∞) such that the following representation holds each for each n ∈ N 0 :

Ψ(α n k) = Ψ n (k), for all k ∈ N 0 i.e. Ψ α n k) k≥0 ∈ CM * (N 0 ) .
For non-bounded completely monotone functions on (0, ∞) an analogous statement is given, but we require a minor correction consisting on shifting the function on the right of zero: Corollary 3. For a function Ψ : (0, ∞) → [0, ∞), the following conditions are equivalent:

(a) Ψ belongs to CM(0, ∞); (b) Ψ is continuous and to every sequence (r n ) n≥0 of positive rational numbers tending to zero, there corresponds a sequence (Ψ n ) n≥0 in CM[0, ∞), such that following representation holds for each n ∈ N 0 :

Ψ(r n (k + 1)) = Ψ n (k), for all k ∈ N 0 i.e. Ψ r n (k + 1)) k≥0 ∈ CM * (N 0 ) ;
(c) Ψ is continuous and there exists a sequence (Ψ n ) n>0 in CM[0, ∞), such that the following representation holds for each n ∈ N:

Ψ( k + 1 n ) = Ψ n (k), for all k ∈ N 0 i.e. Ψ k + 1 n ) k≥0 ∈ CM * (N 0 ) .
Remark 4. (i) By continuity, it is not difficult to see that assertions in Proposition 4 (respectively Corollary 3) are also equivalent to the following:

Ψ is continuous and the sequence Ψ(xk) k≥0 (respectively

Ψ x(k + 1) k≥0 ) be- longs to CM * (N 0 ) for every x ∈ (0, ∞) or for every x ∈ Q ∩ (0, ∞),
The latter is precisely what is stated in Lemma 3.1 in [START_REF] Mai | Two novel characterizations of self-decomposability on the half-line[END_REF] in case Ψ(0) = 1, the minimality condition was somehow occulted.

(ii) The reader could notice that Theorem 4 requires a supplementary assumption of holomorphy and of boundedness compared to Proposition 4 and Corollary 3. The point is that Theorem 4 gives more information since for every function Ψ satisfying condition (a) therein, we have

Ψ ∈ CM(0, ∞) ⇐⇒ σ x Ψ ∈ CM(0, ∞), for some x ∈ (0, ∞) ⇐⇒ Ψ x(k + 1) k≥0 ∈ CM * (N 0 ), for some x ∈ (0, ∞). (12)
The same holds for Ψ ∈ CM[0, ∞) under the additional condition of finiteness of Ψ(0+). The condition of minimality and holomorphy appear to be the lowest price to pay in order to have the condition [START_REF] Nörlund | Leçons sur les Series d'Interpolation[END_REF] expressed for a single x instead of all x. Since every Bernstein functions Φ satisfies λ → Φ(λ)/λ ∈ CM(0, ∞), we immediately deduce from Corollary 2 the following: Corollary 4. Two Bernstein functions coincide on the set of non-negative integers starting from a certain rank if and only if they are equal on [0, ∞).

5.4.

Bernstein property of functions is recognized by their restriction on lattices of the form α n N 0 , where α → 0. As for completely monotone functions, the following two results characterize Bernstein property of functions only in terms of minimal completely alternating sequences, i.e. condition (a) in Theorem 5 would be self contained.

Proposition 5. A function Φ : [0, ∞) -→ [0, ∞) belongs to BF 0
b if and only if it is continuous and for some (and hence for all) sequence (α n ) n≥0 of positive numbers tending to zero, there corresponds a sequence (Φ n ) n≥0 in BF 0 b , such that the following representation holds for each n ∈ N 0 :

Φ(α n k) = Φ n (k), for all k ∈ N 0 i.e. Φ α n k) k≥0 ∈ CA * (N 0 ) .
Corollary 5. For a function Φ : [0, ∞) -→ [0, ∞), the following conditions are equivalent:

(a) Φ belongs to BF ; (b) Φ is continuous and to every sequence (r n ) n≥0 of positive rational numbers tending to zero, there corresponds a sequence (Φ n ) n≥0 in BF, such that the following representation holds for each n ∈ N 0 :

Φ(r n k) = Φ n (k), for all k ∈ N 0 i.e. Φ r n k) k≥0 ∈ CA * (N 0 ) ;
(c) Φ is continuous and there exists a sequence (Φ n ) n>0 in BF, such that the following representation holds for each n ∈ N:

Φ( k n ) = Φ n (k), for all k ∈ N 0 i.e. Φ k n ) k≥0 ∈ CA * (N 0 ) .
Remark 5. As in Remark 4, we can notice the following:

(i) By continuity, assertions in Corollary 5 (respectively Proposition 5) are equivalent to the following assertion:

Φ is continuous and the sequence Φ(xk) k≥0 belongs to ∈ CA * (N 0 ) (respectively belongs to ∈ CA * (N 0 ) and is bounded) for every x ∈ (0, ∞) or for every x ∈ Q + .

(ii) Theorem 5 requires a supplementary assumption of holomorphy and of sub-affinity compared to Proposition 5 and Corollary 5. Theorem 5 gives more information since for every function Φ satisfying condition (a) therein, we have

Φ ∈ BF ⇐⇒ σ x Φ ∈ BF for some x ∈ (0, ∞) ⇐⇒ Φ(xk) k≥0 belongs to
∈ CA * (N 0 ) (respectively belongs to ∈ CA * (N 0 ) and is bounded) for some x ∈ (0, ∞).

SOME PRE-REQUISITE

The following results are crucial in order to conduct our proofs.

6.1. On iterative functional equations and asymptotic of differences. We first present a result of Webster [START_REF] Webster | Log-convex solutions to the functional equation f (x + 1) = g(x)f (x): Γ-type functions[END_REF] which will be used in the proofs of Propositions 1 and 2. Given a log-concave function g : [0, ∞) → [0, ∞), he considered the iterative functional equation f (x + 1) = g(x)f (x), x > 0, and f (1) = 1.

(

) 13
Motivated by the study of generalized gamma functions and their characterization by a Bohr-Mollerup-Artin type theorem, Webster studied equations of type [START_REF] Hausdorff | Moment probleme für ein endliches Intervall[END_REF]. A combination of Theorems 4.1 and 4.2 [START_REF] Webster | Log-convex solutions to the functional equation f (x + 1) = g(x)f (x): Γ-type functions[END_REF] gives results that were stated in [START_REF] Alili | On exponential functionals, harmonic potential measures and undershoots of subordinators[END_REF] under this form: Theorem 6. [Webster, [START_REF] Webster | Log-convex solutions to the functional equation f (x + 1) = g(x)f (x): Γ-type functions[END_REF]] Assume that g : [0, ∞) → [0, ∞) is log-concave and lim a→∞ g(x+a)

g(a) = 1 for every a > 0. For n ≥ 1, let a n = (g -(n) + g + (n))/2g(n) and γ g = lim n→∞ ( n 1 a j -log g(n)). Then, there exists a unique log-convex solution f : [0, ∞) → [0, ∞) to the functional equation ( 13) satisfying f (1) = 1 and given by

f (x) = e -γgx g(x) ∞ n=1 g(n) g(n + x) e anx , x > 0. ( 14
)
If furthermore lim a→∞ g(x) = 1, then the representation simplifies to

f (x) = 1 g(x) ∞ n=1 g(n) g(n + x) , x > 0. (15)
Theorem 1.1.8 p. 5 [START_REF] Bingham | Regular Variation[END_REF] says that if l : R → R is additive (i.e. l(x + y) = l(x) + l(y), ∀x, y ∈ R), and measurable, then l(x) = Cx for some C ∈ R. On the other hand, consider a function l : [0, ∞) → [0, ∞) solution of the iterative equation l(x + 1) = l(x) + l(1), x ∈ (0, ∞).

Take g(x) = e l(1) and f (x) = e l(x)-l (1) in Theorem 6. Clearly, a n = 0 and γ g = -l(1) and ( 14) yields that the unique convex solution is given by l(x) = l(1) x, x ≥ 0. It would be desiderate to have a similar conclusion without the convexity assumption. Karamata's characterization theorem for regularly varying functions (Theorem 1.4.1 p.17 in [START_REF] Bingham | Regular Variation[END_REF]), says that if lim x→∞ h(λ + x) -h(x) = l(λ), then there exists a real number ρ such that lim x→∞ h(λ + x) -h(x) = ρλ for every λ ≥ 0. We propose the following lemma as an improvement of Karamata's characterization: Lemma 1. Suppose two function h, l : [0, ∞) → [0, ∞) are linked for every λ ≥ 0 by the limit

h(λ + n) -h(n) → l(λ), as n → ∞ and n ∈ N.
Then, necessarily l(λ) = λl(1) with l(1) ≥ 0 and

h(λ + x) -h(x) → l(λ), as x → ∞, uniformly in each compact λ-set in [0, ∞) (16)
Proof. The proof goes through the following four steps: a) For every λ ≥ 0, write that

l(λ+1) = lim n→∞ [h(λ+1+n)-h(n)] = lim n→∞ [h(λ+1+n)-h(n+1)]+ lim n→∞ [h(n+1)-h(n)] = l(λ)+l(1)
and retrieve that

l(λ + m) = l(λ) + l(m) = l(λ) + l(1) m, for every λ ≥ 0, m ∈ N 0 . (17)
Since h(n + 1) -h(n) converges to l(1), then, so does its Cesàro mean

l(1) = lim n→∞ 1 n n-1 i=0 [h(i + 1) -h(i)] = lim n→∞ h(n) -h(0) n = lim n→∞ h(n) n ,
and deduce that l(1) ≥ 0.

b) Case where l ≡ 0 (i.e. l ≡ 0): Assume that a function k

: [0, ∞) → [0, ∞) satisfies lim n→∞, n∈N k(λ + n) -k(n) → 0.
Reproduce identically the first proof of Theorem 1.2.1 p. 6 [START_REF] Bingham | Regular Variation[END_REF] (by taking with their notations x = n ∈ N) in order to get k(λ

+ n) -k(n) → 0 uniformly in each compact λ-set in (0, ∞)
k(λ + x) -k(x) = sup λ∈[a,b] k(λ + {x} + [x]) -k({x} + [x]) ≤ sup u∈[a,b+1] k(u + [x]) -k([x]) + sup u∈[0,1] k(u + [x]) -k([x]) goes to zero as [x] → ∞. Finally, get k(λ + x) -k(x) → 0, as x → ∞, uniformly in each compact λ-set in [0, ∞). (18)
c) Case where l ≡ / 0: Taking k(x) = h(x) -l(x) and using [START_REF] Steutel | Infinite divisibility of probability distributions on the real line[END_REF], obtain for every λ > 0

k(λ + n) -k(n) = h(λ + n) -l(λ + n) -h(n) + l(n) = h(λ + n) -h(n) -l(λ) → 0.
as n → ∞. By step b) deduce that k satisfies [START_REF] Webster | Log-convex solutions to the functional equation f (x + 1) = g(x)f (x): Γ-type functions[END_REF]. d) Taking h(x) = log f (e x ) with f as in Theorem 1.4.1 p. 17 [START_REF] Bingham | Regular Variation[END_REF], conclude that necessarily the function l is linear, i.e. l(λ) = l(1)λ.

6.2. On Blaschke's characterization theorem. The second result, due to Blaschke, allows to identify holomorphic functions given their restriction along suitable sequences:

Theorem 7 (Blaschke, Corollary p. 312 in Rudin [START_REF] Rudin | Real And Complex Analysis[END_REF]). If f is holomorphic and bounded on the open unit disc Remark 6. Corollary 6 will be used essentially in the proofs of Theorems 4 and 5 for checking the equality between two functions coinciding along the sequence of positive integers. We are totally aware that Theorems 4 and 5 could be rephrased in a more general setting with different sequences. For clarity's sake, we preferred to state our results there under their current form.

D, if α 1 , α 2 , α 3 , • • • are the zeros of f in D and if ∞ i=1 (1 -|α i |) = ∞, then f (z) = 0 for all z ∈ D.
6.3. On Gregory-Newton development. In the alternative proofs of Theorems 4 and 5, we will also need the concept of Gregory-Newton development that we recall here:

Definition 3.
A function f defined on some domain D of the complex plane is said to admit a Gregory-Newton development if there exists some sequence (a k ) k≥0 such that

f (z) = ∞ k=0 (-1) k a k k! z k , z ∈ D,
where

z 0 = 1 and z k = z(z -1) • • • (z -k + 1) = 1, k ≥ 1.
Remark 7. (i) Notice that the factorial powers z n and the usual powers z k are related through the relations

z n = n k=0 n k (-1) n-k z k and z n = n k=0 n k z k ,
where n k and n k are the Stirling numbers of the first and second kind respectively. These relations allow to swap between Gregory-Newton and power series developments whenever it is possible. This clarifies why a Gregory-Newton development for a holomorphic function is unique.

(ii) For functions f admitting a Gregory-Newton development, Nörlund ([12] p. 103), showed that necessarily

a k = (-1) k ∆ k f (0), k ≥ 0.
(iii) It is worth noting that the transformation

f (l) l=0,•••m → (-1) n ∆ n f (0) n=0,•••m
is the classical binomial transform which is involutive. Since the operators τ and ∆ commute, and so do their iterates, it is immediate that the transformation

f (k + l) l=0,•••m → (-1) n ∆ n f (k) n=0,•••m is also involutive for every fixed k ∈ N 0 . The transformation f (l) l=0,•••m → ∆ n f (0) n=0,•••m is called the Euler transform.
It is not an involution but remains one-to-one (see [START_REF] Flajolet | Mellin transforms and asymptotics: finite differences and rice's integrals[END_REF]). It is now clear that the sequence ∆ k f (0) k≥0 is one-to-one with the sequence

(f (k)) k≥0 . ( 19
)
It is trivial that any function f : D ⊂ C → C could be represented by an interpolating polynomial P n of a degree n ≥ 1, plus a remainder function R n :

f = P n + R n , where P n (z) = n k=0 ∆ k f (0) k! z k .
The following result clarifies when the remainder function goes to zero, i.e. when f could be expanded in a unique way (see point (i) in Remark 7) into a Gregory-Newton series given by

f (z) = ∞ k=0 ∆ k f (0) k! z k . ( 20
)
Theorem 8 (Nörlund,[START_REF] Nörlund | Leçons sur les Series d'Interpolation[END_REF] p. 148). In order that a function f admits a Gregory-Newton development (20), it is necessary and sufficient that f is holomorphic in a certain half-plane Re(z) > α and f is of the exponential type, i.e.

f (z) ≤ Ce D|z| , ( 21
)
where C and D are fixed positive numbers.

As an application, we propose the following: Proposition 6. 1) Every bounded completely monotone function Ψ admits an extension which (i) is bounded, continuous on the half plane Re(z) ≥ 0 and holomorphic on Re(z) > 0;

(ii) is expandable into a Gregory-Newton series on the half plane Re(z) > 0.

2) Every Bernstein function Φ admits an extension which (i) is continuous on the half plane Re(z) ≥ 0 and holomorphic on Re(z) > 0;

(ii) satisfies for some C, D ≥ 0

|Φ(z) -Φ(z )| ≤ C + D |z -z | for every z, z s.t. Re(z) ≥ Re(z ) ≥ 0;
(ii) is expandable into a Gregory-Newton series on the half plane Re(z) > 0.

Proof. 1) Assertion (i) is due to Corollary 9.12 p. 67 [START_REF] Berg | Potential Theory on Locally Compact Abelian Groups[END_REF]. Boundedness of the extension of Ψ insures that Nörlund's condition ( 21) is satisfied and then (ii) is true.

2) Assertion (i) is due to 9.14 p. 68 [START_REF] Berg | Potential Theory on Locally Compact Abelian Groups[END_REF] or to Proposition 3.6 p. 25 [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF], so that the representation (3) extends on Re(z) ≥ 0

Φ(z) = q + d z + (0,∞) (1 -e -zx )µ(dx ).
For 2)(ii), we reproduce some steps of the Proposition 3.6 p. 25 [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF], we observe that for every x ≥ 0 and z, z ∈ C such that Re(z) ≥ Re(z ) ≥ 0, we have

|e -zx -e -z x | ≤ |1 -e -(z-z )x | ≤ 2 ∧ |(z -z )x| ≤ (2 ∨ |z -z |)(1 ∧ x) ≤ (2 + |z -z |)(1 ∧ x).
We deduce

|Φ(z) -Φ(z )| ≤ d |z -z | + (0,∞) |e -zx -e -z x |µ(dx ) ≤ d|z -z | + (2 + |z -z |) (0,∞) (1 ∧ x)µ(dx ) = C + D |z -z | ≤ (C ∨ D) e D|z-z | .
where

C = 2 (0,∞) (1 ∧ x)µ(dx ) and D = d + (0,∞) (1 ∧ x)µ(dx ).
(λ) = (0,∞) e -λx µ Ψ (dx ), λ > 0, then h(λ) := Ψ(λ) -Ψ(λ + c) = (0,∞) e -λx (1 -e -cx ) µ Ψ (dx )
since the measure µ h (dx ) := (1 -e -cx ) µ Ψ (dx ) gives no mass to zero.

For the sufficiency part, take c > 0 and consider the iterative functional equation

Ψ(λ)-Ψ(λ+c) = h(λ) with h ∈ CM(0, ∞) represented by h(λ) = (0,∞) e -λx µ h (dx ).
We would like to show that Ψ ∈ CM(0, ∞), or equivalently (by Remark 2 (ii)) that σ c Ψ ∈ CM(0, ∞). This is the reason why it is sufficient to show that the solution of the iterative functional equation

Ψ(λ) -Ψ(λ + 1) = h(λ)
belongs to CM(0, ∞), i.e. to check things with c = 1. For this purpose, we apply Theorem 6 with the log-concave function g(λ) = e -h(λ) , λ > 0 satisfying lim λ→∞ g(λ) = 1 and f (λ) = e Ψ(λ)-Ψ (1) , λ > 0. We obtain the representation:

Ψ(λ) -Ψ(1) = h(λ) - ∞ n=1 (0,∞) e -nx (1 -e -λx )µ h (dx ) = (0,∞) (e -λx -e -x )µ h (dx ),
which insures that Ψ is differentiable with -Ψ ∈ CM(0, ∞). Because Ψ is non-negative, we conclude that Ψ ∈ CM(0, ∞).

Statement (b) could be extracted from the second proof that follows.

Second proof of the sufficiency part of Proposition 1. Fix c > 0 and write for every n ∈ N and λ > 0,

(-∆ nc )Ψ(λ) = Ψ(λ) -Ψ(λ + nc) = n-1 i=0 Ψ(λ + ic) -Ψ(λ + (i + 1)c) = n-1 i=0 (-∆ c )Ψ(λ + ic)
Obviously, the sequence n → (-∆ nc )Ψ(λ) is increasing for every λ, c > 0, then x → Ψ(x) is decreasing and then converging, since non-negative. We denote Ψ(∞) := lim x→∞ Ψ(x). The function λ → (-∆ nc )Ψ(λ) belongs to CM(0, ∞) and, by Corollary 1.7 p. 6 in [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF], the limiting function (-∆ ∞,c )Ψ := lim n→∞ (-∆ nc )Ψ also belongs to CM(0, ∞), the convergence holds locally uniformly and also for the derivatives. This limit does not depend on c > since it satisfies:

Ψ(λ) = Ψ(∞) + (-∆ ∞,c )Ψ(λ), λ > 0.
Proof of Proposition 2. 1) If Φ ∈ BF is represented by (3), then for every c > 0,

λ → ∆ c Φ(λ) = Φ(λ + c) -Φ(λ) = dc + (0,∞) e -λx (1 -e -cx )µ(dx ), λ ≥ 0,
is non-negative and belongs to CM[0, ∞). By Remark 1 (iii) we deduce that Φ ∈ CA[0, ∞).

Conversely, assume Φ ∈ CA[0, ∞) and non-negative, we will show that Φ is differentiable and that Φ in completely monotone on (0, ∞) which is equivalent to Φ ∈ BF. Remark 1 (iii) and definiteness of Φ in zero yield to

λ → ∆ c Φ(λ) = Φ(λ + c) -Φ(λ) ∈ CM[0, ∞), ∀c > 0.
Inspired by the proof of Proposition 1, we will see, that ∆Φ ∈ CM(0, ∞) (i.e. when taking c = 1) is sufficient for proving that Φ is differentiable and that Φ ∈ CM(0, ∞). Indeed, assume ∆Φ is the Laplace representation µ

∆Φ(λ) = [0,∞) e -λx µ(dx ).
Theorem 6 insures that f (λ) = e Φ(1)-Φ(λ) is the unique solution of the iterative functional equation f (λ + 1) = f (λ)g(λ) and Φ(1) -Φ(λ) has the following representation for every λ > 0:

Φ(1) -Φ(λ) = ∆Φ(λ) - ∞ n=1 ∆Φ(n) -∆Φ(n + λ) = [0,∞) e -λx µ(dx ) - ∞ n=1 (0,∞) (e -nx -e -(n+λ)x ) µ(dx ) = µ({0}) + (0,∞) e -λx µ(dx ) - (0,∞) (1 -e -λx ) e -x 1 -e -x µ(dx ) = µ({0}) + (0,∞) e -x -e -λx
1 -e -x µ(dx ).

Then, for every a > 0, λ → Φ(λ + a) -Φ(a) = (0,∞) (1 -e -λx ) e -ax 1-e -x µ(dx ) is a Bernstein function which is equivalent, by [START_REF] Dubourdieu | Sur un theoreme de M. S. Bernstein relatif ala transformation de Laplace-Stieltjes[END_REF] to Φ ∈ BF.

2) The proof is conducted identically by dropping the positivity condition on Φ.

Proof of Proposition 3. If Φ ∈ BF is represented by (3) and if c > 0, then λ → θ c Φ(λ) = Φ(c) -Φ(0) + Φ(λ) -Φ(λ + c) = (0,∞) (1 -e -λx )(1 -e -cx )µ(dx ).
We deduce that θ c Φ ∈ BF 0 b since (1 -e -cx )µ(dx ) is a measure with finite total mass equal to

Φ(c) -Φ(0) + d c . Conversely, assume λ → θ c Φ(λ) = Φ(c) -Φ(0) -Φ(λ + c) -Φ(λ)] ∈ BF 0
b . The latter is equivalent by (4) to λ → Φ(λ + c) -Φ(λ)] ∈ CM[0, ∞) and we conclude as in the proof of Proposition 2.

Statement (b) could be extracted from the second proof that follows.

Second proof of of the sufficiency part of Proposition 3. Because of the invariance [START_REF] Bingham | Regular Variation[END_REF], it is enough to prove the Proportion in case where c = 1. Since θΦ belongs to BF 0 b , then it is represented with a finite measure µ on (0, ∞) by

θ c Φ(λ) = (0,∞) (1 -e -λx ) µ c (dx ), λ ≥ 0.
We will see that the latter is sufficient to show that φ is differentiable on (0, ∞) and that Φ belongs to CM(0, ∞). First notice that for every n ∈ N 0 and λ ≥ 0,

θ nc Φ(λ) = Φ(nc) -Φ(0) -Φ(λ + nc) -Φ(λ) = n-1 k=0 Φ((k + 1)c) -Φ(kc) -Φ(λ + (k + 1)c) -Φ(λ + kc) = n-1 k=0 Φ(c) -Φ(0) -Φ(λ + (k + 1)c) -Φ(λ + kc) -Φ(c) -Φ(0) -Φ((k + 1)c) -Φ(kc) = n-1 k=0 θ c Φ(λ + kc) -θ c Φ(kc) = n-1 k=0 (0,∞) (1 -e -λx ) e -kcx µ c (dx ) = (0,∞) (1 -e -λx )
1 -e -ncx 1 -e -cx µ c (dx ).

By Corollary 3.9 p. 29 [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF], the sequence θ nc Φ converges locally uniformly, and all its derivatives to a Bernstein function θ ∞,c Φ given by

λ → θ ∞,c Φ(λ) = (0,∞)
1 -e -λx 1 -e -x µ c (dx ) ∈ BF.

We have also showed that for every λ ≥ 0, Φ(nc

) -Φ(λ + nc) → θ ∞,c Φ(λ) + Φ(0) -Φ(λ), when n → ∞.
On the other hand, by ( 16), we get that for every λ ≥ 0

lim x→∞ Φ(λ + x) -Φ(x) = d c λ, for some d c ≥ 0
and we deduce that,

Φ(λ) = Φ(0) + d c λ + θ ∞,c Φ(λ), λ ≥ 0.
Unicity of the triplet of characteristics in the representation (3) of Bernstein functions allows to conclude that d c = lim x→∞ Φ(x)/x and θ ∞,c , both do not depend on c.

Proof of Theorem 4. For the necessity part, use Proposition 6 for (a) and Theorem 3 for (b). For the sufficiency part, use Theorem 3 again which asserts that there is a unique finite measure µ on [0, ∞) such that

Ψ(k) = [0,∞)
e -kx µ(dx ), ∀k ∈ N 0 .

The finiteness of each term Ψ(k), k ∈ N 0 allows to define the function

Ψ(λ) := [0,∞)
e -λx µ(dx ), λ ≥ 0.

Since Ψ(k) = Ψ(k) for every k ∈ N 0 , and since the extensions on Re(z) > 0 of both functions Ψ and Ψ are holomorphic and bounded, then Blaschke's argument given in Theorem 6 insures that the extensions of Ψ and Ψ are equal on Re(z) > 0. We deduce that Ψ and Ψ coincide on (0, ∞) and, by continuity in zero, also on [0, ∞).

Alternative Proof of Theorem 4. We conclude as in the last proof without the use of Blaschke's argument. Because the extensions on Re(z) > 0 of both functions Ψ and Ψ are holomorphic and bounded, they are, by Proposition 6 expandable into Gregory-Newton series as in (20). Since (Ψ(k)) k≥0 = Ψ(k) k≥0 and the sequences ∆ k Ψ(0) k≥0 and (Ψ(k)) k≥0 entirely determine each other by [START_REF] Widder | The Laplace Transform[END_REF], we conclude that ∆ k Ψ(0) = ∆ k Ψ(0) for all k ∈ N 0 . Finally, Ψ and Ψ have the same expansion (20) and then are equal.

Proof of Corollary 1. For the necessity part, do as in the proof of Theorem 4. For the sufficiency part, notice that the sequence of functions τ n Ψ(λ) = Ψ( n + λ), λ ≥ 0, satisfy the conditions of Theorem 4 and converge to Ψ. One concludes with Remark 2 (ii).

Proof of Corollary 2. The necessity part is obvious. For the sufficiency part, consider two functions Ψ 1 and Ψ 2 in CM(0, ∞), represented by their measures ν 1 and ν 2 , and coinciding on {n 0 , n 0 + 1, • • • } for some n 0 ∈ N 0 . By construction, the well defined functions on [0, ∞), τ n 0 Ψ 1 (λ) and τ n 0 Ψ 2 , coincide on N 0 . Using Remark 7 and imitating the end of the proof of Theorem 4, conclude that τ n 0 Ψ 1 and τ n 0 Ψ 2 are equal, that is

[0,∞) e -λx e -n 0 x ν 1 (dx ) = [0,∞)
e -λx e -n 0 x ν 2 (dx ), ∀λ ≥ 0

By injectivity of Laplace transform, conclude that the measures e -n 0 x ν 1 (dx ) and e -n 0 x ν 2 (dx ) are equal and so are ν 1 and ν 2 . One can also use the Gregory-Newton expansion argument as in the alternative proof of Theorem 4. Now, assume

Ψ 1 (0+) < ∞ (that is Ψ 1 ∈ CM[0, ∞)), then, by continuity, necessarily Ψ 1 (0+) = Ψ 2 (0+) and Ψ 1 = Ψ 2 on [0, ∞).
Proof of Proposition 4. The necessity part is obvious by Remark 2 (ii), we tackle the sufficiency part. Using continuity in zero, it is enough to prove that Ψ is completely monotone on (0, ∞). We fix λ > 0 and denote [x] the integer part of the real number x. Notice that α n [ λ αn ] is smaller than λ and tends to λ when n goes to infinity. We claim that we have, for every integer n such that α n < λ and u ≥ 0, that

0 ≤ e n (λ, u) -e -λ u ≤ α n u λ α n -[ λ α n ] e -αn[ λ αn ] u ≤ α n u e -αn[ λ αn ] u ≤ 1 [ λ αn ] ≤ α n λ -α n .
Now, by assumption, we have

Ψ α n [ λ α n ] = Ψ n [ λ α n ] = [0,∞) e -[ λ αn ]u ν n (du) = [0,∞) e n (λ, v) ν n (dv ),
where ν n is the representative measure of Ψ n and ν n is the finite measure with total mass ν n 0, ∞)) = Ψ(0), image of ν n by the change of variable u = α n v. Continuity of Ψ yields

Ψ(λ) = lim n→∞ Ψ α n [ λ α n ] = lim n→∞ [0,∞)
e n (λ, u) ν n (du) and Helly's selection theorem, insures that there exist a subsequence ν np p≥0 and a finite measure ν on [0, ∞) such that ν np converges vaguely (and also weakly) to ν. Taking the limit along the subsequence n p and thanks to the uniformity in (22), we get

Ψ(λ) = [0,∞)
e -λ u ν(du).

Proof of Corollary 3. Since (a) =⇒ (b) is justified by Remark 2 (ii) and (b) =⇒ (c) is immediate, we just need to prove (c) =⇒ (a). In case where Ψ(0+) < ∞, Proposition 4 directly applies. In case where Ψ(0+) = ∞, we claim that for every fixed m ∈ N, the function

τ 1 m Ψ(λ) = Ψ( 1 m + λ), λ ≥ 0,
satisfies the condition of Proposition 4. Indeed, τ 1 m Ψ is continuous and, by assumption, there exists for each n ∈ N, a function Ψ mn ∈ CM[0, ∞), associated to a measure ν nm with finite total mass ν nm [0, ∞)) = Ψ(1/m), such that for every l ∈ N 0 ,

τ 1 m Ψ( l n ) = Ψ( n + ml mn ) = Ψ mn (n + ml) = [0,∞) e -(n+ml) u ν nm (du) = [0,∞) e -n+ml mn v ν n,m (dv ) = [0,∞) e -lv ν n,m (dv ) (23)
where ν n,m is the image of ν nm by the change of variable u = v n . Taking l = 0 in (23), it is immediate that the measure ν n,m (dv ) := e -v m ν n,m (dv ) is also a measure with finite total mass ν n,m [0, ∞) = Ψ( 1 m ). It is now evident, by proposition 4, that for every m, the function τ 1 m Ψ is completely monotone on [0, ∞) for every m ∈ N. Using Remark 2 (ii), we conclude that Ψ ∈ CM(0, ∞).

Proof of Theorem 5. We tackle the proof with the necessity part: the holomorphy condition (i) is in Proposition 6 and the second condition stems from Theorem 2. Proof of the sufficiency part is based on Blaschke's result stated in Corollary 6, used with some care, because Bernstein function are not bounded in general. By Proposition 3, it is enough to check whether the function

λ → θΦ(λ) := ∆ 1 Φ(0) -∆ 1 Φ(λ) = Φ(1) -Φ(0) + Φ(λ) -Φ(λ + 1)
belongs to BF 0 b in order to show that Φ ∈ BF. We argue as follows:

1-representation [START_REF] Mai | Two novel characterizations of self-decomposability on the half-line[END_REF] gives

Φ(k) = q + dk + (0,∞) (1 -e -kx )µ(dx ), k ∈ N 0 ,
and allows to define the function Alternative proof of Theorem 5. As in the alternative proof of Theorem 4, Gregory-Newton expansion approach works. Do as in the proof Theorem 5 until point 3-and use Proposition 6 to conclude in a point 4-that both extensions of θΦ and θΦ share the Gregory-Newton expansion and then are equal.

Φ(λ) = q + dλ + (0,∞) (1 -e -λx )µ(dx ), λ ∈ [0, ∞),
Proof of Proposition 5. The necessity part is comes from [START_REF] Bingham | Regular Variation[END_REF]. The sufficiency part is an adaptation of the proof of Proposition 4. From (22), we have

1 -e n (λ, u) = 1 -e -αn[ λ αn ] u -→ 1 -e -λ u uniformly in u ≥ 0 when n → ∞. Notice that Φ α n [ λ α n ] = Φ n α n [ λ α n ] = [0,∞) 1 -e -[ λ αn ]u ) µ n (du),
where µ n is the representative measure of Ψ n . By the change variable u = α n v, the representation

Φ α n [ λ α n ] = [0,∞) 1 -e n (λ, u) µ n (du)
holds true where µ n being a finite measure with total mass µ n (0, ∞) = lim λ→∞ Φ(λ) < ∞ due to the monotone convergence theorem applied along λ → ∞. The rest of the proof is continued exactly as in proof of Proposition 4 through the limit Φ(λ) = lim n→∞ Φ (α n [λ/α n ]) .

Proof of Corollary 5. The implication (a) =⇒ (b) is justified by ( 5) and (b) =⇒ (c) being immediate, we just need to prove (c) =⇒ (a). By In order to show that Φ ∈ BF, it is enough, by Proposition 3, to check that for every fixed m ∈ N, the function

θ 1 m Φ(λ) = Φ( 1 m ) -Φ(0) + Φ(λ) -Φ( 1 m + λ), λ ≥ 0,
belongs to BF 0 b . By assumption there exists for each n ∈ N, a function Φ mn ∈ BF, having triplet of characteristics (q mn , d mn , µ mn ), such that the following representation holds true for all k ∈ N 0 :

Φ( 1 m + k n ) -Φ( k n ) = Φ( mk + n mn ) -Φ( mk mn ) = Φ mn (mk + n) -Φ mn (mk) = d mn n + (0,∞) e -mku (1 -e -nu ) µ mn (du). (24)
Representation (24) shows that the sequence

k → Φ( 1 m + k n ) -Φ( k n
) is positive and decreasing then is converging. Similarly, we have

θ 1 m Φ( k n ) = Φ( 1 m ) -Φ(0) + Φ( k n ) -Φ( 1 m + k n ) (25) = Φ( n mn ) -Φ(0) + Φ( mk mn ) -Φ( mk + n mn ) = Φ mn (n) -Φ mn (0) + Φ mn (mk) -Φ mn (km + n) = (0,∞) (1 -e -kmu ) (1 -e -nu ) µ mn (du).
Making the change of variable u = v/m in (26), we retrieve with the image µ mn of µ mn that i.e. the measure (1 -e -u m ) µ mn (du) is finite with total mass lim x→∞ θ 1 m Φ(x). We conclude that θ 1 m Φ satisfies the condition of Proposition 5 and then belongs to BF 0 b .

θ 1 m Φ( k n ) = (0,∞) (1 -e -ku ) (1 -e -n m u ) µ mn (du), ∀k ∈ N 0 . (26)

BERNSTEIN SELF-DECOMPOSABILITY PROPERTY OF FUNCTIONS IS ALSO RECOGNIZED BY THEIR RESTRICTION ON N 0

During the redaction of this paper, we felt it important to clarify the probabilistic notion of infinite divisibility and self-decomposability of non0-negative random variables. The probabilistic point of view is well presented in the book Steutel and van Harn in [START_REF] Steutel | Infinite divisibility of probability distributions on the real line[END_REF]. Every Bernstein function Φ, null in zero, is the cumulant function (i.e. Laplace exponent) of an infinitely divisible non-negative random variable Z, i.e.

E[e

-λZ ] := [0,∞) e -λx P(Z ∈ dx ) = e -Φ(λ) , λ ≥ 0.
The latter is equivalent to the existence, for every integer n, of non-negative i.i.d random variables

Z n 1 , • • • , Z n n such that Z d = Z n 1 + • • • + Z n n
, or also to the fact that the function

λ → E[e -λZ ]
t is completely monotone for every t > 0.

In [START_REF] Steutel | Infinite divisibility of probability distributions on the real line[END_REF], Steutel and van Harn present class of non-negative self-decomposable r.v.'s by those random variables X, such that for every c ∈ (0, 1), the function

λ → Ψ c (λ) = E[e -λX ]/ E[e -cλX ], (27)
belongs to CM[0, ∞). The latter is equivalent to the existence, for each c ∈ (0, 1), of a r.v. Y c independent from X such that the folloowing identity in law holds true

X d = c X + Y c
Necessarily the r.v. X is infinitely divisible and is called a self-decomposable r.v. Its cumulant function Φ(λ) = -log E[e -λX ], λ ≥ 0 (necessarily differentiable) satisfies ( 27) or equivalently it satisfies 3)(b) in Proposition 7 below, for this reason, Φ is called a self-decomposable Bernstein function. Another characterization of Φ is a specification of the form (3) with q = 0 and the Lévy measure of the form ν(dx) = x -1 k(x)dx, x > 0 with k a decreasing function (see [START_REF] Sato | Class L of multivariate distributions of its subclasses[END_REF] for more account).

We denote CF the class of cumulant functions of probability measures, i.e.:

CF := {λ → φ(λ) = -log E[e -λZ ] = -log [0,∞)
e -λx P(Z ∈ dx ), Z a non-negative r.v.} .

Remark 8. It is clear that (i) CF is a convex cone, closed under pointwise limits (this is the convergence in distribution) and stable by the operators σ c and τ c introduced in Section 2.

(ii) Φ ∈ BF if and only if t Φ-Φ(0) ∈ CF for every t > 0. φ ∈ CF if and only if 1-e -φ ∈ BF b . The latter yields Φ ∈ CF if and only if 1 -e -tΦ ∈ BF b for every t > 0.

(iii) Observe that Φ ∈ BF if and only if (1 -e -nΦ )/ n ∈ BF b for some positive sequence n tending to zero. To see the claim, use closure property under pointwise limits of BF (Corollary 3.9 p. 29 in [START_REF] Schilling | Bernstein Functions. Theory and applications[END_REF]) together with Φ = lim n→∞ 1 -e -nΦ / n . One can deduce that Φ belongs to BF if and only if n Φ belongs to CF for some positive sequence n tending to zero.

We have the following useful result related to (5): 1) If Φ admits a finite limit at 0 and ρ c Φ ∈ CF (respectively BF) for some c ∈ (0, 1), then Φ belongs to CF (respectively BF).

2) Assume Φ admits a finite limit at 0, then the following assertions are equivalent: belongs to CF (respectively BF). By closure of CF (respectively BF) and using the fact that Φ admits a finite limit at 0, deduce that Φ -Φ(0) = lim n→∞ ρ c n Φ ∈ CF (respectively BF).

2) (a) =⇒ (b): By Remark 8 (ii), ρ c Φ ∈ BF and is null at zero, then ρ c Φ ∈ CF.

(b) =⇒ (c): Since ρ c Φ ∈ CF for all c ∈ (0, 1), then by 2), Φ ∈ BF and then differentiable. Further, by Remark 8 (ii), ρ c Φ ∈ CF for all c ∈ (0, 1) implies to 1 -e -ρcΦ /(1 -c) ∈ BF for all c ∈ (0, 1). Letting c → 1-, we get, by closure of BF again, that the λ → λΦ (λ) = lim c→1-1 -e -ρcΦ(λ) /(1 -c) ∈ BF. x for every c ∈ (0, 1), observe that differentiability under the integral is well justified and the alternating property of the function under the last integral allows to conclude that ρ c Φ ∈ BF.

We are then able to state a Corollary to Theorem 5 and Proposition 7:

Corollary 7. Let function Φ : [0, ∞) → [0, ∞) admitting a finite limit at 0. Then 1) Φ Bernstein function if and only if it admits holomorphic extension on the half plane Re(z) > 0 and (Φ(k) -Φ(ck)) k≥0 is completely alternating and minimal for some c ∈ (0, 1).

2) Φ is a self-decomposable Bernstein function if and only if it admits holomorphic extension on the half plane Re(z) > 0 and one the following holds (a) the sequence (Φ(k) -Φ(ck)) k≥0 is completely alternating and minimal for all c ∈ (0, 1);

(b) the sequence (kΦ (k)) k≥0 is completely alternating and minimal. Remark 9. The main contribution in [START_REF] Mai | Two novel characterizations of self-decomposability on the half-line[END_REF] consists in Theorem 1.1 where it was stated in case Φ(0) = 0: Φ is a self-decomposable Bernstein function if and only if (Φ(xk) -Φ(yk)) k≥0 is completely alternating for every x > y > 0.

No minimality nor holomorphy conditions were required in [START_REF] Mai | Two novel characterizations of self-decomposability on the half-line[END_REF]. In our work, these conditions appeared to be the lowest price to pay in order to fix x = 1 or to have the non parametric characterization 2)(b) which clarifies their discussion at the end of section 1 in [START_REF] Mai | Two novel characterizations of self-decomposability on the half-line[END_REF].

(

  ii) A non-negative function f belongs to CM(D) if and only if -∆ c f belongs to CM(D) for every c ∈ D {0}. (iii) By [3, Lemma 6.3 p. 131], a function f belongs to CA(D) if and only if for every c ∈ D {0}, the function ∆ c f belongs to CM(D).

FUNCTIONS 1 .

 1 AND ADDITIONAL CHARACTERIZATIONS VIA ALGEBRAIC TRANSFORMATIONS 3.Completely monotone functions. Characterization of completely monotone functions is an old story and is due to the seminal works of Bernstein, Bochner and Schoenberg. A nice presentation could be found in the monograph of Schilling et al. [16]: Theorem 1. [16, Proposition 1.2 and Theorem 4.8] The following three assertions are equivalent: (a) Ψ is completely monotone on (0, ∞) (respectively on [0, ∞)); (b) Ψ is represented as the Laplace transform of a unique Radon (respectively finite) measure ν on [0, ∞):

  takes the sign of µ({0})when n is big enough. The above discussion clarifies the concept of minimality initially introduced, with a different approach, in the monograph of Widder[START_REF] Widder | The Laplace Transform[END_REF]:Definition 2. [19,Widder, p. 163] and [2, Athavale-Ranjekar]: Let a = (a k ) k≥0 a completely monotone (respectively alternating) sequence.

  ) through the obvious change of variable u = e -x , we have: Theorem 3. (a) [19, Theorem 14b, p. 14] and [2, Theorem 1] A positive sequence a = (a k ) k≥0 is obtained by interpolating a member of CM[0, ∞) (respectively BF) on N 0 if and only if a belongs to CM * (N 0 ) (respectively CA * (N 0 )).

5. 1 .

 1 Complete monotonicity property of functions is recognized by their restriction on N 0 . Theorem 4. Let Ψ : [0, ∞) -→ [0, ∞) be a bounded function. Then, Ψ is completely monotone if and only if the two following conditions hold: (a) the function Ψ has an holomorphic extension on Re(z) > 0 and remains bounded there; (b) the sequence Ψ(k) k≥0 is completely monotone and minimal. Corollary 1. A function Ψ : (0, ∞) -→ [0, ∞) is completely monotone if and only if the following two conditions hold: for some (and hence all) positive sequence ( n ) n≥0 such that n → 0, (i) the function Ψ has a holomorphic extension on Re(z) > 0 and remains bounded on Re(z) > n ;

5. 3 .

 3 Bernstein property of functions is recognized by their restriction on N 0 . Theorem 5. A function Φ : [0, ∞) → [0, ∞) is a Bernstein function if and only if it (a) the function Φ has an holomorphic extension on Re(z) > 0 and satisfies there |Φ(c+z)-Φ(z)| ≤ M for some c, M > 0; (b) the sequence Φ(k) k≥0 is completely alternating and minimal.

  as n → ∞ and n ∈ N. Denote {x} and [x] the fractional and integer part of x. Then, mimicking the end of the second proof of Theorem 1.2.1 p. 6 [5], take an arbitrary compact interval [a, b] in [0, ∞) and observe that sup λ∈[a,b]

Corollary 6 .

 6 Using the is conformal one-to-one mapping of the open unit disc onto the open right half plane θ(z) = 1 + z 1 -z , one can easily rephrase Blaschke's theorem for function defined on the open right half plane: Two holomorphic functions on the open right half plane P are identical if their difference is bounded and they coincide along a sequence z 1 , z 2 , z 3 , • • • in P , such that the series (1 -| z i -1 z i +1 |) diverge and in particular for z i = i ∈ N.

2 )

 2 (iii) is justified as follows: take z = 0, get that |Φ(z)| ≤ |Φ(0)| + C + D |z| ≤ C + |Φ(0)| ∨ D) e D|z| and deduce Φ satisfies Nörlund's condition (21). 7. THE PROOFS Proof of Proposition 1. (a) For the necessity part, notice that if c > 0 and Ψ is represented by Ψ

  e n (λ, u) := e -αn[ λ αn ] u -→ e -λ u , uniformly in u ≥ 0, when n → ∞. (22) Indeed, using the inequality a e -a ≤ 1 and 0 ≤ e -a -e -b = b a e -u du ≤ (b -a) e -a , 0 ≤ a ≤ b,

and then, by Proposition 3 ,

 3 θΦ ∈ BF 0 b ; 2-the sequences θΦ(k) k≥0 and θΦ(k) k≥0 are equal; 3-boundedness condition in (a) yields boundedness of the function the extension of θΦ, boundedness of the function the extension of θΦ stems from Proposition 6; 4-Corollary 6 insures that the extensions of the functions θΦ and θΦ are equal on (0, ∞) and also on since θΦ(0) = θΦ(0) = 0. Then, θΦ ∈ BF 0 b .

Representation (25) and continuity of θ 1 mΦ justifies that lim x→∞ θ 1 mΦ(x) = lim k→∞ θ 1 m 1 mΦ

 1111 Φ( k n) is finite. Then, the monotone convergence theorem insures applied in (26) gives that(0,∞) (1 -e -n m u ) µ mn (du) = lim x→∞ θ

Proposition 7 .

 7 Let Φ : [0, ∞) -→ [0, ∞) and ρ c Φ(λ) := (σ -σ c )Φ(λ) = Φ(λ) -Φ(cλ), c ∈ (0, 1).

  (a) ρ c Φ ∈ BF for every c ∈ (0, 1);(b) ρ c Φ ∈ CF for every c ∈ (0, 1); (c) Φ is differentiable on (0, ∞) and λ → λΦ (λ) ∈ BF.Proof of Proposition 7. 1) If ρ c Φ belongs to CF (respectively BF), then for every n ∈ N 0 ,λ → ρ c n Φ(λ) = Φ(λ) -Φ(c n λ) = n-1 i=0 Φ(c k λ) -Φ(c k+1 λ) = n-1 i=0 ρ c Φ(c k λ)

1 c

 1 (c) =⇒ (a): The function x → Φ 0 (x) = xΦ (x) ∈ BF. Write λ → ρ c Φ(λ) = Φ 0 (λx) dx


Hasan Abdulrahman

Marc Chaumont

Philippe Montesinos

Baptiste Magnier

Color Image Stegananalysis Using Correlations between RGB Channels

Keywords: Steganalysis, color spatial rich model, channel correlation, co-occurrences, ensemble classifiers

Digital images, especially color images, are very widely used, as well as traded via Internet, e-mail and posting on websites. Images have a large size which allows embedding secret messages of large size, so they are a good medium for digital steganography. The main goal of steganalysis is to detect the presence of hidden messages in digital media. In this paper, we propose a steganalysis method based on color feature correlation and machine learning classification. Fusing features with features obtained from color-rich models allows to increase the detectability of hidden messages in the color images. Our novel proposition uses the correlation between different channels of color images. Features are extracted from the channel correlation and co-occurrence correlation. In this work, all stego images are created with a range of different payload sizes using two steganography S-UNIWARD and WOW algorithms. To validate the proposed method, his efficiency is demonstrated by comparison with color rich model steganalysis.

INTRODUCTION

Steganography is the art of secret communication, by embedding messages within media such as audio, image or video files, in a way to complicate the detection of these hidden messages.

In general, steganography requires two parts, messages are embedded in the first part and they are extracted in the second part [START_REF] Fridrich | Steganography in digital media: principles, algorithms, and applications[END_REF]. Several steganography techniques are used for different purposes, such as, for the most prominent, hiding illegal activities or secret communications [START_REF] Hosmer | Discovering covert digital evidence[END_REF]. Many steganography methods have been developed to enable secret communication. So the real-world needs to develop methods and algorithms rapidly in order to detect the presence of hidden messages and, thus, identify this secret communication. This method is called steganalysis.

We Thank the Iraqi Ministry of Higher Education and Scientific Research for funding this work.

Embed messages inside the digital medium involves some slight changes in this medium. Steganalysis research aims to develop some methods that are effectively able to detect these modifications. Although the real-world uses significantly more color images than grayscale images, there is high development in steganalysis to handle grayscale images, but, while overlooking color images [START_REF] Fridrich | Steganalysis of lsb encoding in color images[END_REF].

In this paper, we propose new features to enrich the color rich model [START_REF] Goljan | Rich model for steganalysis of color images[END_REF], which is formed by co-occurrences of residuals taken across the color channels. Section 2 is dedicated to methods related to color image steganalysis. Section 3 describes the proposed method by recalling the color rich model, color filter array and color channel correlation gradients. Experimental results an comparisons with its tables and curves are given in section 4. Section 5 concludes this paper.

LITERATURE REVIEW

In recent years, only a few methods involving color steganalysis have been proposed.

Westfeld, in [START_REF] Westfeld | Detecting low embedding rates[END_REF], developed a detector of Last Significant Bit (LSB) matching in color images using the 3D color cube representation of the image. The detection method combines the three color components of each pixel to recognize an increased number of neighbor colors. This method estimates a relative frequency of a number of neighbor pixels as a steganalysis feature. LSB typically introduces characteristic tails in the histograms that are not observed in cover images. The authors have shown that this algorithm is efficient for decompressed JPEG images. However, it performs poorly in images with a high number of unique colors, such as never color compressed or resized images.

The detection of LSB Matching from grayscale images was enhanced by Ker, in [START_REF] Ker | Resampling and the detection of lsb matching in color bitmaps[END_REF], to detect LSB Matching in color images. By modifying the Harmsen method [START_REF] Harmsen | Steganalysis of additive-noise modelable information hiding[END_REF] that used the Center Of the Mass (COM ) of the Histogram Characteristic Function (HCF ) for the detection of additive noise based steganography, including LSB Matching. This method especially detects hidden messages in JPEG images but is not very efficient concerning bitmap images which have never been compressed.

Thiyagarajan et al. [START_REF] Thiyagarajan | Steganalysis using color model conversion[END_REF] proposed a universal image steganalysis method based on color model conversion. Indeed, considering color images, they convert Red, Green and Blue (RGB) channels of the images to the Hue Saturation and Intensity (HSI) color model to detect hidden messages. Stego images are then generated by implementing different color image formats, using the last significant bit steganography method. Finally, cover and stego images are recognized using a threshold value which depends on the correlation between pixel pairs in terms of color components.

Lyu et al. [START_REF] Lyu | Steganalysis using color wavelet statistics and one-class support vector machines[END_REF] described a universal steganalysis algorithm that exploits the inherent statistical regularities of natural images. The statistical model consists of first and higher order color wavelet statistics of noise residuals obtained using predictors of coefficients in Quadratic Mirror Filter (QM F ) decomposition of the image from all three color channels. Finally, they estimate that the addition of color statistics provides considerable improvement in overall detection accuracy.

Krichner et al. [START_REF] Kirchner | Steganalysis in technicolor boosting ws detection of stego images from cfainterpolated covers[END_REF] suggested a steganalysis method to detect LSB replacement steganography in color images. These stego images are produced via processing which reveals some traces of Color Filter Array (CF A) interpolations. Here the authors have enhanced the Weighted Stego (W S) image steganalysis method [START_REF] Fridrich | On estimation of secret message length in lsb steganography in spatial domain[END_REF] by replacing the cover predictor in W S with position specific predictors. This technique puzzles out the local predictability of pixels, depending on their position in the CF A interpolation to account for differences between cover (common) images and suspect images.

Fridrich et al. [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF] proposed a general methodology for steganalysis of digital images based on the rich model concept consisting of a high number of diverse submodels. Each submodel considers various types of relationships among neighboring samples of noise residuals obtained by linear filters with compact supports. The submodels are formed from noise residuals, which are computed using high pass filters, then features are extracted using co-occurrence matrices.

From two different components, Goljan et al. [START_REF] Goljan | Rich model for steganalysis of color images[END_REF] developed a color image model which is called the Spatio-Color Rich Model (SCRM Q1, where 'C' represents the color version of the Spatial Rich Model -SRM Q1- [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF], with 'Q1' representing fixed quantization q = 1 ) produced from two different components. The first component is the SRM Q1 with a single quantization step q = 1 and dimensionality of 12753 features. The SRM Q1 feature is computed for each color channel separately, then the three features are merged to keep the same dimensionality, as for grayscale images. The second component is a collection of 3D co-occurrences with a larger threshold T = 3 and q = 1. The final dimensional features obtained from this component are 5404 features. The color rich model is built from the same pixel noise residuals as those used in the SRM Q1, but is formed across the three channels of each pixel.

PROPOSED METHOD

Our proposition is to enrich the SCRM Q1 with an interchannel correlation which is composed of two sets of features. The first set, produced by [START_REF] Goljan | Rich model for steganalysis of color images[END_REF], gives 18157 features. The second set gives 3000 features obtained from the correlation of different R, G, B channel gradients. The following section recalls the color rich model, then section 3.2 explains the basics of color filter array, and section 3.3 gives an explanation about our proposition.

Color Rich Model

The following steps describe the process by which the first set of features is computed. We used the spatial color rich model for color image steganalysis in [START_REF] Goljan | Rich model for steganalysis of color images[END_REF] which produced 18157 features. This method extracts the noise residual from each color channel separately by the following formula:

R ij = Xij (N ij ) -c • X ij . ( 1
)
where

c ∈ N is the residual order, N ij is a local neighborhood of pixel X ij at coordinates (i, j), Xij (•) is a predictor of c•X ij , X ij ∈ N ij , X ij ∈ {0, ...., 255}
. Xij represents a pixel value of an 8-bit grayscale cover image and its corresponding stego image.

All of the submodels (R ij ) ∈ R n1×n2 are formed from noise residual images of size n 1 × n 2 computed using high pass filters of the following form:

R ij ← tranc T round R ij q , (2)
where tranc T represents the truncation function with T > 0 defined for any x ∈ R, tranc T (x) = x for x ∈ [-T, T ] and tranc T (x) = T • sign(x) and otherwise q is the quantization step, and round is a function for rounding to an integer value.

The spatio-color rich model consists of two different components. On one hand, the spatial rich model (SRM Q1) [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF] with a fixed quantization q=1 and truncation T =2 yields a dimensionality of 12753 features. These features are computed from each R, G and B color channel separately. Finally, the three dimensionality features are added together to keep the same dimensionality as for grayscale images. On the other hand, from the same noise residuals (i.e. SRM Q1), the CRM Q1 builds a collection of 3D color co-occurrences, taking three color values at the same position (across the three channels of each pixel). Thus, with fixed truncation T =3 and quantization q=1, CRM Q1 produces 5404 features.

Color Filter Array

Digital cameras capture color images using a single sensor in conjunction with a Color Filter Array (CF A). The CF A allows only one part of the spectrum to pass to the sensor so that only one color is measured at each pixel (red, blue or green) and such images are called mosaic images. To construct a color image, a demosaicking algorithm is used in order to interpolate each color plane (i.e. CF A interpolations). Several patterns exist for the color filter array, with the most common being Bayer CF A [START_REF] Bayer | Color imaging array[END_REF]. During this process, the green color channel is the most important factor which determines the luminance of the color image, 50% of the pixels in the Bayer CF A structure is assigned to the green channel, while 25% is assigned to the red channel and 25% to the blue color channel.

The interpolation introduces specific correlations between the samples of a color image. Hence, a subset of samples, within a color channel, is correlated with their neighboring samples of the same color. Since the color filters in a CF A are typically arranged in a periodic pattern, these correlations are also periodic [START_REF] Wang | New color filter arrays of high light sensitivity and high demosaicking performance[END_REF].

Channel Correlation

In this section, we introduce an inter-channel correlation measure, and demonstrate that it can be listed to first order Euclidean invariants (see Hilbert [START_REF] Hilbert | Theory of algebraic invariants[END_REF] for the invariant theory). Such invariants have mainly been used for stereo-matching [START_REF] Gouet | A fast matching method for color uncalibrated images using differential invariants[END_REF]. In this paper, we show that the information provided by them can enhance steganography detection.

Starting from the local correlation of red and green channels (similarly, correlation of red and blue channels) :

Corr R,G (i, j, k, l) = (i ,j )∈Wi,j X (R) i ,j X (G) k+i ,l+j (3)
with:

• X (R)
i ,j ∈ [0, 255], being a pixel value at position (i, j) in the red channel,

• X (G)
k,l ∈ [0, 255], being a pixel value at position (k, l) in the green channel,

• W i,j , being a small window centred in (i, j).

Considering (k, l) = (0, 0) and a limited development of X (R) and X (G)
around (i, j), then :

Corr R,G (i, j, 0, 0) = h = (i -i, j -j) (i , j ) ∈ W i,j X (R) i,j + ∇X (R) i,j • h X (G) i,j + ∇X (G) i,j • h . (4)
Developing this equation leads to four terms. Three of which are constant or not informative, then there is only one informative term :

∇X (R) i,j • ∇X (G) i,j . (5)
If only one channel has locally been altered, the gradient in this channel is modified. Consequently, the scalar product of two channel gradients reflects the change in the cosine of the difference between the two gradient angles.

Similarly, we can apply the same computation for the red and blue channel and then obtain :

∇X (R) i,j • ∇X (B) i,j (6)
As stated by Gouet et al. [START_REF] Gouet | A fast matching method for color uncalibrated images using differential invariants[END_REF] (and following the Hilbert theory [START_REF] Hilbert | Theory of algebraic invariants[END_REF]), it is unnecessary to investigate the ∇X

(G) i,j • ∇X (B) i,j
term, as it is already implicitly contained in the first two expressions (Eq. 5 and 6).

Our proposition in this paper is to add two new features depending on the normalized correlation between red and green channels, and red and blue channels as :

G RG = ∇X (R) i,j • ∇X (G) i,j |∇X (R) i,j | |∇X (G) i,j | (7)
G RB = ∇X (R) i,j • ∇X (B) i,j |∇X (R) i,j | |∇X (B) i,j | (8)
Fig. 1 illustrates our preprocessing steps to obtain normalized correlations between gradients of each channel. Note that gradients of each channel are estimated by convolution with a [-1; 1] mask (horizontal and vertical).

Then, our features are computed from these correlations G RG and G RB , by computing the co-occurrences as in the rich model [START_REF] Fridrich | Rich models for steganalysis of digital images[END_REF]. We used different values of the quantization q ∈ {0.1, 0.3, 0.5, 0.7, 0.9, 1} with fixed truncation T =1. The reason for using these different values of quantization q is that G RG and G RB belong to [-1, 1]. Moreover, the use of these values gives more accurate features and avoids the generation of too many zero values caused by the truncation step in the co-occurrence vector.

For each quantization, we obtain 12 submodels, one symmetrized spam14h and one spam14v with 25 features each, and one of minmax22h, minmax22v, minmax24, min-max34h, minmax34v, minmax41, minmax34, minmax48h, minmax48v, and one minmax54, with 45 features for each. All submodels are gathered in a one dimension vector to erect a dimensionality of (10 × 45 + 2 × 25) × 6 = 3000 features which are added to 18157 features of color rich models with fixed q = 1 and T = 3 in order to collect a final set of 21157 features.

Ensemble Classifiers

Modern steganalysis methods for digital images are based on extracted features and these methods need machine learning techniques to detect hidden messages. In our work, we choose ensemble classifiers [START_REF] Kodovsky | Ensemble classifiers for steganalysis of digital media[END_REF] because of their efficient classification performance for large scale learning.

Kodovsky's et al. [START_REF] Kodovsky | Ensemble classifiers for steganalysis of digital media[END_REF] proposed ensemble classifiers 1 , which is a new machine learning tool for steganalysis, consisting of many classifier L base learners(B l ).

Each classifier is built from randomly selected subspace d sub -dimensionals from the entire full d-dimension feature space. The authors use Ficher Linear Discriminants (F LD) as base learners and the final decision is made by aggregating the decision of individual base learners.

Let d be a full dimension feature space. First, the classifiers construct a number L of F LD base learners (B l ) with l ∈ {1, ..., L}. Each one performs its learning on a subspace of d sub dimension, where d sub < d. The i th image formed f i ∈ R d , will be mapped R d → {0, 1}, with 0 being a cover image and 1 a stego image.

In the learning phase, each classifier learns to map a feature vector f i , to the correct class number:

F LD l : R d → {0, 1} f i →F LD i (f i )
Each classifier uses the training database to compute the vector orthogonal to the hyperplane separating the two classes. For a test feature, the l th base learner reaches its decision by computing a projection and comparing it to a threshold. After collecting all L decisions, the final classifier selected is the one which has received the most votes. Then the decision threshold of each base learner is adjusted to minimize the total detection error under an equal prior on the training data [START_REF] Kodovsky | Ensemble classifiers for steganalysis of digital media[END_REF]:

P E = min P F A 1 2 [P F A + P M D (P F A )] .
where P F A represents the false alarm probability and P M D the missed detection probability. 1 Ensemble classifiers is available at http://dde.binghamton. edu/download/ensemble. The preprocessing steps for building our database depend on the CF A idea.

EXPERIMENTAL RESULTS

Building Database

A raw image is a class of computer files containing untouched pixel information coming from the digital camera sensor. These files hold a large amount of meta-information about the image generated by the camera [START_REF] Yuan | High quality image reconstruction from raw and jpeg image pair[END_REF]. In our work, the color image database is very carefully built depending on the CF A idea. We collected raw images from two subsets which are the most standard, and have the highest number of images captured (i.e. the Dresden Image Database [START_REF] Gloe | The dresden image database for benchmarking digital image forensics[END_REF] 3500 full-resolution Nikon digital camera raw color images and the Break Our Steganographic System (BOSSbase2 ) 1000 Canon digital camera raw color images).

In order to obtain color images in Portable Pixel Map (P P M ) format of size 512×512, all images take the same CF A map layout, as illustrated in Fig. 2. Two steps are required. The first step consists of using a demosaicking algorithm to convert raw images in demosaicked images . The second step consists of cropping five sub-images from one image. Fig. 3 shows sample images produced by the cropping step.

First we use demosaicking algorithm Patterned Pixel Grouping (PPG) in the dcraw platform 3 to be able to read numerous raw formats used to convert raw images in P P M and T IF F formats. As illustrated in Fig. 2, the obtained images are such that the Bayer Pattern is always of type RGBG (red channel pixel is placed at even position). We wrote a spatial code in C language under Linux to start the crop from the red channel position. Indeed, from one image, this code randomly selected the red channel position and cropped five images of size 512×512, so that all blocks share the same CF A map layout. The final number of images used 10000 P P M color images of size 512×512.

Experimental Protocol

Stego images are obtained using two steganography algorithms: the Spatial-UNIversal WAvelet Relative Distortion (S-UNIWARD 4 ) steganography algorithm [START_REF] Holub | Universal distortion function for steganography in an arbitrary domain[END_REF] and Wavelet Obtained Weights (WOW 5 ) steganography algorithm [START_REF] Holub | Designing steganographic distortion using directional filters[END_REF]. As shown in [START_REF] Holub | Digital image steganography using universal distortion[END_REF], these two algorithms are highly adapted because they are selected for this strong resistance to steganalysis.

The payload is embedded in equal proportion in the three R, G and B channels. The different payload sizes are {0.01, 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5} Bit Per Channel (BPC). Two sets of features are extracted from each image, the first set is the color rich model vector which consists of 18157 features of residuals among neighboring pixels (see section 3.1). The second set of features consists of 3000 proposed features from correlations between channels by applying gradient derivative filters on each channel, as explained in section 3.3. Then we add the two sets of features in a one dimensional vector to get a result of 21157 features of 10000 covers and 10000 stegos which are ready to enter in the classifier. The images are randomly divided into two halves for training and testing. A random subset of images, 5000 covers and 5000 stegos is used to train and test the classifier. The result of the classification is the average testing error over 10 splits PE . The rest of the cover and stego images are tested against the ensemble classifiers [START_REF] Kodovsky | Ensemble classifiers for steganalysis of digital media[END_REF]. Then the decision values are collected for each. Given the decision values, ROC curves are obtained. The area under the ROC curves is calculated as the accuracy of the ensemble classifiers.

Results and Discussion

This section contains the experimental results of our proposed method. We illustrate these results in Tables 1 and2. S-UNIWARD and WOW methods were tested with different relative payloads {0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5} Bits Per channel (BP C) against the method of color rich model [START_REF] Goljan | Rich model for steganalysis of color images[END_REF]. We used the same set of payload values with the same embedding methods. Our method, achieved higher performance by registering 87.54% and 86.63% detection rates for S-UNIWARD and WOW respectively with the payload 0.5 bpc. At the same time, the color rich model method [START_REF] Goljan | Rich model for steganalysis of color images[END_REF] is less efficient because it achieved respectively 86.14% and 85.27% detection rates on the same sample. We noted the same thing with the rest of the payload values for both (the proposed and compared method), as shown in Tables 1 and2.

Curves in Fig. 4 (a) and (b) also illustrate the comparison between the proposed method and the color rich model. As a result, the average testing error of the proposed method is less than that of the color rich model. That proves the importance 4 S-UNIWARD steganography method is available at http://dde. binghamton.edu/download/stego_algorithms/.

5 WOW steganography method is available at http://dde. binghamton.edu/download/stego_algorithms/. of the additional 3000 features derived from the correlation between different channels proposed by our method.

Another experiment involved embedding the entire payload only in one channel of the color image, i.e. with payload 0.2 bpc in the green channel only. In this case, the detection rate records higher than the same payload if it is distributed equally between the three color channels. Table 3 illustrates the comparison of detection rates between the S-UNIWARD and WOW methods with payloads 0.2 bpc and 0.4 bpc embedded in one channel only and in the three channels separately. Fig. 5 (a) and (b), show the ROC curves, illustrating the performance of our method. This experiment revealed that it is easier to detect a hidden massage in only one channel than a message shared in all channels.

CONCLUSION AND FUTURE WORK

Here we proposed and implemented a new method for color image steganalysis. The proposed method estimates correlations between the gradients of red, green and blue channels. Then these correlations are incorporated in the rich model using a co-occurrence matrix in order to obtain 3000 features. These features are added to those that are obtained from the color rich model in order to build a vector of a total of 21157 features. All feature vectors are fed to the ensemble classifiers. We used a quantization step with a set of values that differed from the color rich models. The ensemble classifiers is used to detect steganographic messages. Images for cover and stego are collected using the Dresden database and BOSSbase 1.0. Multiple comparisons are made between the proposed method with the color rich model using WOW and S-UNIWORD steganography methods in different payloads, reflecting the efficiency of the proposed features.

Our future work will focus on developing a new steganalysis method for digital color images, enhance the feature vector and calculate a new correlation between all channels of the color image.

Fig. 1 :

 1 Fig. 1: Diagram illustrating the preprocessing steps to obtain correlations between gradients in each channel.

  Fig.2:The preprocessing steps for building our database depend on the CF A idea.

5 Fig. 3 :

 53 Fig. 3: Sample images of our database built by random cropping from locations of red channel pixels (even position) in a Bayer pattern : a) Original RAW image 3906×2602, b) crop 1 position x=2116, y=1928, c) crop 2 position x=902, y=1182, d) crop 3 position x=3080, y=436, e) crop 4 position x=1866, y=1778, f) crop 5 position x=650, y=1032.

Fig. 4 :

 4 Fig. 4: Detection error PE as a function of the payload for (a) S-UNIWARD and (b) WOW steganography methods illustrates good detectability with our proposed features.

Fig. 5 :

 5 Fig.5: ROC curves of the proposed method for the payloads (a) 0.2 bpc (b) 0.4 bpc using the S-UNIWARD steganography method. The steganalysis technique detects easier hidden messages in only one channel more than equal embedding in the three channels.

Table 3 :

 3 Our steganalysis detection rate of S-UNIWARD and WOW steganography methods at 0.2 bpc and 0.4 bpc payload embedding in the green channel compares with equal embedding in three channels.

	Embedding	Proposed method		Color rich model
	Payload	Average testing error Detection Rate%	Average testing error Detection Rate %
	0.01	0.4830 ± 0.0010	51.70	0.4841 ± 0.0017	51.59
	0.05	0.4010 ± 0.0032	59.90	0.4045 ± 0.0026	59.55
	0.1	0.3203 ± 0.0023	67.97	0.3298 ± 0.0016	67.02
	0.2	0.2370 ± 0.0031	76.30	0.2498 ± 0.0026	75.02
	0.3	0.1808 ± 0.0034	81.92	0.1947 ± 0.0023	80.53
	0.4	0.1470 ± 0.0025	85.30	0.1599 ± 0.0015	84.01
	0.5	0.1246 ± 0.0023	87.54	0.1386 ± 0.0023	86.14

Table 1 :

 1 The comparison between the steganalysis of the proposed method with the color rich model in different relative payloads for the S-UNIWARD steganography method: the average testing error for the proposed is less than for the color rich model method.

	Embedding	Proposed method		Color rich model
	Payload	Average testing error Detection Rate%	Average testing error Detection Rate %
	0.01	0.4836 ± 0.0017	51.64	0.4850 ± 0.0014	51.50
	0.05	0.4042 ± 0.0027	59.58	0.4092 ± 0.0032	59.08
	0.1	0.3317 ± 0.0034	66.83	0.3397 ± 0.0023	66.03
	0.2	0.2502 ± 0.0041	74.98	0.2654 ± 0.0025	73.46
	0.3	0.1918 ± 0.0013	80.82	0.2081 ± 0.0029	79.19
	0.4	0.1574 ± 0.0021	84.26	0.1783 ± 0.0034	82.17
	0.5	0.1307 ± 0.0015	86.63	0.1473 ± 0.0025	85.27

Table 2 :

 2 The comparison between the steganalysis of the proposed method with the color rich model in different relative payloads for the WOW steganography method: the average testing error for the proposed is less than for the color rich model method.

		0.5
												Proposed method
		0.45										Color rich model
		0.4
	Probability of error PE	0.25 0.3 0.35
		0.2
		0.15
		0.1	0	0.05	0.1	0.15	0.2	0.25	0.3	0.35	0.4	0.45	0.5
							Embedding payload

BOSSbase can be accessed at http://www.agents.cz/boss/ BOSSFinal.

dcraw code is available at http://www.cybercom.net/defin/ dcraw.