Brahim Bouya

Mohamed Zarrabi
email: mohamed.zarrabi@math.u-bordeaux1.fr

WEIGHTED BIG LIPSCHITZ ALGEBRAS OF ANALYTIC FUNCTIONS AND CLOSED IDEALS

Keywords: . 2000 Mathematics Subject Classification. primary 46E20, secondary 30C85, 47A15

We give the smallest closed ideal with given hull and inner factor for some weighted big Lipschitz algebras of analytic functions.

Introduction.

Let H ∞ be the space of all bounded analytic functions on the open unit disk D of the complex plane. Various authors have characterized completely the structure of the closed ideals in some separable Banach algebras of analytic functions in D, we refer the reader for instance to [START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF][START_REF] Korenblum | Invariant subspaces of the shift operator in a weighted Hilbert space[END_REF][START_REF] Matheson | Cyclic vectors for invariant subspaces in some classes of analytic functions[END_REF][START_REF] Shamoyan | Closed ideals in algebras of functions that are analytic in the disk and smooth up to its boundary[END_REF][START_REF] Shirokov | Analytic functions smooth up to the boundary[END_REF] and some references therein. They have proved that this structure is standard in the sense of that given by Beurling and Rudin in the usual disk algebra A(D), of functions f ∈ H ∞ that are continuous up to the boundary T, see [START_REF] Hoffman | Banach spaces of analytic functions[END_REF]Page. 85] and [START_REF] Rudin | The closed ideals in an algebra of analytic functions[END_REF]. However, the structure of closed ideals in nonseparable Banach algebras of analytic functions seems much more difficult to characterize, we can see for instance [START_REF] Bouya | On closed ideals in the big Lipschitz algebras of analytic functions[END_REF][START_REF] Gorkin | Higher order hulls in H ∞ . II[END_REF][START_REF] Hedenmalm | Bounded analytic functions and closed ideals[END_REF][START_REF] Hoffman | Bounded analytic functions and Gleason parts[END_REF][START_REF] Pedersen | Ideals in big Lipschitz algebras of analytic functions[END_REF][START_REF] Sherbert | The structure of ideals and point derivations in Banach algebras of Lipschitz functions[END_REF]. In this paper we are interested in the description of the closed ideals in the following weighted big Lipschitz algebra where ω = 0 is a modulus of continuity, i.e., a nonnegative nondecreasing continuous function on [0, 2] with ω(0) = 0 and t → ω(t)/t is nonincreasing. Our main result is stated in Theorem 1 below. It is well-known that Lip ω is a nonseparable commutative Banach algebra when equipped with the norm

f ω,D := f ∞ + sup z,w∈D z =w |f (z) -f (w)| ω(|z -w|) ,
where f ∞ := sup Tamrazov [START_REF] Tamrazov | Contour and solid structural properties of holomorphic functions of a complex variable, (Russian) Uspehi Mat[END_REF] has shown that the algebras Lip ω (D) and Lip ω := Lip ω (T) coincide for any modulus of continuity ω, see also [START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF]Appendix A]. Furthermore, the norms f ω,D and f ω := f ω,T are equivalent, where

f ω,T := f ∞ + sup z,w∈T z =w |f (z) -f (w)| ω(|z -w|) .
In what follows, we denote by E a closed subset of T of Lebesgue measure zero and by U ∈ H ∞ an inner function such that σ(U) ∩ T ⊆ E, where σ(U) := {λ ∈ D : lim inf

z→λ z∈D |U(z)| = 0},
is called the spectrum of U, see for instance [START_REF] Nikolskii | Treatise on the shift operator[END_REF]. It is known that σ(U) = Z U ∪ supp(µ U ), where Z U is the zero set in D of U and supp(µ U ) is the closed support of the singular measure µ U associated with the singular part of U.

For a closed ideal I of Lip ω , we denote by U I the greatest inner common divisor of the inner parts of functions in I \ {0} and by E I the standard hull of I, that is E I := {ξ ∈ T : f (ξ) = 0, ∀f ∈ I}, see [START_REF] Hoffman | Banach spaces of analytic functions[END_REF]Page. 85] and [START_REF] Rudin | The closed ideals in an algebra of analytic functions[END_REF]. We define as it is obtained in Proposition 5 of section 6.1 below. For 0 < α ≤ 1, we set

ω α (t) := t α , 0 ≤ t ≤ 2.
A classical result due to Carleson [START_REF] Carleson | Sets of uniqueness for functions regular in the unit circle[END_REF] A result of type (1.2) was stated first by H. Hedenmalm [START_REF] Hedenmalm | Bounded analytic functions and closed ideals[END_REF] in the algebras H ∞ and Lip ω 1 , for closed ideals I such that E I is a single point. Later, T. V. Pederson [START_REF] Pedersen | Ideals in big Lipschitz algebras of analytic functions[END_REF] has obtained the same result in Lip ωα , for closed ideals I such that E I is countable. In this paper we are interested in the case where ω satisfies the following condition

inf 0<t≤1 ω(t 2 ) ω(t) =: η ω > 0. (1.3)
We obtain the following theorem which will be proved in the next section.

Theorem 1. Let I ⊆ Lip ω be a closed ideal, where ω is a modulus of continuity satisfying (1.3). Then J ω (E I , U I ) ⊆ I.

In particular Theorem 1 provides us with an extension of (1.2) to a large class of smooth algebras between H ∞ and 0<α≤1 Lip ωα , such as Lip χ β and Lip ψ β , where

ψ β (t) := log -β log 2e 1+β t , 0 < t ≤ 2.
As consequence of Theorem 1 we deduce the following corollary.

Corollary 1. We suppose that E or U are nontrivial such that E ⊇ σ(U) ∩ T and

T log ω d(ξ, Z U ∪ E) |dξ| > -∞, (1.4)
where ω is a modulus of continuity satisfying (1.3). Then J ω (E, U) is a nontrivial principal closed ideal with E as hull and U as inner factor.

The estimate (1.4) is in fact a necessary condition guaranteeing that J ω (E, U) is nontrivial, see the proof of Corollary 1. By joining together the results established in Theorem 1 and Corollary 1 we deduce that J ω (E, U) is the smallest closed ideal with E as hull and U as inner factor provided that (1.4) is satisfied.

To prove Theorem 1 we give an adaptation in the space J ω (E) := J ω (E, 1) of Korenblum's functional approximation method, see for instance [START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF][START_REF] Korenblum | Invariant subspaces of the shift operator in a weighted Hilbert space[END_REF][START_REF] Matheson | Cyclic vectors for invariant subspaces in some classes of analytic functions[END_REF]. To do so, we will use some properties enjoyed by the space J ω (E) which we describe in the following theorem.

Theorem 2. Let g ∈ J ω (E) be a function, where ω is an arbitrary modulus of continuity.

Let V ∈ H ∞ be an inner function dividing g, that is g/V ∈ H ∞ . Then V O g ∈ J ω (E), (1.5)
where O g is the outer factor of g.

Let {V n : n ∈ N} ⊂ H ∞ be a sequence of inner functions dividing g such that σ(V n )∩T ⊆ E, for every n ∈ N. If lim n→∞ V n O g -V O g ∞ = 0, (1.6) then lim n→∞ V n O g -V O g ω = 0. (1.7)
In particular, assertion (1.5) shows that J ω (E) possesses the F-property. This kind of factorization property has been proved by Havin, Shamoyan and Shirokov for the whole space Lip ω with respect to any modulus of continuity, see for instance [START_REF] Havin | Analytic functions with a Lipschitzian modulus of the boundary values[END_REF], [START_REF] Shirokov | Analytic functions smooth up to the boundary[END_REF]Chapter I] and also [START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF]Appendix B].

The remainder of this paper is organized as follows: In section 2, we give the proof of Theorem 1 and Corollary 1. Section 3 is devoted to presenting some needed results. Sections 4 and 5 contain respectively the proofs of Propositions 3 and 4. In the last section we give the proof of Theorem 2.

Proofs of Theorem 1 and Corollary 1.

In section 4.2 we give the proof of the following proposition. Proposition 3. Let I ⊆ Lip ω be a closed ideal and let g ∈ J ω (E I ) be a function, where ω is a modulus of continuity satisfying (1.3). We suppose that gf ∈ I for some outer function f ∈ Lip ω . Then g ∈ I.

For f ∈ H ∞ we denote by U f and O f respectively the inner and the outer factor of f . In section 5 we will use both Theorem 2 and Proposition 3 to give the proof of the following proposition.

Proposition 4. Let I ⊆ Lip ω be a closed ideal, where ω is a modulus of continuity satisfying (1.3). There exists a function f ∈ I such that U f = U I .

Proof of Theorem 1. Let I ⊆ Lip ω be a closed ideal. By applying Proposition 4, there exists a function f ∈ I such that U f = U I . Let g ∈ J ω (E I , U I ) be a function. We note that g/U I ∈ Lip ω , by using the F-property of Lip ω . Since gO f = (g/U I )f then gO f ∈ I. Thus g ∈ I, by applying Proposition 3. Therefore J ω (E I , U I ) ⊆ I, which is the desired result.

Proof of Corollary 1. It is known [START_REF] Shirokov | Analytic functions smooth up to the boundary[END_REF]Page. 137] that (1.4) ensure the existence of a function f ∈ Lip ω such that f -1 (0) = Z U ∪ E and f /U ∈ H ∞ . The functions O f and UO f belong to Lip ω , since Lip ω possesses the F-property. Therefore g := UO 2 f ∈ J ω (E, U), as product of two functions in I ω (E) := I ω (E, 1). Hence J ω (E, U) is non trivial and possesses E as hull and U as inner factor. Now, we let [g] be the closed ideal in Lip ω generated by g.

Clearly [g] ⊆ J ω (E, U). Since E [g] = E and U [g] = U, then J ω (E, U) ⊆ [g],
by applying Theorem 1. Therefore J ω (E, U) = [g] and hence J ω (E, U) is principal. The proof of Corollary 1 is completed.

Approximation Lemmas.

We first note that ω is positive on ]0, 2] and since t → ω(t)/t is nonincreasing, The next lemma will be used to simplify the following proofs of Lemma

ω(2t) ≤ 2ω(t), t ∈ [0, 1]. (3.1)
gh n (ξ) -gh n (ζ) ω(|ξ -ζ|) = g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) + h n (ζ) g(ξ) -g(ζ) ω(|ξ -ζ|) , (3.4
lim n→∞ sup ξ,ζ∈T\E(δ) ξ =ζ |gh n (ξ) -gh n (ζ)| ω(|ξ -ζ|) = 0 (3.5)
for each δ ∈]0, 1], and

lim δ→0 sup ξ,ζ∈E(δ) ξ =ζ |gh n (ξ) -gh n (ζ)| ω(|ξ -ζ|) = 0 (3.6)
uniformly with respect to n ∈ N, then gh n ∈ J ω (E), for sufficiently large numbers n ∈ N, and lim n→∞ gh n ω = 0. Indeed, we suppose that (3.5) and (3.6) are satisfied. Let ε be a positive number. Using (3.6), there exists

δ ε ∈]0, 1] such that sup ξ,ζ∈E(δε) ξ =ζ |gh n (ξ) -gh n (ζ)| ω(|ξ -ζ|) ≤ ε, for every n ∈ N. (3.7)
We observe that

T \ E(δ ε ) ⊆ T \ E(δ ε /2).
Then, by using (3.5), there exists

n ε ∈ N such that sup ξ,ζ∈T\E(δε ) ξ =ζ |gh n (ξ) -gh n (ζ)| ω(|ξ -ζ|) ≤ ε, for every n ≥ n ε . (3.8)
On the other hand, for two points

ξ 0 ∈ E(δ ε ) and ζ 0 ∈ T \ E(δ ε ) we can associate a third point z ξ 0 ,ζ 0 in the boundary of E(δ ε ) such that max{|ξ 0 -z ξ 0 ,ζ 0 |, |z ξ 0 ,ζ 0 -ζ 0 |} ≤ |ξ 0 -ζ 0 |.
Then,

|gh n (ξ 0 ) -gh n (ζ 0 )| ω(|ξ 0 -ζ 0 |) ≤ |gh n (ξ 0 ) -gh n (z ξ 0 ,ζ 0 )| ω(|ξ 0 -z ξ 0 ,ζ 0 |) + |gh n (z ξ 0 ,ζ 0 ) -gh n (ζ 0 )| ω(|z ξ 0 ,ζ 0 -ζ 0 |) , (3.9)
since ω is nondecreasing. Combining the estimates (3.7), (3.8) and (3.9), we obtain

sup ξ,ζ∈T\E ξ =ζ |gh n (ξ) -gh n (ζ)| ω(|ξ -ζ|) ≤ 2ε, for every n ≥ n ε . (3.10)
It follows gh n ∈ I ω (E), for every n ≥ n ε , and hence gh n ∈ J ω (E), since we have assumed (3.6). We now observe that for each δ ∈]0, 1], there exists

n δ ∈ N such that T \ E(δ) ⊆ ∆ n , for all numbers n ≥ n δ , since n∈N ∆ n = T \ E.
Then, for each δ ∈]0, 1], the sequence {h n :

n ∈ N} converges uniformly to 0 on T \ E(δ), by using the hypothesis i 3 . Furthermore, by using hypothesis i 1 , lim

δ→0 sup ξ∈E(δ) n∈N |gh n (ξ)| = 0.
We deduce lim n→∞ gh n ∞ = 0. Hence lim n→∞ gh n ω = 0, by using once more (3.10).

So, it remains to show that (3.5) and (3.6) are fulfilled if (3.3) holds. For this we fix two distinct points ξ, ζ ∈ T \ E and a number δ ∈]0, 1]. We consider the following cases:

Case A. We suppose that ξ, ζ ∈ T \ E(δ). A 1 . If |ξ -ζ| ≥ 1 2 d(ξ, E), then ω(|ξ -ζ|) ≥ ω( 1 2 d(ξ, E)) ≥ 1 2 ω(d(ξ, E)),
by using the properties of ω. Thus In all what follows we use the notation X Y to mean that there exists a positive constant c such that X ≤ cY, where X and Y are two nonnegative functions. Since t → t/ω(t) is nondecreasing,

g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) ≤ 4 g ω sup z∈T\E(δ) |h n (z)|, n ≥ n δ . ( 3
g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) ≤ |g(ξ)| |h n (ξ) -h n (ζ)| |ξ -ζ| |ξ -ζ| ω(|ξ -ζ|) 2 g ∞ ω(2) sup z∈(ξ,ζ) |h ′ n (z)| g ∞ sup z∈T\E(δ) |h ′ n (z)|, n ≥ n δ . (3.12)
From (3.11) and (3.12) we deduce that for each δ ∈]0, 1], there exists

n δ ∈ N such that sup ξ,ζ∈T\E(δ) ξ =ζ g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) g ω sup z∈T\E(δ) |h n (z)| + sup z∈T\E(δ) |h ′ n (z)| , (3.13)
for all numbers n ≥ n δ . On the other hand it is plain that

sup ξ,ζ∈T\E(δ) ξ =ζ h n (ζ) g(ξ) -g(ζ) ω(|ξ -ζ|) g ω sup z∈T\E(δ) |h n (z)|, n ∈ N. ( 3
g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) ≤ 4 sup n∈N h n ∞ sup ξ∈E(δ) |g(ξ)| ω(d(ξ, E)) = o(1), as δ → 0, (3.15)
by using the hypothesis i 1 and the fact that g ∈ J ω (E). It follows

lim δ→0 sup ξ,ζ∈E(δ)\E ξ =ζ and |ξ-ζ|≥ 1 2 d(ξ,E) g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) = 0, (3.16)
and hence

lim δ→0 sup ξ,ζ∈E(δ)\E ξ =ζ g(ξ) h n (ξ) -h n (ζ) ω(|ξ -ζ|) = 0, (3.17)
uniformly with respect to n ∈ N, since we have assumed (3.3). On the other hand, by using again the hypothesis i 1 and the fact that g ∈ J ω (E), it is easily seen that

lim δ→0 sup ξ,ζ∈E(δ)\E ξ =ζ h n (ζ) g(ξ) -g(ζ) ω(|ξ -ζ|) = 0, (3.18)
uniformly with respect to n ∈ N. Hence, the equality (3.6) is deduced by combining together (3.4), (3.17) and (3.18). The proof of Lemma 1 is completed.

Let f ∈ A(D) be an outer function and let Γ ∈ Ω E , for some closed subset E of T. We define the following Korenblum's outer function

f Γ (z) := exp 1 2π Γ ξ + z ξ -z log |f |(ξ)|dξ| , z ∈ D. (3.19)
We observe that f Γ belongs to H ∞ with boundary values satisfying

|f Γ |(ξ) = |f |(ξ), if ξ ∈ Γ, 1, if ξ ∈ T \ Γ.
We also observe that f Γ = f × f -1 T\Γ and that f Γ has an extension both to an analytic function on C \ Γ and to a continuous one on D \ E. For a function f ∈ A(D), we denote by E f the boundary zero set of f. The following lemma will be used in the proofs of Proposition 3 and Proposition 4.

Lemma 2. Let f ∈ Lip ω be an outer function, where ω is a modulus of continuity satisfying (1.3). Let g ∈ Lip ω be a function and let E be a closed subset of E g . Then gf Γ ∈ Lip ω and

gf Γ ω ≤ c ω,f g ω , for every Γ ∈ Ω E , (3.20)
where c ω,f > 0 is a constant not depending on both Γ and g. If moreover g ∈ J ω (E), then

gf Γ ∈ J ω (E), for every Γ ∈ Ω E , (3.21)
and

lim n→∞ gf Γn -g ω = 0, (3.22)
where

Γ n := m>n (a m , b m ), for each n ∈ N. Proof. Let g ∈ Lip ω be a function. Let Γ ∈ Ω E , where E is a fixed closed subset of E g . Since f Γ ∈ H ∞ and has a continuous extension on D \ E, then gf Γ ∈ A(D) with vanishing values on E. Let ξ, ζ ∈ T \ E be two distinct points such that d(ξ, E) ≤ d(ζ, E). Two cases are possible; Case A. We suppose that |ξ -ζ| ≥ 1 4 d 2 (ξ, E). Since ω satisfies (1.3), g(ξ) f Γ (ξ) -f Γ (ζ) ω(|ξ -ζ|) ≤ 2 max{1, f ∞ } |g(ξ)| ω(|ξ -ζ|) ≤ 4η -1 ω max{1, f ∞ } |g(ξ)| ω(d(ξ, E)) (3.23) ≤ 4η -1 ω max{1, f ∞ } g ω . (3.24) Case B. We suppose that |ξ -ζ| ≤ 1 4 d 2 (ξ, E). In this case, we let (ξ, ζ) be the arc such that (ξ, ζ) ⊆ T \ E. We observe that d(z, E) ≥ d(ξ, E), for all z ∈ (ξ, ζ). B 1 . We assume that (ξ, ζ) ⊆ T \ Γ. We simply compute f Γ (ξ) -f Γ (ζ) ξ -ζ sup z∈(ξ,ζ) |f ′ Γ (z)| ≤ c f d 2 (ξ, E) ,
where c f > 0 is a constant not depending on both Γ and the points ξ and ζ. Since t → t/ω(t)

is nondecreasing, ω satisfies (1.3) and |ξ -ζ| ≤ 1 4 d 2 (ξ, E), we get g(ξ) f Γ (ξ) -f Γ (ζ) ω(|ξ -ζ|) = |g(ξ)||ξ -ζ| ω(|ξ -ζ|) |f Γ (ξ) -f Γ (ζ)| |ξ -ζ| ≤ c f |g(ξ)||ξ -ζ| ω(|ξ -ζ|)d 2 (ξ, E) ≤ c f η -1 ω 2 |g(ξ)| ω(d(ξ, E)) (3.25) ≤ c f η -1 ω 2 g ω . (3.26) B 2 . We now assume that (ξ, ζ) ⊆ Γ. We have |f T\Γ (ξ)| = |f T\Γ (ζ)| = 1.
The inequality (3.25) and the following equality

f Γ (ξ) -f Γ (ζ) = f (ξ)f -1 T\Γ (ξ) -f (ζ)f -1 T\Γ (ζ) = f -1 T\Γ (ξ) f (ξ) -f (ζ) + f (ζ) f -1 T\Γ (ξ) -f -1 T\Γ (ζ) = f -1 T\Γ (ξ) f (ξ) -f (ζ) -f (ζ)f -1 T\Γ (ξ)f -1 T\Γ (ζ) f T\Γ (ξ) -f T\Γ (ζ) (3.27)
give the following estimate

g(ξ) f Γ (ξ) -f Γ (ζ) ω(|ξ -ζ|) ≤ |g(ξ)| |f (ξ) -f (ζ)| ω(|ξ -ζ|) + |f (ζ)||g(ξ)| |f T\Γ (ξ) -f T\Γ (ζ)| ω(|ξ -ζ|) ≤ |g(ξ)| f ω + c f η -1 ω 2 f ∞ |g(ξ)| ω(d(ξ, E)) (3.28) ≤ (1 + c f η -1 ω 2 ) g ω f ω . (3.29)
Joining together the estimates (3.24), (3.26) and (3.29), we obtain

sup ξ,ζ∈T\E ξ =ζ g(ξ) f Γ (ξ) -f Γ (ζ) ω(|ξ -ζ|) ≤ C ω,f g ω , Γ ∈ Ω E ,
where C ω,f > 0 is a constant not depending on Γ and g. On the other hand, it is obvious that

sup ξ,ζ∈T\E ξ =ζ f Γ (ζ) g(ξ) -g(ζ) ω(|ξ -ζ|) ≤ f ∞ g ω , Γ ∈ Ω E . Therefore sup ξ,ζ∈T\E ξ =ζ gf Γ (ξ) -gf Γ (ζ) ω(|ξ -ζ|) ≤ C ω,f + f ∞ g ω , Γ ∈ Ω E ,
which proves (3.20), and so gf Γ ∈ Lip ω .

We now suppose additionally that g ∈ J ω (E). Then, by using (3.23), (3.25) and (3.28),

lim δ→0 sup ξ,ζ∈E(δ)\E ξ =ζ g(ξ) f Γ (ξ) -f Γ (ζ) ω(|ξ -ζ|) = 0, (3.30)
uniformly with respect to all Γ ∈ Ω E . On the other hand, it is plain that

lim δ→0 sup ξ,ζ∈E(δ)\E ξ =ζ f Γ (ξ) g(ξ) -g(ζ) ω(|ξ -ζ|) = 0, (3.31)
uniformly with respect to all Γ ∈ Ω E , since we have supposed g ∈ J ω (E). Therefore gf Γ ∈ J ω (E). Taking account of the estimate (3.30), we can now apply Lemma 1 with ∆ n = T \ {Γ n ∪ E} and h n = 1 -f Γn to deduce (3.22). This finishes the proof of Lemma 2.

4. Proof of Proposition 3.

The proof of the proposition will be given in subsection 4.2. Before this we need to point out some technical results. 4.1. Technical Lemmas. We start with the following classical lemma. Lemma 3. Let g ∈ J ω (E) be a function, where ω is an arbitrary modulus of continuity. Let A := {a k : 0 ≤ k ≤ K} be a finite sequence of points in E not necessarily distinct, where K ∈ N. Then

lim ρ→0 + ϕ ρ,K g -g ω = 0, (4.1)
where

ϕ ρ,K (z) := k=K k=0 za k -1 za k -1 -ρ , z ∈ D,
and ρ ≤ 1 is a positive number.

Proof. We will apply Lemma 1 with ∆ n = T \ E, for every n ∈ N. To prove (4.1) it is sufficient to show that

lim δ→0 sup ξ,ζ∈E(δ)\E ξ =ζ and |ξ-ζ|≤ 1 2 d(ξ,E) g(ξ) h ρ (ξ) -h ρ (ζ) ω(|ξ -ζ|) = 0, (4.2)
uniformly with respect to all positive numbers ρ ≤ 1, where h ρ := 1 -ϕ ρ,K . For this, we let ξ, ζ ∈ E(δ) \ E be two distinct points such that |ξ -ζ| ≤ 1 2 d(ξ, E). We consider the arc

(ξ, ζ) satisfying (ξ, ζ) ⊆ T \ E. A simple calculation gives d(z, E) ≥ 1 2 d(ξ, E), for all z ∈ (ξ, ζ),
and

|ϕ ′ ρ,K (z)| ≤ K d(z, E) , z ∈ T \ E.
Since t → t/ω(t) is nondecreasing, we obtain

g(ξ) h ρ (ξ) -h ρ (ζ) ω(|ξ -ζ|) = |g(ξ)| |ϕ ρ,K (ξ) -ϕ ρ,K (ζ)| |ξ -ζ| |ξ -ζ| ω(|ξ -ζ|) |g(ξ)| sup z∈(ξ,ζ) |ϕ ′ ρ,K (z)| |ξ -ζ| ω(|ξ -ζ|) ≤ 2K |g(ξ)| d(ξ, E) |ξ -ζ| ω(|ξ -ζ|) ≤ 2K |g(ξ)| ω(d(ξ, E)) . (4.3)
Since by hypothesis g ∈ J ω (E), the desired result (4.2) follows from (4.3).

The next lemma will be used to simplify the proof of the following one.

lim ε→0 φ 2 ρ,0 φ ρ,ε f γ\γε -φ 3 ρ,0 ω = 0,
where ρ is a fixed positive number and

φ ρ,ε (z) := zae -iε -1 zae -iε -1 -ρ zbe iε -1 zbe iε -1 -ρ , z ∈ D. Proof. Using (3.20), φ ρ,0 φ ρ,ε f γ\γǫ ω ≤ c ω,f φ ρ,0 ω φ ρ,ε ω , (4.4)
where c ω,f > 0 is a constant not depending on ε. Clearly

φ ρ,ε ω ≤ zae -iε -1 zae -iε -1 -ρ ω zbe iε -1 zbe iε -1 -ρ ω = z -1 z -1 -ρ 2 ω . (4.5) Thus φ ρ,0 φ ρ,ε f γ\γǫ ω ≤ c ω,f z -1 z -1 -ρ 4 ω . (4.6)
We set F := {a, b}. We have φ 2 ρ,0 φ ρ,ε f γ\γǫ ∈ J ω (F), as product of two functions φ ρ,0 and φ ρ,0 φ ρ,ε f γ\γǫ from I ω (F). Taking account of (4.5), (4.6) and the fact that φ ρ,ε ∞ ≤ 1, we calculate, for all distinct points ξ, ζ ∈ T, that

|φ 2 ρ,0 φ ρ,ε f γ\γǫ (ξ) -φ 2 ρ,0 φ ρ,ε f γ\γǫ (ζ)| ω(|ξ -ζ|) ≤ |φ ρ,0 φ ρ,ε f γ\γǫ (ξ)| |φ ρ,0 (ξ) -φ ρ,0 (ζ)| ω(|ξ -ζ|) + |φ ρ,0 (ζ)| |φ ρ,0 φ ρ,ε f γ\γǫ (ξ) -φ ρ,0 φ ρ,ε f γ\γǫ (ζ)| ω(|ξ -ζ|) ≤ |φ ρ,0 (ξ)|( f ∞ + 1) φ ρ,0 ω + |φ ρ,0 (ζ)| φ ρ,0 φ ρ,ε f γ\γǫ ω ≤ c ω,f,ρ |φ ρ,0 (ξ)| + |φ ρ,0 (ζ)| , (4.7)
where c ω,f,ρ > 0 is a constant not depending on ε. Then

lim δ→0 sup ξ,ζ∈F(δ) ξ =ζ |φ 2 ρ,0 φ ρ,ε f γ\γǫ (ξ) -φ 2 ρ,0 φ ρ,ε f γ\γǫ (ζ)| ω(|ξ -ζ|) = 0, (4.8)
and since φ 3 ρ,0 ∈ J ω (F), we get

lim δ→0 sup ξ,ζ∈F(δ) ξ =ζ |k ε (ξ) -k ε (ζ)| ω(|ξ -ζ|) = 0, (4.9)
uniformly with respect to ε, where k ε := φ 2 ρ,0 φ ρ,ε f γ\γε -φ 3 ρ,0 . We will now show that

lim ε→0 sup ξ,ζ∈T\F(δ) ξ =ζ |k ε (ξ) -k ε (ζ)| ω(|ξ -ζ|) = 0, (4.10)
for each number δ ∈]0, 1]. We first remark that the function k ε has an analytic extension across T \ F(δ), for sufficiently small numbers ε. A simple calculation gives For two distinct points ξ, ζ ∈ T \ F(δ), we clearly have two possible cases:

Case 1. |ξ -ζ| ≤ 1 2 d(ξ, F) or Case 2. |ξ -ζ| ≥ 1 2 d(ξ, F).
In the first case, by using the fact that t → t/ω(t) is nondecreasing

|k ε (ξ) -k ε (ζ)| ω(|ξ -ζ|) sup z∈(ξ,ζ) |k ′ ε (z)| |ξ -ζ| ω(|ξ -ζ|) sup z∈T\F(δ) |k ′ ε (z)|, (4.12)
where (ξ, ζ) is the open arc joining the points ξ and ζ, and such that (ξ, ζ) ⊆ T \ F(δ). In the second case, by using the fact that ω is nondecreasing and (3.1), we get

|k ε (ξ) -k ε (ζ)| ω(|ξ -ζ|) ≤ 2 sup z∈T\F(δ) |k ε (z)| ω( 1 2 d(ξ, F)) ≤ 4 ω(δ) sup z∈T\F(δ) |k ε (z)|. (4.13)
Hence the desired equality (4.10) is deduced from (4.11), (4.12) and (4.13). As we have done in the proof of Lemma 1 (see formulas from (3.5) to (3.10) above), we derive from the estimates (4.9) and (4.10) that lim ε→0 k ε ω = 0, which is the desired result of our lemma.

The following lemma will be used in the proof of both Proposition 3 and Proposition 4. Let ε be a nonzero real number such that γ ε := (ae iε , be -iε ) ⊂ γ. According to Lemma 2, the functions φ ρ,ε f γε and φ ρ,ε f T\γε belong to Lip ω , since φ ρ,ε ∈ I ω ({ae iε , be -iε }). Now, for π : Lip ω → Lip ω /I being the canonical quotient map, we clearly have

0 = π φ 2 ρ,ε gf = π φ ρ,ε gf T\γε π φ ρ,ε f γε . Then π(φ ρ,ε gf T\γε ) = 0, since the function π φ ρ,ε f γε is invertible. It follows φ ρ,ε gf T\γε ∈ I. Since g ∈ J ω (E I ) then gf Γ ∈ J ω (E I )
, by applying again Lemma 2. Using Lemma 4 and the following inequality

φ 2 ρ,0 φ ρ,ε gf T\γε -φ 3 ρ,0 gf Γ ω ≤ gf Γ ω × φ 2 ρ,0 φ ρ,ε f γ\γε -φ 3 ρ,0 ω , we obtain lim ε→0 φ 2 ρ,0 φ ρ,ε gf T\γε -φ 3 ρ,0 gf Γ ω = 0. (4.14)
We deduce that φ 3 ρ,0 gf Γ ∈ I, for all ρ > 0. On the other hand lim

Proof of Proposition 4.

To prove the proposition we need first to establish some lemmas.

Lemma 6. Let f ∈ Lip ω be a function, where ω is a modulus of continuity satisfying (1.3). Let E be a closed subset of E f and let S ∈ H ∞ be a singular inner function such that σ(S) ⊆ E and T dµ S (e iθ ) ≤ M, where M > 0 is a constant. Then Sf ∈ Lip ω and

Sf ω ≤ c ω,M f ω , (5.1)
where c ω,M > 0 is a constant not depending on f. If moreover f ∈ J ω (E) then Sf ∈ J ω (E).

Proof. We let ξ, ζ ∈ T \ E be two distinct points such that d(ξ,

E) ≤ d(ζ, E). Since Sf (ξ) -Sf (ζ) ω(|ξ -ζ|) = f (ξ) S(ξ) -S(ζ) ω(|ξ -ζ|) + S(ζ) f (ξ) -f (ζ) ω(|ξ -ζ|) (5.2) then Sf (ξ) -Sf (ζ) ω(|ξ -ζ|) ≤ f (ξ) S(ξ) -S(ζ) ω(|ξ -ζ|) + f ω . (5.3)
We have two situations:

Case A. We suppose that |ξ -ζ| ≥ 1 4 d 2 (ξ, E). Since ω satisfies (1.3), f (ξ) S(ξ) -S(ζ) ω(|ξ -ζ|) ≤ 2 |f (ξ)| ω( 1 4 d 2 (ξ, E)) ≤ 4η -1 ω |f (ξ)| ω(d(ξ, E)) (5.4) ≤ 4η -1 ω f ω . ( 5
(ξ, E) ≤ d(ζ, E), then d(z, E) ≥ d(ξ, E), z ∈ (ξ, ζ).
We have

|S(ξ) -S(ζ)| |ξ -ζ| sup z∈(ξ,ζ) |S ′ (z)| sup z∈(ξ,ζ) a S (z),
where

a S (z) := 1 π T 1 |e iθ -z| 2 dµ S (e iθ ), z ∈ T \ E. Then |S(ξ) -S(ζ)| |ξ -ζ| M d 2 (ξ, E) ,
and consequently

f (ξ) S(ξ) -S(ζ) ω(|ξ -ζ|) M |f (ξ)| d 2 (ξ, E) |ξ -ζ| ω(|ξ -ζ|) .
Therefore, by using the facts that t/ω(t) is nondecreasing and that ω satisfies (1.3),

f (ξ) S(ξ) -S(ζ) ω(|ξ -ζ|) Mη -1 ω |f (ξ)| ω(d(ξ, E))
(5.6)

Mη -1 ω f ω . (5.7)
Using the inequalities (5.3), (5.5) and (5.7) we deduce that Sf ω ≤ c ω,M f ω , and hence Sf ∈ Lip ω . It remains to prove that if f ∈ J ω (E) then so is for Sf. Using (5.2)

sup ξ,ζ∈E(δ) ξ =ζ Sf (ξ) -Sf (ζ) ω(|ξ -ζ|) ≤ sup ξ,ζ∈E(δ)\E ξ =ζ f (ξ) S(ξ) -S(ζ) ω(|ξ -ζ|) + sup ξ,ζ∈E(δ) ξ =ζ f (ξ) -f (ζ) ω(|ξ -ζ|) , (5.8)
for every δ ∈]0, 1]. The desired result follows by combining (5.4), (5.6) and (5.8) and using the hypothesis that f ∈ J ω (E).

For a function f ∈ H ∞ , we denote by B f and S f respectively the Blaschke product associated to Z U f , and the singular part of U f . Similarly, B I and S I are respectively the Blaschke product constructed from Z U I , and the singular part of U I . For completeness, we give the proof of the following classical lemma. Lemma 7. Let I ⊆ Lip ω be a closed ideal, where ω is an arbitrary modulus of continuity. There exists a sequence of functions {g k : k ∈ N} ⊆ I satisfying the following properties:

i. The greater common divisor of the singular functions S g k , k ∈ N, is equal to S I and

lim k→∞ µ Sg k -µ S I = 0.
ii. The sequence {B g k : k ∈ N} converges uniformly to B I on each compact subset of D.

Proof. We denote by I the closedness of I in A(D). According to Beurling-Rudin Theorem, we have I := {g ∈ A(D) : g/U I ∈ H ∞ and g(z) = 0, for every z ∈ E I }.

(5.9) Let now O ∈ A(D) be an outer function vanishing on E I . Since E I ⊇ σ(U I ) ∩ T, then U I O ∈ A(D). Thus U I O ∈ I, by using (5.9). We suppose that U I O / ∈ I, otherwise the lemma is obvious. By applying again (5.9), there exists a sequence of functions {f k : k ∈ N} ⊆ I converging uniformly to U I O. As it is done in [4, Lemma 1.3], we can extract a subsequence {g k : k ∈ N} of {f k : k ∈ N} satisfying the desired results of the lemma.

We also give the proof of the following simple lemma. Lemma 8. Let {S k : k ∈ N} be a sequence of singular functions such that supp(µ S k ) ⊆ F, for every k ∈ N, where F is a closed subset of T. We suppose that lim k→∞ µ S k -µ S = 0, for some singular function S. Then {S k : k ∈ N} converges uniformly to S on each compact subset of D \ F.

Proof. We let K be a compact subset of D \ F. It is obvious that

T e iθ + z e iθ -z dµ S k (e iθ ) - T e iθ + z e iθ -z dµ S (e iθ ) ≤ 2 µ S k -µ S δ K , z ∈ K, (5.10)
where δ K := inf z∈K d(z, F). We remark that δ K > 0. Using the estimate (5.10) and the simple fact that there exists a neighborhood U of 0 such that

|e z -1| ≤ 2|z|, z ∈ U,
we deduce that for sufficiently large numbers k ∈ N,

|S k (z) -S(z)| ≤ exp 1 2π T e iθ + z e iθ -z d(µ S -µ S k )(e iθ ) -1 ≤ 4 µ S k -µ S δ K , (5.11)
for all points z ∈ K. Which gives the desired result of the lemma.

With a singular function S ∈ H ∞ and a closed subset K ⊆ T we associate the following singular function

S K (z) := exp - 1 2π K e iθ + z e iθ -z dµ S (e iθ ) , z ∈ D.
Proof of Proposition 4. Let f ∈ I be a function. We define B {f,n} and B {I,n} to be respectively the Blaschke product with zeros Z U f ∩ D n and Z U I ∩ D n , where

D n := {z ∈ D : |z| < n n + 1 }, n ∈ N \ {0}.
For a fixed n ∈ N \ {0}, we set

J n := {g ∈ Lip ω : gB {I,n} ∈ I}.
Let {g k : k ∈ N} ⊆ I be the sequence of Lemma 7. Since Z B {I,n} is finite, then B {g k ,n} = B {I,n} for a sufficiently large number k ∈ N, by using the property (ii) of Lemma 7. Then g k /B {g k ,n} ∈ J n , for a large k ∈ N. Thus the zero set g∈Jn g -1 (0) of the ideal J n is contained in D \ D n , since g k /B {g k ,n} does not vanish on D n . Therefore B {f,n} is invertible in the quotient algebra Lip ω /J n .

We now let π n : Lip ω → Lip ω /J n be the canonical quotient map. Since f ∈ I then f ∈ J n , and hence

0 = π n f = π n f /B {f,n} π n B {f,n} .
Therefore π n f /B {f,n} = 0, and consequently f /B {f,n} ∈ J n . It follows that

B {I,n} (f /B {f,n} ) ∈ I. Since the sequence {B {I,n} (B f /B {f,n} ) : n ∈ N} converges uniformly on compact subsets of D \ E f to B I , and since S f O 2 f is continuous on D and vanishes on E f , then lim n→∞ B {I,n} (B f /B {f,n} )S f O 2 f -B I S f O 2 f ∞ = 0. Thus lim n→∞ B {I,n} (B f /B {f,n} )S f O 2 f -B I S f O 2 f ω = 0,
by using Theorem 2 and the fact that

f O f ∈ J ω (E f ), as product of two functions f, O f ∈ I ω (E f ). Therefore B I S f O 2 f ∈ I.
We let ε be a nonzero real number such that γ ε := (ae iε , be It is obvious that

E Lε = {ae iε , be -iε } ∪ γ ε ∩ E f . Since σ((S f ) γε ) ⊆ γ ε ∩ E f ⊆ E Lε , then S f γε L ε ∈
Lip ω , by applying Lemma 6. Furthermore, we remark that B I S f T\γε O f ∈ Lip ω , by using the F-property of Lip ω . We just proved above that

B I S f O 2 f ∈ I, then 0 = π L ε B I S f O 2 f = π S f γε L ε × π B I S f T\γε O 2 f ,
B I S f T\∆ N O 2 f ∈ I, where ∆ N := n≤N (a n , b n ) ∈ Ω E I
. As it is just done above, a simple application of Lemma 8 gives lim

N →∞ B I S f T\∆ N O 2 f -B I S f E I O 2 f ∞ = 0. Then lim N →∞ B I S f T\∆ N O 2 f -B I S f E I O 2 f ω = 0,
by using again Theorem 2. Therefore

B I S f E I O 2 f ∈ I.
We deduce

B I S g k E I O 2 g k ∈ I, for every k ∈ N,
and hence

B I S g k E I O 2 g k O 2 f ∈ I, for every k ∈ N, (5.12)
where {g k : k ∈ N} ⊆ I is the sequence of Lemma

B I S g k E I O 2 f -U I O 2 f ∞ = 0,
by using Lemma 8 and also the simple fact that B I O 2 f is continuous on D and vanishes on E I . Then

lim k→∞ B I S g k E I O 2 f -U I O 2 f ω = 0,
by using once more Theorem 2. Hence U I O 2 f ∈ I, which gives the desired result of the proposition.

Proof of Theorem 2.

Before giving the proof of the theorem, we need first to establish some needed results. Some of these results are actually inspired from [START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF][START_REF] Shirokov | Analytic functions smooth up to the boundary[END_REF][START_REF] Havin | Analytic functions with a Lipschitzian modulus of the boundary values[END_REF].

6.1. Some properties of functions from J ω (E). The next lemma will simplify the proof of the following one. Lemma 9. Let g ∈ J ω (E) be a function, where ω is an arbitrary modulus of continuity. For every ν ≥ 0, we have

exp 1 2π ξ∈T 1 -|z| 2 |ξ -z| 2 log |g(ξ) -g(z/|z|)| + ν |dξ| ≤ o(ω(1 -|z|)) + Aν, as d(z, E) → 0 and z ∈ D, (6.1)
where A ≥ 1 is an absolute constant.

Proof. Since g ∈ J ω (E),

lim δ→0 sup ξ,ζ∈E(δ) ξ =ζ |g(ξ) -g(ζ)| ω(|ξ -ζ|) = 0. (6.2)
Let ε ≤ 1 be a positive number. By using (6.2), there exists a positive number c ε such that

|g(ξ) -g(z/|z|)| ≤ ε ω(|ξ -z/|z||)
for every z ∈ D for which d(z, E) ≤ c ε and for every ξ ∈ T satisfying |ξ -z/|z|| ≤ c ε . For a point z ∈ D, we divide T into the following three parts

Γ 1 (z) := ξ ∈ T : |ξ -z/|z|| ≤ 1 -|z| ≤ c ε , Γ 2 (z) := ξ ∈ T : 1 -|z| ≤ |ξ -z/|z|| ≤ c ε , Γ 3 (z) := ξ ∈ T : c ε ≤ |ξ -z/|z|| .
As in the proof of [1, Lemma A.2] (with δ = ν ) we deduce that, for all points z ∈ D sufficiently close to E,

1 2π ξ∈T 1 -|z| 2 |ξ -z| 2 log |g(ξ) -g(z/|z|)| + ν |dξ| ≤ 1 2π Γ 1 (z) + 1 2π Γ 2 (z) + 1 2π Γ 3 (z) ≤ log ε ω(1 -|z|) + ν + c t≥1 log(t) t 2 dt -ε log(ε), (6.3)
where c > 0 is a constant. Hence (6.1) is deduced from (6.3).

An application of Lemma 9 provide us with the following proposition.

Proposition 5. Let ω be an arbitrary modulus of continuity. Then J ω (E) = K ω (E), where

K ω (E) := f ∈ I ω (E) : lim δ→0 sup d(z,E),d(w,E)≤δ z,w∈D and z =w |f (z) -f (w)| ω(|z -w|) = 0 .
Proof. It is obvious that J ω (E) ⊇ K ω (E). We now let g ∈ J ω (E) be a function and ε be a fixed positive number. We remark that, for z, w ∈ D such that min{|z|, |w|} ≥ We let z, w ∈ D be two distinct points such that min{|z|, |w|} ≥ 1 2 . In the first case, by using the facts that both ω and t → t/ω(t) are nondecreasing functions, and (3.1),

|g(z) -g(w)| ω(|z -w|) ≤ 2 g((z) -g( z |z| ) ω(1 -|z|) + 4 g( z |z| ) -g( w |w| ) ω(| z |z| -w |w| |) + 2 g(w) -g( w |w| ) ω(|1 -|w||) . (6.4)
Furthermore, by applying Lemma 9 with ν = 0,

g(z) -g( z |z| ) ≤ exp 1 2π ξ∈T 1 -|z| 2 |ξ -z| 2 log g(ξ) -g( z |z| ) |dξ| = o(ω(1 -|z|)), as d(z, E) → 0. (6.5)
Thus, by combining the estimates (6.4) and (6.5), and using our assumption that g ∈ J ω (E), we obtain

|g(z) -g(w)| ω(|z -w|) ≤ ε, (6.6)
if z, w ∈ D are two distinct points sufficiently close to E, and satisfying |z -w| ≥ 1 2 max{1-|z|, 1 -|w|}.

In the second case, it can be assumed without loss of generality that max{1 -|z|, 1 -|w|} = 1 -|z|. Since t → t/ω(t) is nondecreasing, we clearly have

|g(z) -g(w)| ω(|z -w|) ≤ sup |q-z|≤ 1 2 (1-|z|) |g ′ (q)| |z -w| ω(|z -w|) ≤ sup |q-z|≤ 1 2 (1-|z|) |g ′ (q)| 1 -|z| ω(1 -|z|) . (6.7)
On the other hand, for a point q ∈ D satisfying |q -z| ≤ 1 2 (1 -|z|), the classical Cauchy formula gives

g ′ (q) = 1 2iπ |p-z|= 3 4 (1-|z|) f (p) -f (z) (p -q) 2 dp. It follows |g ′ (q)| ≤ 1 2π |p-z|= 3 4 (1-|z|) |f (p) -f (z)| |p -q| 2 |dp|. (6.8) Then |g ′ (q)| ≤ 12 1 -|z| sup |p-z|= 3 4 (1-|z|) |f (p) -f (z)|. (6.9)
Thus, by using (6.6) and since ω is nondecreasing, We now arrive to the conclusion that g ∈ K ω (E), by joining together the estimates (6.6) and (6.11). Hence J ω (E) ⊆ K ω (E), which completes the proof of the proposition.

|g ′ (q)| ≤ ε ω(1 -|z|) 1 -|z| , ( 6

For an inner function

U ∈ H ∞ we set a U (ξ) := n 1 -|z n | 2 |ξ -z n | 2 + 1 π T 1 |e iθ -ξ| 2 dµ U (θ), ξ ∈ T \ E f ,
where z n , n ∈ N, are the zeros of U, each z n is repeated according to its multiplicity, and µ U is the positive singular measure associated with the singular factor of U, with the understanding that a U = 0 if U is constant. In [START_REF] Shirokov | Ideals and factorization in algebras of analytic functions that are smooth up to the boundary[END_REF] Shirokov proved

sup ζ∈T\E f |f (ζ)| ω(min{1, a -1 U f (ζ)})
< +∞, (6.12)

for every function f ∈ Lip ω , see also [START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF]Lemma B.5]. However, functions from the space J ω (E) admit more precise control than (6.12); Lemma 10. Let g ∈ J ω (E) be a function, where ω is an arbitrary modulus of continuity. Then ≤ 24Aε ω(a -1 Ug (ζ)). The lemma follows by joining together the results of the above two cases. 6.2. Proof of Theorem 2. Let g ∈ Lip ω be a function and V ∈ H ∞ (D) be an inner function dividing g. We have O g ∈ Lip ω and V O g ∈ Lip ω , by the F-property of Lip ω . Now, we suppose additionally that g ∈ J ω (E). We will first show that

  Lip ω (D) := f ∈ H ∞ : sup z,w∈D z =w |f (z) -f (w)| ω(|z -w|) < +∞ ,

  z∈D |f (z)| is the supremum norm. Similarly the weighted big Lipschitz algebra Lip ω (T) is defined by Lip ω (T) := f ∈ A(D) : sup z,w∈T z =w |f (z) -f (w)| ω(|z -w|) < +∞ .

I

  ω (E, U) := {f ∈ Lip ω : f |E ≡ 0 and f /U ∈ H ∞ }, andJ ω (E, U) := f ∈ I ω (E, U) : lim δ→0 sup ξ,ζ∈E(δ) ξ =ζ |f (ξ) -f (ζ)| ω(|ξ -ζ|) = 0 , where E(δ) := {ξ ∈ T : d(ξ, E) ≤ δ}, δ > 0,and d(z, E) designs the Euclidian distance from the point z ∈ D to the set E. It should be mentioned that for any ω, the closed ideal J ω (E, U) coincides with K ω (E, U) := f ∈ I ω (E, U) : lim δ→0 sup d(z,E),d(w,E)≤δ z,w∈D and z =w |f (z) -f (w)| ω(|z -w|) = 0 ,

For a closed

  subset E ⊆ T, we let {(a m , b m ) : m ∈ N} be a sequence of distinct arcs such that T \ E = m∈N (a m , b m ), where (a m , b m ) is an open arc of T \ E joining the points a m , b m ∈ E. We set Ω E := { m∈M (a m , b m ) : M ⊆ N}.

. 11 )A 2 .

 112 In the case where |ξ -ζ| ≤ 1 2 d(ξ, E), we consider the open arc (ξ, ζ) of T joining the points ξ and ζ, and such that (ξ, ζ) ⊆ T \ E(δ).

Lemma 4 .

 4 Let f ∈ Lip ω be an outer function, where ω is a modulus of continuity satisfying (1.3). Let γ := (a, b) be an open arc joining two points a, b ∈ E f such that γ ⊆ T \ E f . Let ε be a nonzero real number such that γ ε := (ae iε , be -iε ) ⊂ γ. Then

Lemma 5 .

 5 Let I ⊆ Lip ω be a closed ideal and let g ∈ J ω (E I ) be a function, where ω is a modulus of continuity satisfying(1.3). Let Γ ∈ Ω E I be such that T \ {Γ ∪ E I } isa finite union of arcs (a, b) ⊆ T \ E I (a, b ∈ E I ). We suppose that gf ∈ I for some outer function f ∈ Lip ω . Then gf Γ ∈ I. Proof. For simplicity we suppose that T \ {Γ ∪ E I } = (a, b) =: γ, where a, b ∈ E I and (a, b) ⊆ T \ E I .

ρ→0 φ 3 4 . 2 .

 342 ρ,0 gf Γ -gf Γ ω = 0, by applying Lemma 3. Therefore gf Γ ∈ I. Proof of Proposition 3. Let I ⊆ Lip ω be a closed ideal and let g ∈ J ω (E I ) be a function. We suppose that gf ∈ I, for some outer function f ∈ Lip ω . We set T \ E I = n∈N (a n , b n ), where (a n , b n ) ⊆ T \ E I and a n , b n ∈ E I . We have lim n→+∞ gf Γn -g ω = 0, by applying Lemma 2 with Γ n = m>n (a m , b m ). Then g ∈ I, since, by applying Lemma 5, gf Γn ∈ I for all n ∈ N.

. 5 ) 4 d 2

 542 Case B. We now suppose that |ξ -ζ| ≤ 1 (ξ, E). In this case we let (ξ, ζ) be the open arc joining the points ξ and ζ, and such that (ξ, ζ) ⊆ T \ E. Since we have assumed d

  -iε ) ⊂ γ := (a, b), where (a, b) ⊆ T \ E I is the open arc joining the points a, b ∈ E I . Using Lemma 2, we have L ε := p ε × O f γε ∈ Lip ω , where p ε (z) := (zae -iε -1)(zbe iε -1), z ∈ D.

  where π : Lip ω → Lip ω /I is the canonical quotient map. The function S f γε L ε is invertible in the quotient algebra Lip ω /I, since its zero set E Lε does not intersect with the spectrum σ(U I )∪E I of the Banach algebra Lip ω /I. Thus π B I S f T\γε O 2 f = 0, and hence B I S f T\γε O 2 f ∈ I. Using Lemma 8, we deduce that the sequence of elements S f T\γε converges uniformly on compact subsets of D \ E f to S f T\γ , when ε goes to 0. This fact and the clearly fact that B I O 2 f is continuous on D and vanishes on E f , give simply lim ε→0 B I S f T\γε O 2 f -B I S f T\γ O 2 f ∞ = 0. Then lim ε→0 B I S f T\γε O 2 f -B I S f T\γ O 2 f ω = 0, by using Theorem 2. Thus B I S f T\γ O 2 f ∈ I. We now argue similarly to arrive at

From Proposition 5 ,-ρ ζ 8 a

 58 2 |ξ -z| 2 log |g(ξ) -g(z/|z|)| + |g(z/|z|)| |dξ| , z ∈ D. (6.13) Then, using Lemma 9 with ν = |g(z/|z|)|, |O g (z)| ≤ o(ω(1 -|z|)) + A|g(z/|z|)|, as d(z, E) → 0. (6.14) |g(z) -g(ζ)| = o(ω(|z -ζ|)), as d(z, E), d(ζ, E) → 0. (6.15)We let ε ≤ 1 be a positive number. Using (6.14) and (6.15), there exists a positive numberδ ε such that |O g (z)| ≤ εω(|1 -|z|) + A|g(z/|z|)|,(6.16)and|g(z) -g(ζ)| ≤ εω(|z -ζ|),(6.17)whenever 1 -|z| ≤ δ ε , d(z/|z|, E) ≤ δ ε and d(ζ, E) ≤ δ ε . Let ζ ∈ E(δ ε ) \ E g be a point satisfying 8Aa -1 Ug (ζ) < 1.Two cases are possible;Case A. We assume that d(ζ, Z g ) ≤ 8Aa -1 Ug (ζ). Using (6.17)|g(ζ)| ≤ εω(d(ζ, Z g )). Then |g(ζ)| ≤ εω(8Aa -1 Ug (ζ)) ≤ 8Aεω(a -1 Ug (ζ)). Case B. We now assume that d(ζ, Z g ) ≥ 8Aa -1 Ug (ζ). Then 1 -ρ ζ ≤ d(ζ, Z g ), where ρ ζ := 1 -8Aa -1Ug (ζ). We have|U g (ρ ζ ζ)| ≤ exp -1 Ug (ζ) = e -A ,(6.18)as it is already computed in[START_REF] Bouya | Closed ideals in analytic weighted Lipschitz algebras[END_REF] Lemma B.4]. We observe that1 -ρ ζ ≤ d(ζ, Z g ) ≤ d(ζ, E g ) ≤ d(ζ, E) ≤ δ ε .From (6.16) and (6.18)|g(ρ ζ ζ)| = |U g (ρ ζ ζ)||O g (ρ ζ ζ)| ≤ Ae -A ε ω(1 -ρ ζ ) + |g(ζ)| . (6.19) Therefore |g(ζ)| ≤ |g(ζ) -g(ρ ζ ζ)| + |g(ρ ζ ζ)| ≤ ε ω(1 -ρ ζ ) + Ae -A ε ω(1 -ρ ζ ) + |g(ζ)| . Hence |g(ζ)| ≤ ε(1 -Ae -A ) -1 (1 + Ae -A )ω(1 -ρ ζ ) ≤ 3ε ω(1 -ρ ζ )

  to any inner function V dividing U g . Let ξ, ζ ∈ T\E g be two distinct points. We have two possibilities;Case A. We suppose that |ξ -ζ| ≥ 1 2 d(ζ, Z g ). Since g ∈ J ω (E), then |g(ζ)| ω(d(ζ, Z g )) = o(1), as d(ζ, E) → 0. (6.21)by using(6.15). It followsg(ζ) V (ξ) -V (ζ) ω(|ξ -ζ|) ≤ 2 |g(ζ)| ω(d(ζ, Z g )) = o(1), as d(ζ, E) → 0. (6.22) Case B. We now suppose that |ξ -ζ| ≤ 1 2 d(ζ, Z g ). Let (ξ, ζ) ⊂T \ E g be the arc joining the points ξ and ζ. We have1 2 |ζ -w| ≤ |z -w| ≤ 3 2 |ζ -w|, z ∈ (ξ, ζ) and w ∈ Z g . Since |V (ξ) -V (ζ)| |ξ -ζ| sup z∈(ξ,ζ) |V ′ (z)| sup z∈(ξ,ζ) n 1 -|z n | 2 |z -z n | 2 + 1 π T 1 |e iθ -z| 2 dµ Ug (θ) ,where {z n : n ∈ N} is the zero set of g in D and µ Ug is the positive singular measure associated with the singular factor of U g . Then|V (ξ) -V (ζ)| |ξ -ζ| a Ug (ζ).(6.23)Case B.1. We assume that |ξ -ζ| ≤ a -1 Ug (ζ). Since t → t/ω(t) is nondecreasing, and by applying Lemma 10, we have|g(ζ)|a Ug (ζ) |ξ -ζ| ω(|ξ -ζ|) ≤ |g(ζ)| ω(min{1, a -1 Ug (ζ)}) = o(1), as d(ζ, E) → 0. (6.24) Using (6.23) and (6.24), g(ζ) V (ξ) -V (ζ) ω(|ξ -ζ|) = |g(ζ)| |V (ξ) -V (ζ)| |ξ -ζ| |ξ -ζ| ω(|ξ -ζ|) = o(1), as d(ζ, E) → 0. (6.25)

  says that E is a boundary zero set of a function f ∈ Lip ωα \ {0} if and only if In this case E is called a Beurling-Carleson set. Recently, it is proved [2, Theorem 1] that if I is a closed ideal of Lip ωα , then J ωα (E I , U I ) ⊆ I. (1.2) The proof of this result uses the following stronger property: If E is a Beurling-Carleson set then there exists a function f holomorphic in D and continuously differentiable in D such that both f and f ′ vanish exactly on E. On the other hand, Shirokov [17, Chapter III] has shown that generally the boundary zero set of a function f ∈ Lip ω does not necessarily satisfy (1.1) as in the case of Lip χ

T log d(ξ, E)|dξ| > -∞.

(1.1) β , where

χ β (t) := log -β 2e β t , 0 < t ≤ 2,

and β > 0 is a fixed real number. According to these facts, we see that the method used in [2, Proof of Theorem 1] to prove (1.2) does not work for Lip χ β .

  Let g ∈ I ω (E) be a function, where ω is an arbitrary modulus of continuity. Let {∆ n ∈ Ω E : n ∈ N} be a sequence such that ∆ n ⊆ ∆ n+1 , for every n ∈ N, and We consider a sequence of functions {h n ∈ H ∞ : n ∈ N} such that; For each n ∈ N, the function h n has an extension both to a continuous function on D \ E and to an analytic function across ∆ n , i 3 . For each p ∈ N, the both sequences {h n : n ≥ p} and {h By definition, a function g ∈ I ω (E) is continuous on D and vanishes on E. Let {h n ∈ H ∞ : n ∈ N} be a sequence of functions satisfying the hypotheses i 1 , i 2 and i 3 of the lemma. Since h n is bounded and has an extension to a continuous function on D \ E, the product gh n possesses an extension to a function in A(D) with vanishing values on E, for every n ∈ N. Using the facts that h n ∈ H ∞ , g ∈ Lip ω and the following equality

	where n ∈ N. If moreover we suppose that g ∈ J ω (E) and
		lim δ→0	sup ξ,ζ∈E(δ)\E 2 d(ξ,E) ξ =ζ and |ξ-ζ|≤ 1	g(ξ)	h n (ξ) -h n (ζ) ω(|ξ -ζ|)	= 0,	(3.3)
	uniformly with respect to all numbers n ∈ N, then gh n ∈ J ω (E) for sufficiently large
	numbers n ∈ N, and lim n→∞	gh n ω = 0.
	Proof.
							2, Lemma 3 and
	Theorem 2.
	Lemma 1. i 1 . sup	h n ∞ < +∞,
	n∈N
	i 2 . ′ n : n ≥ p} converge
	uniformly to 0 on each compact subset K ⊆ ∆ p .
	Then gh n ∈ I ω (E) if and only if
			sup ξ,ζ∈T\E ξ =ζ	g(ξ)	h n (ξ) -h n (ζ) ω(|ξ -ζ|)	< +∞,	(3.2)

n∈N

∆ n = T \ E.

  .14) Taking account of the hypothesis i 3 , the equality (3.5) is deduced by combining together (3.4), (3.13) and (3.14). Case B. Now, we suppose that ξ, ζ ∈ E(δ) \ E. If we suppose additionally that |ξ -ζ| ≥

	1 2	d(ξ, E), then

  7. Using respectively the F-Property of Lip ω and Lemma 6, we obtain B I O f ∈ Lip ω and S g k E I O f ∈ Lip ω . Since this two functions vanish on E I , we obtain B I S g k E I

								O 2 f ∈ J ω (E I ). Thus, by applying Proposition
	3,
		B I S g k E I	O 2 f ∈ I,	for every k ∈ N,
	since we proved (5.12). We deduce that the singular functions S g k E I S I , and since	are all divided by
	µ	Sg k E I	-µ S I	≤ µ Sg k	-µ S I	,	for every k ∈ N,
	then, by using Lemma 7,
			lim k→∞	µ	Sg k E I	-µ S I	= 0.
	Thus
			lim k→∞

This paper was partially supported by ANR FRAB: ANR-09-BLAN-0058-02. The first author wishes to thank the ICTP of Trieste, where a part of this paper was achieved during his stay as a post-doctoral researcher in mathematics section.

Case B.2. We assume that |ξ -ζ| ≥ a -1 Ug (ζ). Then, by applying once more Lemma 10,

Hence, the equality (6.20) is deduced by combining (6.22), (6.25) and (6.26). We now clearly have

for all distinct points ξ, ζ ∈ T \ E g . By considering the particular case V = U g in the equality (6.27), we obtain

for all distinct points ξ, ζ ∈ T\ E g . Therefore V O g ∈ J ω (E), since by hypothesis g ∈ J ω (E), and combining the estimates (6.20) and (6.29). Which finishes the proof of (1.5). In particular we deduce that O g ∈ J ω (E).

We now let {V n : n ∈ N} ⊂ H ∞ be a sequence of inner functions dividing g, satisfying (1.6) and such that σ(V n ) ∩ T ⊆ E, for every n ∈ N. Then σ(V ) ∩ T ⊆ E. The equality (6.20) gives the following one,

uniformly with respect to n ∈ N, where h n := V n -V. Tacking account of (6.30) and the fact that O g ∈ J ω (E), the equality (1.7) is deduced by applying Lemma 1 with ∆ n = T \ E, for every n ∈ N. This completes the proof of the theorem.


Célestin C Kokonendji
email: celestin.kokonendji@univ-fcomte.fr

Tristan Senga Kiessé
email: tristan.senga-kiesse@onf.fr

Discrete associated kernels method and extensions

Keywords: Asymmetric kernel, dispersion, cross-validation, discrete distribution, finite difference, mean integrated squared error, nonparametric estimation, variable kernel, zero-proportion. MSC 2010: Primary 62G07 ; Secondary 62G99

Discrete kernel estimation of a probability mass function (p.m.f.) often mentioned in the literature has been far less investigated in comparison with continuous kernel estimation of a probability density function (p.d.f.). In this paper, we are concerned with a general methodology of discrete kernels for smoothing a p.m.f. f . We give a basic of mathematical tools for further investigations. First, we point out a generalizable notion of discrete associated kernel which is defined at each point of the support of f and built from any parametric discrete probability distribution. Then, some properties of the corresponding estimators are shown, in particular pointwise and global (asymptotical) properties. Other discrete kernels are constructed from usual discrete probability distributions such Poisson, binomial and negative binomial. For small samples sizes, underdispersed discrete kernel estimators are more interesting than the empirical estimator; thus, an importance of discrete kernels is illustrated. The choice of smoothing bandwidth is classically investigated according to cross-validation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning general probability function is discussed.

Introduction

Let X 1 , . . . , X n be independent and identically distributed (i.i.d.) random variables with an unknown probability density function (p.d.f.) f on R. A continuous kernel estimator f n of f can be defined in the two following ways:

f n (x) = 1 nh n i=1 K x -X i h (1) = 1 n n i=1 K x,h (X i ) , x ∈ R, (2)
where K(•) is the continuous kernel function which is typically a bona-fide p.d.f. with zero mean and unit variance, h = h(n) > 0 is an arbitrary sequence of smoothing parameters (or bandwidths) that fulfills lim n→∞ h(n) = 0, and K x,h (•) will be the "continuous associated kernel" with the target x and the bandwidth h (cf. Section 1 for details in discrete case). Following the well-known expression [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF] for unbounded supports of f , K(•) is classically symmetric and, therefore, the associated kernel is written as

K x,h (•) = 1 h K x -• h . (3)
But the way from ( 2) to ( 1) is not always possible like for asymmetric associated kernels with respect to the target x. In the expressions (1) and ( 2), the bandwidth plays the role of a dispersion parameter around the target; this can be easily illustrated through the symmetric gaussian associated kernel N x,h with mean x (the target) and standard deviation h (the bandwidth) where K = N 0,1 (e.g. Senga Kiessé [START_REF] Kiessé | Nonparametric Approach by Discrete Associated-Kernel for Count Data[END_REF], pages 172-174). The expression (1) is known since Rosenblatt [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF] and Parzen [START_REF] Parzen | On estimation of a probability density function and mode[END_REF]. For recent references, one can see Tsybakov [START_REF] Tsybakov | Introduction à l'Estimation Non-Paramétrique[END_REF]. The works usually cited of Devroye [START_REF] Devroye | A Course in Density Estimation[END_REF], Scott [START_REF] Scott | Multivariate Density Estimation -Theory, Practice, and Visualization[END_REF] and Silverman [START_REF] Silverman | Density Estimation for Statistics and Data Analysis[END_REF] concern some generalities on (supposed) continuous data. For functional data, one can refere to Ferraty and Vieu [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF]. The contributions of Simonoff [START_REF] Simonoff | Smoothing Methods in Statistics[END_REF] and Simonoff and Tutz [START_REF] Simonoff | Smoothing methods for discrete data[END_REF] are concerned with ordered categorical and discrete data always using the continuous kernels. The second expression [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF], that we will use in this paper, is from Chen [START_REF] Chen | Beta kernels estimators for density functions[END_REF][START_REF] Chen | Gamma kernel estimators for density functions[END_REF] for adapting a "type of continuous kernel" generally asymmetric (such beta and gamma) to the support of f ; see also Scaillet [START_REF] Scaillet | Density estimation using inverse and reciprocal inverse Gaussian kernels[END_REF] for inverse Gaussian and reciprocal inverse Gaussian kernels. The case of a bounded support (from two or one end) of f to estimate induces a choice of type of asymmetric kernel, while the symmetric continuous kernels K does not have any important proper effects and can be used indifferently for smoothing functions on unbounded supports.

In order to estimate a probability mass function (p.m.f.) on T (e.g. N + pN for p ≥ 0, Z d , {0, 1, . . . , N } d , d ∈ N\{0}) using a discrete kernel method, the empirical or naive estimator is often used because of its good asymptotical properties. However, this Dirac type kernel estimator is not appropriate with small samples sizes. Furthermore, its great default is that it does not take into account observations around the target because its bandwidth is null or does not exist. Except the naive estimator, Aitchison and Aitken [START_REF] Aitchison | Multivariate binary discrimination by the kernel method[END_REF] have been the pioneers of discrete kernel estimators in the sense of [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF]; see our Example 3. But the discrete kernel used has a unique shape and is appropriate for categorical data and finite discrete distributions; see also Li and Racine [START_REF] Li | Nonparametric Econometrics: Theory and Practice[END_REF] and references therein. Thus the case of discrete kernels for count data is not investigated in the sense of our works since [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF], except a first attempt of Marsh and Mukhopadhyay [START_REF] Marsh | Discrete Poisson kernel density estimation with an application to wildcat coal strikes[END_REF]. That attempt is only experimental and applied on univariate count data (i.e. T = N). A necessity of a discrete smoothing using discrete kernels out of the Dirac kernel is illustrated in Figure 3; for example, a binomial discrete kernel estimator is more interesting than the empirical estimator for a small sample size. Thus, we define and build a discrete associated kernel which asymptotically tends to the Dirac type kernel. It results in many applications of the discrete associated kernel method in literature such that nonparametric estimations of discrete weighted function (Kokonendji et al. [START_REF] Kokonendji | Semiparametric estimation for count data through weighted distributions[END_REF]) and regression count function (Kokonendji et al. [START_REF] Kokonendji | Appropriate kernel regression on a count explanatory variable and applications[END_REF]). The present work is concentrated on providing some theoretical mathematical tools and asymptotical results for the discrete associated kernel estimator. In this way, it completes the papers cited previously which practically show the usefulness of the discrete associated kernel approach.

In this paper, we review more generally the estimator (2) with discrete associated kernels for a p.m.f. f (x) = Pr(X i = x), for all x in T. We point out some ingredients necessary to the construction and study of discrete kernel estimators. The rest of the paper is organized as follows. Section 2 presents the definition of a discrete associated kernel with some examples. In Section 3, the basic properties of the corresponding estimator f n of f are given. The pointwise then global consistencies of f n in quadratic mean are therefore proved. Section 4 illustrates some aspects of the estimators using standard discrete kernels of Poisson, binomial and negative binomial; the underdispersed ones will be more interesting for small samples sizes than the consistent estimators. Section 5 is devoted to some remarks on the bandwidth selection and the importance of the kernel choice. A simulation study and some possible extensions to continuous kernel in view of unified method are presented.

Discrete associated kernel

In order to simplify we assume that the support T of the p.m.f. f , to be estimated, is the count set N. We then consider for T = N the topology inherited from the standard one of the real line number R. The easy and straightforward methodology of discrete associated kernels requires a clarification of the notions of integral, continuity and derivation of any discrete function f on T.

Let us consider on T the counting measure µ = y∈T δ y , where δ y denotes the Dirac mass at y. The integral on T 1 ⊆ T is the following summation:

T 1 f (x)µ(dx) = x∈T 1 f (x).
The continuity of f : T → R at x ∈ T ⊂ R can be defined as follows:

∀ ǫ > 0, ∃ η > 0 : ∀ y ∈]x -η, x + η[∩T ⇒ |f (y) -f (x)| < ǫ. (4)
Thus, we are not restrained to a particular form of continuation of discrete function. Moreover, we easily observe that any p.m.f. is bounded and continuous in the sense of (4). Note that, for η > 0 in (4), the notion of discrete neighbourhood ]x-η, x + η[∩T of x can be reduced to the single point {x}.

Finally, the finite difference of f on T is used instead of derivation on R; see, for example, Schumaker ([25], page 343), Agarwal and Bohner [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF] for other definitions.

In the case T = N of this work, we consider the finite difference f (k) (x) of order k ∈ N \ {0} at x ∈ N by the recursive relation:

f (k) (x) = f (k-1) (x) (1) with f (1) (x) =      {f (x + 1) -f (x -1)}/2 if x ∈ N {0} f (1) -f (0) if x = 0.
The f (k) (x) always exist and are some linear combinations of f (x ± j) for j ∈ {0, 1, • • • , k} and x ± j ∈ N. For instance, the particular case with k = 2 is given by

f (2) (x) =              {f (x + 2) -2f (x) + f (x -2)}/4 if x ∈ N {0, 1} {f (3) -3f (1) + 2f (0)}/4 if x = 1 {f (2) -2f (1) + f (0)}/2 if x = 0. (5)
We use these finite differences in the discrete Taylor expansion of f (x) at any point a ∈ T such that

f (x) = k j=0 f (j) (a) j! (x -a) j + o{(x -a) k } (6)
(see, for example, Schumaker [START_REF] Schumaker | Spline Functions: Basic Theory[END_REF], Theorem 8.61 -page 351). Note that if a / ∈ T then we have f (a) = 0. However, denoting by ⌊a⌋ ∈ T the nearest value (in the sense of usual topology of R) to a ∈ R T such that a = ⌊a⌋ ± η with η > 0, we extend the definition of f (a) as

f (a) = f (⌊a⌋ ± η) = f (⌊a⌋) ± ǫ, ǫ > 0 = f (⌊a⌋) ± ηf (1) (⌊a⌋) + o(η). (7)
Thus, the expansion (6) of f (x) can be done at any point a / ∈ T by using (7) of order 1. This is useful for studying a p.m.f. at its mean which would not belong in T.

Definition

The discrete associated kernel introduced in (2) is defined as follows.

Definition 1 Let T be the discrete support of the p.m.f. f , to be estimated, x a fixed target in T and h > 0 a bandwidth. A p.m.f. K x,h (•) on support S x (not depending on h) is said to be an associated kernel, if it satisfies the following conditions:

x ∈ S x , (8)
lim h→0 E(K x,h ) = x, (9)
lim h→0 V ar(K x,h ) = 0, ( 10
)
where K x,h is the discrete random variable whose p.m.f. is K x,h (•).

Note that the condition "h → 0 as n → ∞" does not need to be included in Definition 1 because the sample size n is not yet required.

In order to construct a discrete associated kernel K x,h from a parametric discrete probability distribution K θ , θ ∈ Θ ⊂ R d , on the support S θ such that S θ ∩ T = ∅, we need to establish a correspondence between (x, h) ∈ T × (0, ∞) and θ ∈ Θ. In what follows, we will call K ≡ K θ the type of discrete kernel to make a difference from the classical notion of continuous kernel [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF]. In this context, the choice of the discrete associated kernel becomes important as well as that of the bandwidth. Moreover, we distinguish the discrete associated kernels said sometimes of "second order" of those said of "first order" which verify the two first conditions ( 8) and [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF]; see Section 4 below.

Remarks. (i) Given a type of discrete kernel K, the construction of any discrete associated kernel is obviously not unique.

(ii) The condition (8) can be replaced by x∈T S x ⊇ T. This implies that the discrete associated kernel takes into consideration the support T of the p.m.f. f , to be estimated. We sometimes have S x = S (not depending on x) and, therefore, S x = T for all x. If x∈T S x is not equal to T then we have a problem of boundary bias.

(iii) The condition [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF] expresses that the information around the target is taken into account such that if h → 0 then we find again the kernel of the naive estimator at the limit of the mean. This basical condition points out that the discrete associated kernel of the estimator (2) is a kind of variable kernel. It allows for more flexibility to construct different discrete associated kernel from any discrete distribution K; for example, E(K x,h ) = x + h or E(K x,h ) = x. Thus, all the discrete associated kernels verifying [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF] share the property that the shape of kernel changes according to the value of the target x where they are calculated. The variable shape does not concern only the point mass at x but at each y ∈ S x such that the modal value stays in x as h → 0; see Figure 1 for the shape of the binomial kernel presented in Section 4. The amount of smoothing obtained changes depending on the behaviour of the variance Var(K x,h ) with respect to the target x.

(iv) The bandwidth parameter h > 0 allows to take into account the observations X i near to the target x ∈ T (in the sense of stochastic distance of the discrete kernel K). The local dispersion Var(K x,h ) at each x ∈ T shows the importance of the discrete associated kernel K x,h chosen which imposes its variance property for a consistency.

(v) The behaviour desired in [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF] allows the discrete associated kernel to tend to the kernel of the naive estimator which is the Dirac type kernel.

Note finally that, to specify the rate of convergence of the discrete random variable K x,h to x ∈ T, we may write the mean and the variance of K x,h as

E(K x,h ) = x + hA(x; h), with A(x; h) = o(1) (eventually uniformly), or E(K x,h ) = x + h 1/r o(1), with r ≥ 1, and V ar(K x,h ) = hB(x; h) + O(h 2 ), with B(x; h) = 0;
thus, the assumptions ( 8)-( 10) are still satisfied. However, we do not investigate here these previous hypotheses and we concentrate on discrete associated kernels satisfying the general assumptions of Definition 1.

Figure 1 about here

Examples

We present below some examples of discrete kernels that fullfils conditions ( 8)- [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF] for some counting, categorical or finite T.

Example 1. The well-known discrete empirical (or naive) estimator might be viewed as a particular case of the discrete associated kernel (2) upon taking

K x,h (y) = I y=x =      1 if y = x 0 if y = x, for any x ∈ T and any h ≥ 0, (11)
where I A denotes the indicator function of any given event A that takes the value 1 if the event A occurs and 0 otherwise. Note that the smoothing parameter h plays no role here and that the degenerate random variable D(x) associated to this discrete kernel satisfies (8), ( 9) and [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF] with

S x = {x}, E{D(x)} = x, Var{D(x)} = 0.
Example 2. The following class of symmetric discrete kernels has been proposed

by Kokonendji et al. [17]. It generalizes the classical triangular kernel and might be constructed as follows. First, the support T of the p.m.f f , to be estimated, can be unbounded (e.g. N, Z) or finite (e.g. {0, 1, • • • , N }). Then, suppose that h is a given bandwidth parameter and a is an arbitrary and fixed integer. For any x fixed in T, consider the random variable T a;x,h defined on S x = {x, x ± 1, • • • , x ± a} and whose p.m.f. is given by

K x,h (y) = (a + 1) h -|y -x| h P (a, h) , ∀y ∈ S x ,
where P (a, h) = (2a + 1)(a + 1) h -2 a k=0 k h is the normalizing constant. Since K x,h is symmetric around x, assumptions ( 8) and ( 9) are satisfied. As for the variance term [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF], note that, for a ∈ N fixed, one has

V (a, h) = 1 P (a, h) a(2a + 1)(a + 1) h+1 3 -2 a k=0 k h+2 (12)
≃ a(2a 2 + 3a + 1) 3 log(a + 1) -2 a k=1 k 2 log(k) h + O(h 2 ),
which does not depend on x = E(T a;x,h ) and tends to 0 when h → 0. The last approximation holds for h sufficiently small. Note that a R package1 for discrete triangular distributions is available. Recently, Kokonendji and Zocchi [START_REF] Kokonendji | Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions[END_REF] have introduced a general version of discrete triangular distributions which helps for solving problems of boundary bias.

Example 3. Aitchison and Aitken [START_REF] Aitchison | Multivariate binary discrimination by the kernel method[END_REF] have introduced a discrete kernel estimator for categorical or finite discrete distribution (see also, Li and Racine [START_REF] Li | Nonparametric Econometrics: Theory and Practice[END_REF]). Hence, we deduce its asymmetric discrete associated kernel that we present as follows. First, the support T of the p.m.f f , to be estimated, is finite with fixed size c ∈ N {0, 1}.

If the random variable X under investigation takes c different values, i.e. T := {0, 1, . . . , c -1} (say), then, the discrete kernel in (2) might be

K x,h (y) = (1 -h)I y=x + h c -1 I y =x , ∀y ∈ S x , (13)
where h belongs to (0, 1]. In addition, the target x can be considered as the reference point of X and the smoothing parameter h is such that 1-h is the success probability of the reference point. Finally, if the bandwidth h goes to 0, then, the random variable A c;x,h associated to K x,h will satisfy ( 8), ( 9), [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF]. Indeed, its support S x = S coincides with T and, its mean and variance are such that

E(A c;x,h ) = x + h 1 -x - x c -1 + hc 2 , Var(A c;x,h ) = - c 2 (-2x + c -1) 2 4(c -1) 2 h 2 + c(6x 2 + 2c 2 -3c + 1 -6xc + 6x) 6(c -1) h. (14)
The distribution of A c;x,h presents a uniform weight function on S x {x} outside the reference point x ∈ S x . Some graphs of the distribution of A c;x,h are presented in Senga Kiessé ([27], page 181).

Example 4. An extension of the discrete kernel in [START_REF] Kokonendji | Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions[END_REF] to the integers set Z has been proposed by Wang and Van Ryzin [START_REF] Wang | A class of smooth estimators for discrete distributions[END_REF]. More precisely, suppose that T = Z and, for any x in T and h in (0, 1), denote by Z x,h the random variable with support S x = Z and p.m.f.

K x,h (z) = (1 -h)I z=x + 1 2 (1 -h)h |z-x| I |z-x|≥1 , ∀z ∈ Z.
Then, provided that h → 0, this discrete kernel fulfills assumptions ( 8), ( 9) and ( 10) since

T = S x , E(Z x,h ) = x, and Var(Z x,h ) = h (1 + h) (1 -h) 2 .

Discrete associated kernel estimator

Let us give the first properties of the estimator (2) with a discrete associated kernel.

Proposition 1 Let X 1 , X 2 , . . . , X n be an n random sample i.i.d. from the unknown p.m.f. f on T. Let f n = f n,h,K be an estimator (2) of f with a discrete associated kernel. Then, for all x ∈ T and h > 0, we have

E{ f n (x)} = E{f (K x,h )},
where K x,h is the random variable associated to the p.m.f. K x,h on S x . Furthermore, we have f n (x) ∈ [0, 1] for all x ∈ T, and

x∈T f n (x) = C, where C = C(n; h, K) is a positive and finite constant if x∈T K x,h (y) < ∞ for all y ∈ T.
Proof : First, for all x ∈ T, we successively have

E{ f n (x)} = y∈T∩Sx K x,h (y)f (y) = y∈T∩Sx f (y) Pr(K x,h = y) = E{f (K x,h )}, because f (y)K x,h (y) = 0 for y / ∈ T ∩ S x ; thus, the formula is proved. Then, f n (x) ∈ [0, 1] proceeds immediately from K x,h (X i ) ∈ [0, 1]
for all X i . Finally, by writing

C = n -1 n i=1 { x∈T K x,h (X i )} for all h > 0 and by noting that x∈T K x,h (y) = x∈T∩Sx K x,h (y)
for all y ∈ T = support(X i ), it follows from it that:

-on the one hand C > 0 because K x,h (x) > 0 for y = x ∈ T ∩ S x and -on the other hand C < ∞ because 0 ≤ K x,h (y) < 1 for all y ∈ T ∩ S x and x∈T K x,h (y) < ∞ for all y ∈ T.

Notice that C = 1 for the estimators (2) with discrete Aitchison-Aitken associated kernel (Example 3) and symmetric continuous associated kernels such Gaussian. In general we have C = 1 for the estimators (2), as with discrete triangular associated kernels (Example 2) and standard discrete kernels in Section 4. In practice, we calculate the constant C depending on observations before normalizing f n to be a p.m.f. Without loss of generality, from now we assume C = 1.

In the following section, the asymptotic behaviour of the mean squared error (MSE) of f n (x) is examined together with its strong consistency and asymptotic normality; see Abdous and Kokonendji [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF] for details. Then the global consistency of f n (x) in the sense of the mean integrated squared error (MISE) is investigated with some illustrations using discrete kernels of Examples 1-3 in Section 2.2. A detailed study of the asymptotic bias and variance of the discrete triangular associated kernel estimator is already provided in Kokonendji et al. [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF].

Pointwise consistency and asymptotic normality

The first consistency result concerns the MSE of f n (x).

Theorem 1 Under assumptions ( 8)-( 10), for any fixed x in T, one has

lim n→∞ E{ f n (x) -f (x)} 2 = 0.
Proof: The usual bias-variance decomposition of MSE gives :

E{ f n (x) -f (x)} 2 = V ar{ f n (x)} + E{ f n (x)} -f (x) 2 .
Next, the bias term satisfies

E{ f n (x)} -f (x) = y∈T∩Sx [f (y) -f (x)] Pr(K x,h = y) -f (x) y∈T∩S x Pr(K x,h = y) = : B 1 + B 2 ,
where T denotes R \ T. In the first quantity B 1 , the sum runs over y's belonging to T ∩ S x and such that y = x. Thus, since both T and S x are discrete sets, one can find a finite constant η x > 0 such that |y -x| ≥ η x for any y in T ∩ S x \ {x}. Consequently, we can write

|B 1 | ≤ 2 Pr K x,h ∈ T ∩ S x and |K x,h -x| ≥ η x ≤ 2 Pr |K x,h -x| ≥ η x ≤ 2 η 2 x E{(K x,h -x) 2 }.
The last inequality follows from the Tchebychev-Markov inequality. To conclude that B 1 goes to 0 as n → ∞, note that

E{(K x,h -x) 2 } = V ar(K x,h ) + {E(K x,h ) -x} 2
and use assumptions ( 9) and [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF].

Next, to show that the second quantity B 2 converges to 0, remark that if y ∈ T ∩ S x then necessarily y = x and consequently it satisfies |y -x| ≥ η x for some finite η x > 0. Similar arguments as those used above enable to conclude.

The convergence of the variance term stems from

V ar{ f n (x)} ≤ 1 n y∈T∩Sx f (y){Pr(K x,h = y)} 2 ≤ 1 n .
Remark. The assumptions under which the MSE of the discrete kernel estimator converges to zero, might, at first glance, appear striking. Indeed, we do not impose any apparent assumption on the bandwidth h = h(n), but as we saw in the examples presented in Section 2, assumptions ( 9) and ( 10) hold provided that lim Theorem 2 Under assumptions ( 8)- [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF], for any fixed x in T, we have

f n (x) a.s. -→ f (x) as n → ∞, (15)
where a.s.

-→ denotes almost sure convergence.

Theorem 3 Under assumptions ( 8)- [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF], for any fixed x in T such that f (x) > 0, we have

f n (x) -E{ f n (x)} V ar f n (x) 1/2 d -→ N (0, 1) as n → ∞, (16)
where d -→ denotes convergence in distribution and N (0, 1) is the standard normal distribution.

Global consistency and illustrations

The criterion to use for this consistency is the MISE of f n = f n,h,K defined as

M ISE = x∈T Var{ f n (x)} + x∈T bias 2 { f n (x)} = M ISE(n, h, K, f ). ( 17
)
Theorem 4 Let f be a p.m.f on T with lim x→∞ f (x) = 0. Then, the estimator (2) f n = f n,h,K of f with any discrete associated kernel is such that, for n → ∞ and h = h(n) → 0, we have the behaviour

M ISE(n, h, K, f ) = 1 n x∈T f (x)[{Pr(K x,h = x)} 2 -f (x)] + x∈T f {E(K x,h )} -f (x) + 1 2 Var(K x,h )f (2) (x) 2 + o 1 n + h 2 ,
where f (2) is the finite difference of second order given in [START_REF] Chen | Beta kernels estimators for density functions[END_REF].

Proof: Without loss of generality, we assume S x ⊆ T for all x ∈ T. The pointwise variance of f n can be written around the target x (which realizes the modal probability of K x,h ) as:

Var{ f n (x)} = 1 n    y∈Sx f (y) {Pr(K x,h = y)} 2 -    y∈Sx f (y) Pr(K x,h = y)    2    = 1 n f (x){Pr(K x,h = x)} 2 - 1 n f 2 (x) + R n (x; h), (18)
where the rest

R n (x; h) = 1 n y∈Sx\{x} f (y) {Pr(K x,h = y)} 2 + 1 n f 2 (x) - 1 n   f (x) + y∈Sx {f (y) -f (x)} Pr(K x,h = y)   2
is o(1/n) under the hypothesis of discrete associated kernel; thus for all x ∈ T, R n (x; h) → 0 when n → ∞ and h = h(n) → 0. Indeed, let y ∈ S x \ {x} we can find a constant η = η(y) > 0 such that

0 ≤ Pr(K x,h = y) ≤ Pr(|K x,h -x| > η) ≤ 1 η 2 E{(K x,h -x) 2 } = 1 η 2 [Var(K x,h ) + {E(K x,h ) -x} 2 ] → 0 when h → 0,
and for y = x we deduce the asymptotic modal probability Pr(K x,h = x) → 1 when h → 0. Furthermore, we successively have:

-f (x){Pr(K x,h = x)} 2 ≤ 0, - 1 n x∈T f (x){Pr(K x,h = x)} 2 + 1 n x∈T f 2 (x) ≤ 1 n x∈T f 2 (x), x∈T R n (x; h) ≤ 1 n x∈T Var{ f n (x)} + 1 n x∈T f 2 (x).
Using Var{ f n (x)} < ∞ and lim n→∞ Var{ f n (x)} = 0 with 0 < x∈T f 2 (x) < 1 and R n (x; h) ≥ 0, we get x∈T R n (x; h) → 0 when n → ∞ and h = h(n) → 0; hence, the result. Concerning the pointwise bias of f n , we successively use the formula of Proposition 1 then the discrete Taylor expansion (6) of f (K x,h ) at the point E(K x,h ) with ( 7) to obtain bias{

f n (x)} = E{f (K x,h )} -f (x) = f {E(K x,h )} -f (x) + 1 2 Var(K x,h )f (2) (x) + o(h). (19)
Thus, the theorem ensues from ( 18) and [START_REF] Marron | A comparison of cross-validation techniques in density estimation[END_REF] in the criterion of MISE [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF].

Application 1. In the very particular case of the Dirac type kernel [START_REF] Johnson | Univariate Discrete Distributions[END_REF] estimator and unbiased f n,0,D , the MISE ( 17) is equal to the integrated variance

M ISE(n, 0, D, f ) = 1 n x∈T f (x) {1 -f (x)} = 1 n    1 - x∈T f 2 (x)    .
This exact result is used as reference in comparison to the MISE of the others discrete associated kernel estimators, because 0 ≤ x∈T f 2 (x) < 1 and therefore we have the global consistency of the naive estimator as M ISE(n, 0, D, f ) → 0 when n → ∞.

Application 2. For an estimator (2) with a discrete triangular associated kernel (cf. Example 2) with a ∈ N * fixed, the exact

M ISE(n, h, T a , f ) is obtained from bias{ f n (x)} = f (x) (a + 1) h P (a, h) -1 + y∈Sx\{x} f (y) Pr(T a;x,h = y)
which tends to 0 when h → 0, and

Var{ f n (x)} = 1 n   f (x) (a + 1) h P (a, h) 2 + y∈Sx\{x} f (y) {Pr(T a;x,h = y)} 2   - 1 n   f (x) + y∈Sx {f (y) -f (x)} Pr(T a;x,h = y)   2
which tends to the variance n -1 f (x) {1f (x)} of the naive estimator when h → 0. Theorem 4 leads to the expression of M ISE(n, h, T a , f ) as

M ISE(n, h, T a , f ) = 1 n x∈T f (x)   (a + 1) h P (a, h) 2 -f (x)   + 1 4 {V (a, h)} 2 x∈T f (2) (x) 2 +o 1 n + h 2 .
Thus, we have:

M ISE(n, h, T a , f ) = o(1/n + h 2 ) → 0 when n → ∞ and h = h(n) → 0; because lim h→0 (a + 1) h /P (a, h) = 1, 0 ≤ x∈T f (x) {1 -f (x)} < 1, lim h→0 V (a, h) = 0 and x∈T f (2) (x)
2 < ∞ from (5). Hence, the estimators [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF] with discrete triangular associated kernels are consistent in the sense of MISE.

Application 3. Concerning the estimator (2) with a discrete Aitchison-Aitken associated kernel (see Example 3) with c ∈ N {0, 1} fixed, the exact

M ISE(n, h, A c , f ) is obtained from bias{ f n (x)} = -hc c -1 f (x) + h c -1 c-1 i=0 f (i),
which also tends to 0 when h → 0, and

Var{ f n (x)} = 1 n f (x)(1 -h) 2 + h 2 (c -1) 2 c-1 i=0 f (i) -f (x) - 1 n f (x)(1 -h) + h c -1 c-1 i=0 f (i) -f (x) 2
which also tends to the variance n -1 f (x) {1f (x)} of the naive estimator when h → 0. Using Theorem 4, the M ISE(n, h, A c , f ) is given by

M ISE(n, h, A c , f ) = 1 n x∈T f (x){(1 -h) 2 -f (x)} + x∈T f {E(A c;x,h )} -f (x) + 1 2 Var(A c;x,h )f (2) (x) 2 + o 1 n + h 2 .
Consequently, we have:

M ISE(n, h, A c , f ) = o(1/n + h 2 ) → 0 as n → ∞ and h = h(n) → 0; because, for the variance part, 0 ≤ x∈T f (x) {1 -f (x)} < 1 and, for the bias part, lim h→0 f {E(A c;x,h )} -f (x) + (1/2)Var(A c;x,h )f (2) (x) = 0.
Hence, the global consistency in quadratic mean of the estimator (2) with discrete

Aitchison-Aitken associated kernels is provided.

Other discrete kernels

In this section, we examine the case of the so-called standard discrete kernels which are discrete associated kernels of the first order (i.e. not verifying the condition [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF] in Definition 1). They are built from usual discrete probability distributions of Poisson, binomial and negative binomial (see Johnson et al. [START_REF] Johnson | Univariate Discrete Distributions[END_REF]). They are also useful for smoothing a p.m.f f on T = N or distributions of count data with small samples (see also Senga Kiessé [START_REF] Kiessé | Nonparametric Approach by Discrete Associated-Kernel for Count Data[END_REF], Chapter 1).

For all x ∈ N and h > 0, the discrete random variable K x,h of standard discrete kernels satisfies, among other, the condition

lim h→0 Var(K x,h ) ∈ V(0) (20)
which takes the place of ( 10) and where V(0) is a neighbourhood of 0 which does not depend on x. Here we present some standard discrete kernels such that E(K x,h ) = x + h; thus, the condition (9) holds for K x,h . In fact, it is more appropriate at the bound x = 0 and, in general, the target x is obviously not the mean of K x,h which is asymmetric but rather its mode. From Theorem 4, the condition (20) does not allow the consistency in the sense of MISE of the corresponding estimators (2).

However, the standard discrete kernels estimators (2), a fortiori with small variances or underdispersed (variance ≤ mean), can be more interesting (in the sense of small MISE) for small samples sizes than the estimators (2) with discrete associated kernels or Dirac type kernel for which they have some good asymptotical properties. We give an element of proof by a graphic comparison of different exact MISE of a p.m.f. First, we present each of the three standard discrete kernels; then we give the bias and the variances of the corresponding estimators (2) which are sufficient to deduce their MISE using [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF].

Poisson kernel

Consider a Poisson distribution P(λ) with λ > 0. For any x fixed in T = N and h > 0, the corresponding random variable P x,h associated to the Poisson kernel P x,h follows the distribution P(x + h) with support S x = N and p.m.f.

P x,h (y) = (x + h) y e -(x+h) y! , ∀y ∈ N.
Note that the discrete kernel proposed by Marsh and Mukhopadhyay (1999) inverts x and y in the expression P x,h (y) above and does not allow any mathematical study of properties. Our Poisson kernel P x,h is equidispersed (i.e. E(P x,h ) = Var(P x,h ) = x+h)

and has its mode between x + h -1 and x + h. Then, the discrete kernel P x,h fullfills assumptions ( 8) and ( 9) except [START_REF] Marsh | Discrete Poisson kernel density estimation with an application to wildcat coal strikes[END_REF]. The corresponding estimator f n has the pointwise bias bias{

f n (x)} = f (x) {P x,h (x) -1} + y∈N\{x} f (y)P x,h (y)
which does not tend to 0 when h → 0. Its pointwise variance can be written as

nVar{ f n (x)} = f (x)P 2 x,h (x)+ y∈N\{x} f (y)P 2 x,h (y)-   f (x) + y∈N {f (y) -f (x)}P x,h (y)   2 .
This f n is not consistent in the sense of small MISE but can be more interesting than the naive estimator, for small or moderate samples sizes; see Figure 2 .

Binomial kernel

If we consider a binomial distribution B(N, p) with N ∈ N {0} and p ∈ (0, 1], we associate the random variable B x,h corresponding to the binomial kernel B x,h following the distribution B{x + 1, (x + h)/(x + 1)} on S x = {0, 1, • • • , x + 1} for any x ∈ N and h ∈ (0, 1]:

B x,h (y) = (x + 1)! y!(x + 1 -y)! x + h x + 1 y 1 -h x + 1 x+1-y , ∀y ∈ S x ⊆ N.
It is an underdispersed discrete kernel (i.e. Var(B x,h ) = (x+h)(1-h)/(x+1) smaller than E(B x,h ) = x + h) having its mode around x + h. The binomial kernel satistfies the three assumptions ( 8), ( 9) and ( 20) with V(0) = [0, 1). The bias and the variance of the corresponding estimator (2), for any x ∈ N, are written as:

bias{ f n (x)} = f (x) {B x,h (x) -1} + y∈Sx\{x} f (y)B x,h (y)
and

nVar{ f n (x)} = f (x)B 2 x,h (x)+ y∈Sx\{x} f (y)B 2 x,h (y)-   f (x) + y∈Sx {f (y) -f (x)}B x,h (y)   2 .
The MISE of this estimator is not consistent but can be more smaller than those of estimators with discrete associated kernels and Dirac type kernel for some sample sizes not so large (see also Figure 2 below).

Binomial negative kernel

In the case of the negative binomial distribution BN (λ, p) with λ > 0 and p > 0, we define the binomial negative kernel BN x,h with the random variable BN x,h following the distribution BN {x + 1, (x + 1)/(2x + 1 + h)} on S x = N for any x ∈ N and h > 0:

BN x,h (y) = (x + y)! x!y! x + h 2x + 1 + h y x + 1 2x + 1 + h x+1 , ∀y ∈ N.
It is an overdispersed discrete kernel (i.e. Var(BN x,h ) = (x + h){1 + (x + h)/(x + 1)} greater than E(BN x,h ) = x + h) having its mode around x + h. This dicrete kernel satisties assumptions (8), [START_REF] Ferraty | Nonparametric Functional Data Analysis: Theory and Practice[END_REF] but not [START_REF] Marsh | Discrete Poisson kernel density estimation with an application to wildcat coal strikes[END_REF]. For an estimator (2) with a negative binomial kernel, the pointwise bias is given as

bias{ f n (x)} = f (x) {BN x,h (x) -1} + y∈N\{x} f (y)BN x,h (y)
and the pointwise variance can be written as

nVar{ f n (x)} = f (x)BN 2 x,h (y)+ y∈N\{x} f (y)BN 2 x,h (y)-   f (x) + y∈N {f (y) -f (x)}BN x,h (y)   2 .
Similarly to the previous cases, this estimator f n is not consistent in the sense of small MISE; but, it can be more interesting than the naive estimator for some small samples sizes (see Figure 2 below).

Remark. For standard discrete kernel estimators, there exists a problem of boundary bias which can be solved as for continuous kernel estimators; see Zhang and Karunamuni [START_REF] Zhang | Some improvements on a boundary corrected kernel density estimator[END_REF] and also Chen [START_REF] Chen | Beta kernels estimators for density functions[END_REF][START_REF] Chen | Gamma kernel estimators for density functions[END_REF]. Here, a solution is to use the Dirac type kernel at the bound x = 0 and a standard discrete kernel for any x ∈ N \ {0}.

For discrete triangular associated kernel estimators, this problem has arleady been solved in [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF]; see also Kokonendji and Zocchi [START_REF] Kokonendji | Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions[END_REF].

Comparisons of MISE: Figure 2 presents the comparative results of MISE of the estimators (2) with discrete kernels for the p.m.f.

f (x) = 0.4 e -0.5 0.5 x x! + 0.6 e -10 10 x x! , ∀x ∈ N,
which is a mixture of two Poisson distributions with respective means µ 1 = 0.5 and µ 2 = 10. All computations were done by using the R software [START_REF] Development | R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing[END_REF]. The intersection points between the MISE curves of each estimators (2) with discrete kernels and those of the MISE of the naive estimator point out the superior limit of n for which these estimators are more efficient than the naive. Beyond this limit, the naive estimator is better and its MISE tends to 0 as for the estimator (2) with a discrete triangular associated kernel under the condition that h is very small (→ 0). Among estimators (2) with standard discrete kernels, the binomial one seems consistent for h small (here 0.1) but it is only a visual effect. However, the binomial kernel estimator is more interesting than the others here for small or moderate samples sizes and h ≤ 0.3. The last observation is also illustrated through some discrete smoothing using appropriate bandwidth selection in Figure 3.

Bandwidth selection

The bandwidth choice is generally realized in the sense of MISE ( 17) by approaching the ideal value of the bandwidth defined as

h id = arg min h>0 M ISE(n, h, K, f ) = h id (n, K, f ).
Several methods already existing for continuous kernels can be adapted to the discrete case as the classical least-squares cross-validation method; see, for example, Bowman [START_REF] Bowman | An alternative method of cross-validation for the smoothing of density estimates[END_REF], Marron [START_REF] Marron | A comparison of cross-validation techniques in density estimation[END_REF] and references therein. We simply propose two choices of bandwidth without making here a study on their consistencies.

Thus, for a given discrete kernel K x,h with x ∈ T and h > 0, we can prove that the optimal bandwidth h cv of h is obtained by cross-validation as

h cv = arg min h>0 CV (h),
where

CV (h) = x∈T { f n (x)} 2 - 2 n n i=1 f n,-i (X i ) = x∈T 1 n n i=1 K x,h (X i ) 2 - 2 n(n -1) n i=1 j =i K X i ,h (X j ) with f n,-i (y) = (n -1) -1 j =i K y,h (X j
) being computed as f n (y) by excluding the observation X i . This method is applied to all the estimators (2) with discrete kernels cited in this paper, independently on the support T of f to be estimated.

Let us consider the particular situation of count data (T = N) for which there exists a proportion of excess zeros n 0 = ♯ (X i = 0) in the sample X 1 , X 2 , . . . , X n . That corresponds to a well-known phenomenon (e.g. Kokonendji et al. [START_REF] Kokonendji | Connections of the Poisson weight function to overdispersion and underdispersion[END_REF], and references therein) for that we can use the adapted bandwidth h 0 = h 0 (n; K, f ) of h as solution of n i=1 Pr (K X i ,h 0 = 0) = n 0 .

This last equation ensues from the expression E{ f n (x)} = y∈N f (y) Pr(K x,h = y), where we take y = 0 and f (0) = 1 to identify the number of theoretical zeros into the empirical number of zeros n 0 . According to the importance of zero-proportion in the sample, this bandwidth h 0 can be interesting (see Figure 3).

Table 1 about here

Finally, this selection method of h 0 by excess zeros does not apply to all discrete kernels. Indeed, h 0 exists for the three standard discrete kernels (see Table 1: explicit for Poisson and implicit for the other ones); but h 0 does not exist for discrete triangular associated kernels (see Kokonendji et al. [17] and Senga Kiessé [START_REF] Kiessé | Nonparametric Approach by Discrete Associated-Kernel for Count Data[END_REF], pages 84-85) and the method is not applicable to discrete Aitchison-Aitken associated kernel for categorical data.

Simulation study

We consider simulated data from a Poisson distribution with mean µ = 2. In order to measure the performance of estimators, we use the practical

ISE = x∈N f n (x) -f (x) 2 = ISE(n, h, K, f )
and the theoretical MISE in [START_REF] Kokonendji | Discrete triangular distributions and non-parametric estimation for probability mass function[END_REF]. In Table 2, we calculate the optimal average of ISE and their standard errors for the estimators based on 1000 replications. For each simulation, the optimal discrete smoothing bandwidths are given by the cross-validation method. The optimal ISE are determined by using the optimal bandwidths. In general, for small or moderate samples sizes, the results in Table 2 show that discrete kernel estimators are more interesting that the empirical estimator. In addition, the simulated sum of the rest R n in ( 18) is calculated in Table 3. On the one hand, we have x∈N R n (x; h) → 0 as n → ∞ for discrete triangular associated kernels with a = 1 and a = 2; on the other hand, the binomial kernel has a sum x∈N R n (x; h) smaller than those of the two others standard discrete kernels.

Tables 2 and 3 about here

About discrete kernel choice

Similarly, the ideal discrete kernel satisfies

K id = arg min K M ISE(n, h, K, f ) = K id (n, h, f ).
Since the discrete (associated) kernel K x,h depends on the support T of f and also on each target x ∈ T, we have to restrict us to a specifical class of discrete kernels for realizing the optimization.

Thus, without loss of generality, we consider two random variables K [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF] x,h and K [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF] x,h connecting to discrete associated kernels (of first or second order) K [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF] x,h and K [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF] x,h on comparable supports S [1] x and S [2] x , respectively. Up to

E K [1] x,h = E K [2]
x,h , ∀x ∈ T and h > 0, [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] the discrete kernel K [1] is said to be better than the discrete kernel K [2] if and only if Var K [START_REF] Abdous | Consistency and asymptotic normality for discrete associated kernel estimator[END_REF] x,h ≤ Var K [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF] x,h , ∀x ∈ T and h > 0. The ingredients of this criterion of efficiency are partially presented in the Theorem 4 for n large and h small. For example, the better discrete kernel in the family of triangular T a with a ∈ N * is obtained for a = 1. As for the class of standard discrete kernels, the better discrete kernel is the binomial one which is underdispersed. However, it is not possible with this indicator to compare systematically these two discrete kernel families; because of ( 21), we should take into consideration n, h and f , and therefore the MISE (see, for example, Figures 2,3 and Table 2). After all, for h > 0 and a p.m.f f , the choice of a discrete kernel depends on the sample size n. For n large, the Dirac type kernel or a discrete associated kernel will be sufficient to get a good discrete smoothing. However, an other discrete kernel which is not associated kernel but satisfying [START_REF] Marsh | Discrete Poisson kernel density estimation with an application to wildcat coal strikes[END_REF] can be also efficient. Concerning small samples sizes for which the Dirac type kernel is not appropriate, the use of a discrete kernel of first order verifying [START_REF] Marsh | Discrete Poisson kernel density estimation with an application to wildcat coal strikes[END_REF] or a discrete associated kernel is more interesting.

Table 4 about here

Extensions

The estimator (2) opens another way for nonparametric approach by continuous or discrete associated kernel and therefore mixed, because Definition 1 concerning the discrete associated kernels can be extended. Indeed, the symmetric continuous kernels (3) and the asymmetric ones in Tables 5 and6, respectively, fullfill assumptions (8)-(10) of associated kernels; see Senga Kiessé [START_REF] Kiessé | Nonparametric Approach by Discrete Associated-Kernel for Count Data[END_REF], pages 172-178. Thus, we give the extension of Definition 1 for continuous associated kernels as follows.

Definition 2 Let T be a continuous support of f , to be estimated, x a fixed target in T and h > 0 a bandwidth. A p.d.f. K x,h (•) on support S x,h is said to be an associated kernel, if it satisfies assumptions (8)- [START_REF] Hille | Functional Analysis and Semi-Groups[END_REF] with

E(K x,h ) = S x,h ∩T yK x,h (y)dy, ( 22
) V ar(K x,h ) = S x,h ∩T {y -E(K x,h )} 2 K x,h (y)dy, (23)
where K x,h is the continuous random variable whose p.d.f. is K x,h (•).

The following result provides an interpretation for the well-known symmetric continuous kernels (3); see, for example, Table 5.

Proposition 2 If K(•) is a symmetric continuous kernel function which is a bonafide p.d.f. with zero mean and unit variance. Then, for x fixed in T and h > 0, the associated kernel (1/h)K{(x -•)/h} = K x,h (•) on support S x,h is also symmetric with mean x and standard deviation h.

Proof: Without loss of generality, we assume that S x,h ⊆ T. By writing t = (yx)/h then dy = hdt in ( 22) and ( 23), it is easy to check the results using K(-t) = K(t), S x,h tK(t)dt = 0 and S x,h t 2 K(t)dt = 1.

Tables 5 and6 about here Some properties which are avalaible in discrete case can be applied to continuous case, like the choice of the kernel depending on the support T of the function to be estimated and also the consistencies. The first extensions of this work will be to build an associated kernel estimator on time scales T (e.g. Agarwal and Bohner [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF]), then to detail in this case some notions as integral, continuity, derivation of a function and also boundary bias (e.g. Chen [START_REF] Chen | Beta kernels estimators for density functions[END_REF][START_REF] Chen | Gamma kernel estimators for density functions[END_REF], Zhang and Karunamuni [START_REF] Zhang | Some improvements on a boundary corrected kernel density estimator[END_REF]). A basic of mathematical tools is given for discrete kernels summarized in Table 4; for example, the discrete Taylor expansion can be improved in order to unify the theory. Several points are still to be proved such that the good behaviour of the binomial kernel for small samples sizes, other optimal choices of discrete smoothing bandwidth and the crucial problem of an optimal associated kernel (discrete or continuous). New discrete probability distributions may also be constructed, as the class of discrete triangular (e.g. [START_REF] Kokonendji | Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions[END_REF]), for serving as discrete associated kernels.

Finally, the discrete associated kernels methodology is useful in various domains as in actuarial or demography. Moreover, two others extensions of this associated kernels method that will be of interest are to consider the multidimentional case in the estimation of a discrete or mixed regression function and the nonparametric weighted Poisson regression problems. Some works in these direction are in progress with respect to the discrete associated kernel approach.

  n→∞ h(n) = 0. That said, we still do not have the usual assumptions encountered in kernel probability density function, i.e. lim n→∞ [h(n) + {nh(n)} -1 ] = 0. Now, we simply state the strong consistency of f n (x) and then the asymptotic normality of f n (x); see Abdous and Kokonendji [1, Theorem 2.4] and [1, Theorem 2.5], respectively.

Figure 2 about here 5

 5 Figure 2 about here

Figure 3

 3 Figure 3 about here

Fig. 1 .

 1 Fig. 1. Shape of the binomial kernel for h = 0.1 and some values of x

Fig. 2 .

 2 Fig.2. Comparison of the exact MISE of discrete kernel estimators for f = 0.4P(0.5) + 0.6P[START_REF] Hille | Functional Analysis and Semi-Groups[END_REF]

Fig. 3 .

 3 Fig. 3. Discrete smoothing (black line) by discrete kernel estimators for simulated data of f = 0.4P(0.5) + 0.6P(10) (grey line) with n = 50

Table 1

 1 Solutions h 0 for standard discrete kernels Type of kernel h 0 such that n i=1 Pr (K X i ,h 0 = 0) = n 0 Poisson h 0 = log 1

			n 0		n i=1 e -X i
	Binomial	n i=1	1-h 0 X i +1	X i +1 = n 0
	Negative binomial	n i=1	X i +1 2X i +1+h 0	X i +1 = n 0

Table 2

 2 Simulated MISE and optimal average ISE and their standard errors (in parentheses) for discrete kernel and empirical estimators. The results multiplied by 10 3 are given

	Kernel	Triangular a = 1	Triangular a = 2	Binomial
	n M ISE	E(ISE) M ISE	E(ISE) M ISE	E(ISE)
	20	10.82 14.10 (15.94)	15.37 14.28 (13.14)	12.81 21.52 (23.36)
	50	5.47	6.81 (7.07)	9.89	8.63 (5.91)	7.13	7.10 (7.87)
	80	4.02	4.80 (4.37)	5.60	6.34 (4.23)	4.92	4.46 (3.68)
	100	3.63	4.14 (3.55)	4.95	5.42 (3.43)	4.60	3.94 (3.41)
	200	2.64	2.57 (1.83)	3.94	2.82 (1.98)	3.47	2.72 (2.01)
	500	1.22	1.28 (0.87)	1.57	1.46 (0.94)	2.86	2.03 (1.12)
	700	0.93	0.93 (0.62)	1.12	1.17 (0.71)	2.75	1.91 (0.92)
	Kernel		Poisson	Negative binomial	Empirical
	n M ISE	E(ISE) M ISE	E(ISE) M ISE	E(ISE)
	20	21.15 17.88 (9.42)	27.83 27.60 (7.58)	39.65 36.48 (24.73)
	50	15.73 15.76 (5.76)	26.76 27.73 (5.39)	15.86 15.84 (10.76)
	80	14.68 15.02 (4.09)	26.34 27.63 (4.31)	9.91 10.03 (6.97)
	100	13.80 15.07 (3.68)	24.85 27.78 (4.07)	7.93	8.13 (5.97)
	200	14.04 14.95 (2.50)	25.80 28.26 (2.99)	3.96	3.89 (2.76)
	500	13.62 13.75 (1.59)	25.00 26.76 (2.09)	1.59	1.57 (1.09)
	700	13.58 12.95 (1.07)	24.94 25.96 (1.87)	1.13	1.14 (0.78)

Table 3 Simulated

 3 The results multiplied by 10 3 are given Kernel Triang. a = 1 Triang. a = 2 Binomial Poisson Negative binomial

	n			x∈N R n (x; h)
	50	3.95	3.78	2.71	3.11	5.65
	200	0.19	0.13	0.67	0.77	1.53
	500	0.057	0.016	0.26	0.31	0.61
	700	0.035	0.013	0.18	0.22	0.44

x∈N R n (x; h) of discrete kernel estimators for f = P(2).

Table 4

 4 Summary of some properties of discrete kernel estimators Type of discrete E(K x,h ) Var(K x,h ) lim

	h→0	Var(K x,h ) Convergence	Cross-	Excess Symmetry Remarks

(n ր ∞ and h ց 0) bias

Preprint submitted to Statistical Methodology (revised)

http://cran.r-project.org/web/packages/TRIANG/index.html

Acknowledgements. We thank the anonymous referee and Associate Editor for their valuable comments. We are grateful to Belkacem Abdous for several discussions and his careful reading.

Table 5 Examples of classical symmetric continuous kernels (Wand and Jones [32], page 31)

Kernel

Density Support Efficiency

Summary of some properties of asymmetric continuous associated kernel Kernel Support Associated kernel

RIG(a,b) [START_REF] Scaillet | Density estimation using inverse and reciprocal inverse Gaussian kernels[END_REF] (0, +∞) a = (xh) -1 x xh + h 2 and b = h -1