Philippe Rambour
email: philippe.rambour@math.u-psud.fr

Asymptotic of the terms of the Gegenbauer polynomials on the unit circle and applications to the inverse of Toeplitz matrices

Keywords: Mathematical Subject Classification (2000) Primary 15B05, 33C45; Secondary 33D45, 42C05, 42C10 Orthogonal polynomials, Gegenbauer polynomials, inverse of Toeplitz matrices

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Asymptotic of the terms of the Gegenbauer polynomial on the unit circle and applications to the inverse of Toeplitz matrices

Introduction

The study of the orthogonal polynomials on the unit circle is an old and difficult problem (see [START_REF] Simon | Orthogonal polynomials on the unit circle, Part 1: classical theory[END_REF], [START_REF] Simon | Orthogonal polynomials on the unit circle, Part 2: spectral theory[END_REF] or [START_REF] Szegö | Orthogonal polynomials[END_REF]). The Gegenbauer polynomials on the torus are the orthogonal polynomials on the circle with respect to a weight of type f α : θ → 2 2α (cos θ -cos θ 0 ) 2α c 1 with α > -1 2 and c 1 a positive integrable function. In this paper we assume -1 2 < α ≤ 1 2 and c 1 sufficiently smooth regular function. It is said that a function k is regular if k(θ) > 0 for all θ ∈ T and k ∈ L 1 (T). In a first part we are interested in the asymptotic of the coefficients of these polynomials (see Corollary 3). The main tool to compute this is the study of the Toeplitz matrix with symbol f . Given a function h in L 1 (T) we denote by T N (h) the Toeplitz matrix of order N with symbol h the (N + 1) × (N + 1) matrix such that (T N (h)) i+1,j+1 = ĥ(ji) ∀i, j 0 ≤ i, j ≤ N where m(s) is the Fourier coefficient of order s of the function m (see, for instance [START_REF] Böttcher | Toeplitz matrices and determinants with Fisher-Hartwig symbols[END_REF] and [START_REF] Böttcher | Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity[END_REF]). There is a close connection between Toeplitz matrices and orthogonal polynomials on the complex unit circle. Indeed the coefficients of the orthogonal polynomial of degree N with respect of h are also the coefficients of the last column of T -1 N (h) except for a normalisation (see [START_REF] Landau | Maximum entropy and the moment problem[END_REF]). Here we give an asymptotic expansion of the entries (T N (f α )) -1 k+1,1 (Theorem 4). Using the symmetries of the Toeplitz matrix T N (f α ), we deduce from this last result an asymptotic of (T N (f α )) -1 N -k+1,N +1 (corollary 3). The proof of Theorem 4 often refers to results of [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF]. In this last work we have treated the case of the symbols h α defined by θ → (1 -cos θ) α c whith -1 2 < α ≤ 1 2 and the same hypothesis on c as on c 1 . We have stated the following Theorem which is an important tool in the demonstration of Theorem 4.

Theorem 1 ( [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF]) If - 1 2 < α ≤ 1 2 , α = 0 we have for c ∈ A(T, 3 2 ) and 0 < x < 1

c(1) (T N (h α )) -1 [N x]+1,1 = N α-1 1 Γ(α) x α-1 (1 -x) α + o(N α-1 ).
uniformly in x for x ∈ [δ 1 , δ 2 ] with 0 < δ 1 < δ 2 < 1, with the definition Définition 1 For all positive real τ we denote by A(T, τ ) the set

A(T, τ ) = {h ∈ L 2 (T)| s∈Z |s τ ĥ(s)| < ∞}
This theorem has also been proved for α ∈ N * in [START_REF] Rambour | Formulas for the inverses of Toeplitz matrices with polynomially singular symbols[END_REF] and for α ∈] 1 2 , +∞[\N * in [START_REF] Rambour | Inversion des matrices de Toeplitz dont le symbole admet un zéro d'ordre rationnel positif,valeur propre minimale[END_REF].

The second part of the present paper is devoted to the inversion of a class of Toeplitz matrices. We give an asymptotic expansion of (T N (f α )) -1 k+1,l+1 for α ∈]0, 1 2 ] and k N → x, l N → y and 0 < x = y < 1. First we obtain these entries as a function of cos(lk)θ 0 and (T N (h α )) -1 k+1,l+1 . It is Theorem 6. With the same hypothesis as for Theorem 1 we have stated in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] the following Theorem Theorem 2 ( [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF]) For 0 < α < 1 2 we have

c(1) (T N (h α )) -1 [N x]+1,[N y]+1 = N 2α-1 1 Γ 2 (α) G α (x, y) + o(N 2α-1 )
uniformly in (x, y) for 0 < δ 1 ≤ x = y < 1.

Theorem 2 has been proved for α ∈ N * in [START_REF] Rambour | Formulas for the inverses of Toeplitz matrices with polynomially singular symbols[END_REF], for α = 1 2 in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] and for α ∈] 1 2 , +∞[\N * in [START_REF] Rambour | Inversion des matrices de Toeplitz dont le symbole admet un zéro d'ordre rationnel positif,valeur propre minimale[END_REF]. The quantities G α (x, y) is the integral kernel on L 2 (0, 1) of Corollaries 5. A direct consequence of theorems 6 and 2 is that, for α > 0 the entries of (T N (f α )) -1 are functions of cos(lk)θ 0 and the integral kernel G α (x, y) (see corollaries 5). The results of this paper are of interest in the analysis of time series. Indeed it is known that the n-th covariance matrix of a time series is a positive Toeplitz matrix. If φ is the symbol of this Toeplitz matrix, φ is called the spectral density of the time series. The time series with spectral density is the function f α are also called GARMA processes. For more on this processes we refer the reader to [START_REF] Beran | Statistics for long memory process[END_REF][START_REF] Beaumont | Robust estimation of GARMA model parameters with an application to cointegration among interest rates of industrialized country[END_REF][START_REF] Cheng | A k-factor GARMA long-memory model[END_REF] and to [START_REF] Dahlhaus | Efficient parameter estimation for self-similar processes[END_REF][START_REF] Doukhan | Theory and applications of long-range dependence[END_REF][START_REF] Grenander | Toeplitz forms and their applications[END_REF][START_REF] Beran | Statistics for long memory process[END_REF][START_REF] Brockwell | Times series: theory and methods[END_REF][START_REF] Kirman | Long memory in economic[END_REF][START_REF] Lu | On the complexity of the preconditioned conjugate gradient algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity[END_REF] for Toeplitz matrices in times series.

Predictor polynomial

Now we have to precise the deep link between the orthogonal polynomials and the inverse of the Toeplitz matrices. Let T n (f ) a Toeplitz matrix with symbol f and (Φ n ) n∈N the orthogonal polynomials with respect to f ( [START_REF] Landau | Maximum entropy and the moment problem[END_REF]). To have the polynomial used for the prediction theory we put

Φ * n (z) = n k=0 (T n (f )) -1 k+1,N +1 (T n (f )) -1 N +1,N +1 z k , | z |= 1. (1)
We define the polynomial Φ * n (see [START_REF] Simon | Orthogonal polynomials on the unit circle, Part 1: classical theory[END_REF]) as

Φ * n (z) = z n Φn ( 1 z ), (2)
that implies, with the symmetry of the Toeplitz matrix

Φ * n (z) = n k=0 (T n (f )) -1 k+1,1 (T n (f )) -1 1,1 z k , | z |= 1. (3)
The polynomials

P n = Φ * n (T n (f )) -1
1,1 are often called predictor polynomials. As we can see in the previous formula their coefficients are, up to a normalisation, the entries of the first column of T n (f ) -1 .

The proof of Theorem 6 uses the important following theorem ( [START_REF] Landau | Maximum entropy and the moment problem[END_REF]), Theorem 3 If h a non negative symbol with a finite set of zeroes, and P n the predictor polynomial of degree n of h, we have, for all integers s such that -n ≤ s ≤ n,

1 |P n | 2 (s) = ĥ(s).

It implies

Corollary 1 For a fonction h as in Theorem 3 we have

T n 1 |P n | 2 = T n (h).

Main results

Main notations

In all the paper we consider the symbol defined by θ → 2 2α (cos θ -cos θ 0 ) 2α c 1 where

c 1 = |P | |Q| with P, Q ∈ R[X]
, without zeros on the united circle and -1 2 < α < 1 2 and 0 < θ 0 < π. We have

c 1 = c 1,1 c1,1 with c 1,1 = P Q . Obviously c 1,1 ∈ H 2+ (T) since H 2+ (T) = {h ∈ L 2 (T)|u < 0 =⇒ ĥ(u) = 0}. If χ is the function θ → e iθ and if χ 0 = e iθ 0 we put g α,θ 0 ,c 1 = (χ -χ 0 ) α (χ -χ 0 ) α c 1,1 and g α,θ 0 = (χ -χ 0 ) α (χ -χ 0 ) α since(2(cos θ -cos θ 0 )) 2α = |χ -χ 0 | 2α |χ -χ 0 | 2α . Clearly g α,θ 0 ,c 1 , g α,θ 0 ∈ H 2+ (T) and 2 2α (cos θ -cos θ 0 ) 2α c 1 = g α,θ 0 ,c 1 g α,θ 0 ,c 1 , 2 2α (cos θ -cos θ 0 ) 2α = g α,θ 0 g α,θ 0 . Then we denote by β (α) k,θ 0 ,c 1 the Fourier coefficient of g -1
α,θ 0 ,c 1 and by β (α) k,θ 0 the one of g -1 α,θ 0 . Without loss of generality we assume β (α) 0,θ 0 ,c 1 = 1. We put also β(α)

k = g-1 α (k) with gα = (1 -χ) α .

Orthogonal polynomials

Theorem 4 Assume θ 0 ∈]0, π[ and -1 2 < α < 1 2 . Then we have for all integers k, k N → x, 0 < x < 1, the asymptotic

T -1 N |χ -χ 0 | 2α |χ -χ0 | 2α c 1 k+1,1 = =K α,θ 0 ,c 1 cos (kθ 0 + ω α,θ 0 ) T -1 N |χ -1| 2α k+1,1 (1 + o(1)) uniformly in k for x ∈ [δ 0 , δ 1 ], 0 < δ 0 < δ 1 < 1, and with ω α,θ 0 = αθ 0 + arg (c 1,1 (θ 0 )) -πα 2 and K α,θ 0 ,c 1 = 2 -α+1 (sin θ 0 ) -α c -1
1 (χ 0 ) . Then the following statement is an obvious consequence of Theorems 4 and 1.

Corollary 2 With the same hypotheses as in Theorem 4 we have

T -1 N |χ -χ 0 | 2α |χ -χ0 | 2α c 1 k+1,1 = = K α,θ 0 ,c 1 Γ(α) cos (kθ 0 + ω α,θ 0 ) k α-1 (1 - k N ) α + o(N α-1 ) uniformly in k for x ∈ [δ 0 , δ 1 ] 0 < δ 0 < δ 1 < 1.
Moreover the equalities (2) and (3) provide

Corollary 3 Let Φ N = N j=0
δ j χ j be the orthogonal polynomial of degree N (Gegenbauer polynomial) with respect to the weight θ → 2 2α (cos θ -cos θ 0 )c 1 (θ), with -1 2 < α < 1 2 . Then we have, for j N → x, 0 < x < 1,

δ j = N α-1 K α,θ 0 ,c 1 Γ(α) cos (N -jθ 0 + ω α,θ 0 ) j α (1 - j N ) α-1 + o(N α-1 ). uniformly in j for x ∈ [δ 0 , δ 1 ], 0 < δ 0 < δ 1 < 1.
We can also point out the asymptotic of the coefficients of order k of the predictor polynomial when k N → 0.

Theorem 5 With the same hypotheses as in Theorem 4 we have, if k N → 0 when N goes to the infinity

T -1 N |χ -χ 0 | 2α |χ -χ0 | 2α c 1 k+1,1 = β (α) k,θ 0 ,c 1 + O( 1 N
).

Lastly when α approaches 1 2 we obtain the entries of the last column of T N (2(cos θ -cos θ 0 )c 1 ). Corollary 4 Assume θ 0 ∈]0, π[. Then for all integers k for k N → x, 0 < x < 1, we have the asymptotic

T -1 N (|χ -χ 0 ||χ -χ0 |c 1 ) k+1,1 = =K 1/2,θ 0 ,c 1 cos kθ 0 + ω 1/2,θ 0 1 k - 1 N + o( √ N ) uniformly in k for x ∈ [δ 0 , δ 1 ], 0 < δ 0 < δ 1 < 1.
Remark 1 This corollary implies that the coefficient of order k of the orthogonal polynomial with respect of θ → 2(cos θ -

cos θ 0 )c 1 (θ) is K 1/2,θ 0 ,c 1 cos kθ 0 + ω 1/2,θ 0 1 k -1 N -1 + o( √ N )

Application to Toeplitz matrices

Theorem 6 Assume θ 0 ∈]0, π[ and 0 < α < 1 2 . For k N → x, l N → y and 0 < x = y < 1, we have asymptotic

T -1 N |χ -χ 0 | 2α |χ -χ0 | 2α c 1 k+1,l+1 = =|K α,θ 0 ,c 1 | 2 cos (θ 0 (k -l)) T -1 N |χ -1| 2α k+1,l+1 + o(N 2α-1 )
uniformly for k, l such that 0 < δ 1 < x = y < δ 2 < 1.

At it has been said in the introduction this statement and the results of [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] provides the next corollary.

Corollary 5 Assume α ∈]0, 1 2 ] and θ 0 ∈]0, π[. Let G α be the function defined on 0 < x = y < 1 by

G α (x, y) = x α y α Γ 2 (α) 1 max(x,y) (t -x) α-1 (t -y) α-1 t 2α dt.
With the same hypothesis as in Theorem 6 we have the asymptotic

T -1 N |χ -χ 0 | 2α |χ -χ 0 | 2α c 1 [N x]+1,[N y]+1 = =N 2α-1 |K α,θ 0 ,c 1 | 2 cos (θ 0 ([N x] -[N y])) G α (x, y) + o(N 2α-1 )
uniformly in k, l for 0 < δ 1 ≤ x = y ≤ δ 2 < 1.

Jacobi polynomial (in a particular case)

We note that in Theorem 6 one passes from the zeroes χ 0 and χ 0 to two zeroes χ 1 = e iθ 1 and χ 2 = e iθ 2 with |θ 1θ 2 | ∈]0, π[. Namely it is easy to see that

T -1 N (|χ -χ 1 | 2α |χ -χ 2 | 2α c 1 ) = ∆(χ 1/2 1 χ 1/2 2 )T -1 N |χ 1/2 1 χ -1/2 2 -ψ| 2α |χ -1/2 1 χ 1/2 2 -ψ| 2α c 1,ψ -1 ∆ -1 (χ 1/2 1 χ 1/2 2 )
with ∆(χ

1/2 1 χ 1/2
) is the diagonal matrix defined by ∆(χ

1/2 1 χ 1/2 2 ) i,j = 0 if i = j and (∆(ψ)) j,j = (χ 1/2 1 χ 1/ 2
2 ) j . From this and Equation (2) we deduce the following proposition Proposition 1 Let Φ 1,2 = j=0 δj χ j be the orthogonal polynomial (Jacobi polynomial) with re-

spect to the weight |χ-χ 1 | 2α |χ-χ 2 | 2α , with α ∈]-1 2 , 1 2 ]. Let K α,θ 1 ,θ 2 be the real 2 -α+1 | sin(θ 1 - θ 2 )| -α . Then we have, for j N → x, 0 < x < 1. δj = N α-1 K α,θ 1 ,θ 2 ((χ 1 χ 2 ) 1/2 ) N -j cos ( θ 1 -θ 2 2 )(N -j) + ω α,θ 1 -θ 2 j α (1 -j N ) α-1 Γ(α) +o(N α-1 ), uniformly in j for x ∈ [δ 0 , δ 1 ], 0 < δ 0 < δ 1 < 1.
3 Inversion formula

Definitions and notations

Let H 2+ (T) and H 2-(T) the two subspaces of L 2 (T) defined by

H 2+ (T) = {h ∈ L 2 (T)|u < 0 =⇒ ĥ(u) = 0} and H 2-(T) = {h ∈ L 2 (T)|u ≥ 0 =⇒ ĥ(u) = 0}
. We denote by π + the orthogonal projector on H 2+ (T) and π -the orthogonal projector on H 2-(T). It is known (see [START_REF] Grenander | Toeplitz forms and their applications[END_REF]) that if f ≥ 0 and ln

f ∈ L 1 (T) we have f = gḡ with g ∈ H 2+ (T). Put Φ N = g ḡ χ N +1 . Let H Φ N and H *
Φ N be the two Hankel operators defined respectively on H 2+ and H 2-by

H Φ N : H 2+ (T) → H 2-(T), H Φ N (ψ) = π -(Φ N ψ),
and

H * Φ N : H 2-(T) → H 2+ (T), H * Φ N (ψ) = π + ( ΦN ψ).

A generalised inversion formula

We have stated in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] for a precise class of non regular functions which contains cos α (θθ 0 )c 1 and (cos θ -cos θ 0 ) α c 1 the following lemma (see the appendix of [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] for the demonstration),

Lemma 1 Let f be an almost everywhere positive function on the torus T such that ln f , f , and 1 f are in L 1 (T). Then f = gḡ with g ∈ H 2+ (T). For all trigonometric polynomials P of degree at most N , we define G N,f (P ) by

G N,f (P ) = 1 g π + P ḡ - 1 g π + Φ N ∞ s=0 H * Φ N H Φ N s π + ΦN π + P ḡ .
For all P we have

• The serie ∞ s=0 H * Φ N H Φ N s π + ΦN π + P ḡ converges in L 2 (T). • det (T N (f )) = 0 and (T N (f )) -1 (P ) = G N,f (P ).

An obvious corollary of Lemma 1 is

Corollary 6 With the hypotheses of Lemma 1 we have

(T N (f )) -1 l+1,k+1 = π + χ k ḡ χ l ḡ - ∞ s=0 H * Φ N H Φ N s π + ΦN π + χ k ḡ ΦN χ l ḡ . Lastly if γ u,α,θ = g α,θ 0 g α,θ 0
(u) we obtain as in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] the formal result

H * Φ N H Φ N m π + ΦN π + χ k ḡ = k u=0 β (α) u,θ 0 ,c 1 ∞ n 0 =0 ∞ n 1 =1 γ-(N+1+n 1 +n 0 ),α,θ 0 ∞ n 2 =0 γ -(N +1+n 1 +n 2 ),α,θ 0 • • • ∞ n 2m-1 =1 γ-(N+1+n 2m-1 +n 2m-2 ),α,θ 0 ∞ n 2m =0 γ -(N +1+n 2m-1 +n 2m ),α,θ 0 γ-(u-(N+1+n 2m ),α,θ 0 ) χ n 0

Application to the orthogonal polynomials

With the corollary 6 and the hypothesis on β (α) 0,θ 0 ,c 1 the equality in the corollary 6 becomes, for l = 1, and for

f = |χ -χ0 | 2α |χ -χ0 | 2α c 1 (T N (f )) -1 1,k+1 = β (α) k,θ 0 ,c 1 - k u=0 β (α) k-u,θ 0 ,c 1 H N (u) (4)
with

H N (u) = +∞ m=0 ∞ n 0 =0 γ N +1+n 0 ,α,θ 0 ∞ n 1 =0 γ-(N+1+n 1 +n 0 ),α,θ 0 ∞ n 2 =0 γ -(N +1+n 1 +n 2 ),α,θ 0 • • • ∞ n 2m-1 =0 γ-(N+1+n 2m-1 +n 2m-2 ),α,θ 0 ∞ n 2m =0 γ -(N +1+n 2m-1 +n 2m ),α,θ 0 γ(u-(N+1+n 2m ),α,θ 0
Our proof consists in the computation of the coefficients β

(α)
u,θ 0 ,c 1 , γ u,α,θ and H N (u) which appear in the inversion formula. For each step we obtain the corresponding terms for the symbol 2 α (1 -cos θ)c 1 multiplied by a trigonometric coefficient. That provides the expected link with the formulas in Theorems 1, 2.

H 2+ (T) such that c 1 = c 1,1 c 1,1 .
In all this proof we put φ 0 = arg (c 1,1 (θ 0 )) Property 1 For -1 2 < α < 1 2 and θ 0 ∈]0, π[ we have, for sufficiently large k and for the real

β defined by β = α -1 2 if α < 0 and β = α if α > 0, β (α) k,θ 0 ,c 1 = K α,θ 0 ,c 1 cos(kθ 0 + ω α,θ 0 ) k α-1 Γ(α) + o(k β-1 ) uniformly in k. With K α,θ 0 ,c 1 = 1 √ c 1 (χ 0 )
2 -α+1 (sin θ 0 ) -α and ω α,θ 0 as in the statement of Theorem 4.

First we have to prove the lemma

Lemma 2 For -1 2 < α < 1 2 and θ 0 ∈]0, π[ we have, for a sufficiently large k.

β (α) k,θ 0 = K α,θ 0 cos((k + α)θ 0 + ω α ) k α-1 Γ(α) + o(k β-1 ).
uniformly in k, with K α,θ 0 = 2 -α+1 (sin θ 0 ) -α , ω α = -πα 2 , and β as in Property 1.

Remark 3 In these two last statements"uniformly in k " means that for all ǫ > 0 we have an integer k ǫ such that for all k ≥ k ǫ

β (α) k,θ 0 -K α,θ 0 cos((k + α)θ 0 + ω α ) k α-1 Γ(α) < ǫk β-1 and β (α) k,θ 0 ,c 1 -K α,θ 0 ,c 1 cos((k + α)θ 0 + ω α ) k α-1 Γ(α) < ǫk β-1 .
Proof : With our notations we can write

β (α) k,θ 0 = k u=0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u . Put k 0 = k γ with 0 < γ < 1 such that for u > k 0 we have β(α) u = u α-1 Γ(α) + O(k α-2 ) (5)
uniformly in u (see [START_REF] Zygmund | Trigonometric series[END_REF]). Writting for

k ≥ k 0 k u=0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u = k 0 u=0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u + k-k 0 -1 u=k 0 +1 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u + k u=k-k 0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u .
The first sum is also

k 0 u=0 β(α) u (χ 0 ) u β(α) k-u - β(α) k + β(α) k (χ 0 ) k-u .
We observe that

k 0 u=0 β(α) u (χ 0 ) u β(α) k-u - β(α) k ≤ 1 Γ(α) k 0 u=0 |(k -u) α-1 -k α-1 || β(α) u |. (6)
Consequently

k 0 u=0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u = k 0 u=0 β (α) u (χ 0 ) 2u χk 0 k α-1 Γ(α) + R 1 =   +∞ u=0 β(α) u (χ 0 ) 2u - +∞ u=k 0 β(α) u (χ 0 ) 2u   χk 0 k α-1 Γ(α) + R 1 with R 1 = O(k α-2+γ ) if α < 0 and R 1 = O(k α-2+γα ) if α > 0. Then Lemma 9 implies | +∞ u=k 0 β(α) u (χ 0 ) 2u | ≤ | β(α) k 0 χ 2u 0 | + ∞ u=k 0 | β(α) u+1 - β(α) u | Γ(α) , (7)
that is

+∞ u=k 0 β(α) u (χ 0 ) 2u = O(k α-1 0 ).
Finally we get

k 0 u=0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u = k α-1 Γ(α) χ 0 k (1 -χ 2 0 ) -α + O k (α-1)(γ+1) + R 1 .
Analogously we obtain

k u=k-k 0 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u = χ k 0 k α-1 Γ(α) (1 -χ2 0 ) -α + O k (α-1)(γ+1) + R 2 , with R 2 as R 1 .
For the third sum an Abel summation provides

k-k 0 -1 u=k 0 +1 β(α) u (χ 0 ) u β(α) k-u (χ 0 ) k-u = χ 0 k β(α) k 0 β(α) k-k 0 σ k 0 -1 + k-k 0 -2 u=k 0 ( βα u β(α) k-u - β(α) u+1 β(α) k-u-1 )σ u   + β(α) k-k 0 -1 β(α) k 0 σ k-k 0 with σ v = 1 + χ 2 0 + • + χ 2v 0 . This last sum is also equal to χ 0 k (A + B) , with |A| = O β(α) k 0 β(α) k = O(k α-1 0 k α-1 ) = o(k (α-1)(γ+1) )
and

B = - k-k 0 -2 u=k 0 1 Γ 2 (α) u α-1 (k -u) α-1 -(u + 1) α-1 (k -u -1) α-1 χ 2u+2 0 1 -χ 2 0 .
The main value Theorem implies

|B| ≤ M k   k-k 0 v=k 0 v α-2 (k -v) α-2   , (8)
with M no depending from k. With the Euler and Mac-Laurin formula it is easyly seen that

k-k 0 v=k 0 v α-2 (k -v) α-1 ∼ k α-2 0 (k -k 0 ) α-1 + k α-1 0 (k -k 0 ) α-2 + k-k 0 k 0 t α-2 (k -t) α-2 dt.
The decomposition

k-k 0 k 0 t α-2 (k -t) α-2 dt = k/2 k 0 t α-2 (k -t) α-2 dt + k-k 0 k/2 t α-2 (k -t) α-2 dt provides the estimation |B| = O(k α-1 0 k α-1 ) = O(k (α-1)(γ+1)
). If α > 0 and 0 < γ < 1 we have

β (α) k,θ 0 = k α-1 Γ(α) χ k 0 (1 -χ 2 0 ) -α + χ k 0 (1 -χ 2 0 ) -α + o(k α-1 )
If α < 0 and γ = 1 2 we get

β (α) k,θ 0 = k α-1 Γ(α) χ k 0 (1 -χ 2 0 ) -α + χ k 0 (1 -χ 2 0 ) -α + o(k β-1 )
with β = α -1 2 . On the another hand we have

β (α) k,θ 0 = 2 k α-1 Γ(α) ℜ e -ikθ 0 (1 -cos(2θ 0 ) -i sin(2θ 0 )) -α + o(k β-1 ) = 2 1-α k α-1 Γ(α) ℜ e -ikθ 0 (sin(θ 0 ) (sin θ 0 -i cos θ 0 )) -α + o(k β-1 ) Since θ 0 ∈]0, π[ we have (sin θ 0 (sin θ 0 -i cos θ 0 )) -α = (sin θ 0 ) -α e iα( π 2 -θ 0 )
This last remark gives the definition of ω α . The equations ( 6), ( 7), [START_REF] Doukhan | Theory and applications of long-range dependence[END_REF], imply the uniformity that completes the proof of the lemma. ✷

To ends the proof of the property we need to obtain β

(α) k,θ 0 ,c 1 from β (α)
k,α,θ 0 for a sufficiently large k. We can remark that a similar case has been treated in [START_REF] Rambour | Inversion des matrices de Toeplitz dont le symbole admet un zéro d'ordre rationnel positif,valeur propre minimale[END_REF] for the function (1χ) α c 1 .

Here we develop the same idea than in this last paper. Let c m the coefficient of Fourier of order m of the function c -1 1,1 . The hypotheses on c 1,1 imply that c -1

1,1 is in A(T, p) = {h ∈ L 2 (T)| u∈Z u p | ĥ(u)| < ∞} for all positive integer p. We have, β (α) m,θ 0 ,c 1 = m s=0 β (α) m,θ 0 c m-s . For 0 < ν < 1 we can write m s=0 β (α) s,θ 0 c m-s = m-m ν s=0 β (α) s,θ 0 c m-s + m s=m-m ν +1 β (α) s,θ 0 c m-s . Lemma 2 provides m s=m-m ν +1 β (α) s,θ 0 c m-s = K α,θ 0 m s=m-m ν s α-1 Γ(α) (cos ((s + α)θ 0 + ω α ) c m-s + o(m β-1 ) m s=m-m ν +1 |c m-s | and, since s∈Z |c s | < ∞, we have m s=m-m ν +1 β (α) s,θ 0 c m-s = K α,θ 0 s α-1 Γ(α) m s=m-m ν s α-1 Γ(α) (cos ((s + α)θ 0 + ω α ) c m-s + o(m β-1 ).
We have always

m s=m-m ν (s α-1 -m α-1 )c m-s ≤ (1 -α)m ν+α-2 m s=m-m ν |c m-s |. (9)
The convergence of (c s ) implies

K α,θ 0 m s=m-m ν s α-1 -m α-1 + m α-1 Γ(α) (cos ((s + α)θ 0 + ω α ) c m-s = K α,θ 0 m α-1 Γ(α) m s=m-m ν (cos ((s + α)θ 0 + ω α ) c m-s + O(m α-2+ν ).
For all positive integer p the function c 1,1 A(p, T)). Hence one can prove first

∞ v=m ν +1 e -ivθ c v ≤ (m -pν ) s∈Z |c s | (10)
and secondly

m s=m-m ν (cos ((s + α)θ 0 + ω α ) c m-s = 1 2 m s=m-m ν e isθ 0 c m-s e i(θ 0 α+ωα) + 1 2 m s=m-m ν e -isθ 0 c m-s e -i(θ 0 α+ωα) = 1 2 c -1
1,1 (e -iθ 0 )e i(mθ 0 +θ 0 α+ωα) + c -1 1,1 (e iθ 0 )e -i(mθ

0 +θ 0 α+ωα) + O(m -pν ). Since c -1 1,1 (e iθ 0 ) = c -1 1,1 (e -iθ 0 ) that last formula provides m s=m-m ν (cos ((s + α)θ 0 + ω α ) c m-s = c -1 1 (χ 0 ) cos ((m + α)θ 0 + ω α + φ 0 ) + O(m -pν ) (11)
and

m s=m-m ν +1 β (α) s,θ 0 c m-s = K α,θ 0 m α-1 Γ(α) c -1 1 (χ 0 ) cos ((m + α)θ 0 + ω α + φ 0 ) + O(m α-1-pν ) + O(m α-2+ν ) + o(m β-1 ).
On the other hand we have (because

c -1 1,1 in A(T, p)) m-m ν s=0 β s,α c m-s ≤ 1 m 2ν v∈Z v p |c v | max s∈N (|β (α) s,θ 0 |).
For a good choice of p and ν we obtain the expected formula for β α,θ 0 ,c 1 . The uniformity is provided by Lemma 2 and the equation ( 9) and (10).

Estimation of the Fourier coefficients of

g α,θ 0 g α,θ 0 Property 2 Assume -1 2 < α < 1 2 and θ 0 ∈]0, π[ then we have for all integer k ≥ 0 sufficiently large

g α,θ 0 g α,θ 0 (-k) = 2 k + α sin(πα) π cos(θ 0 k + 2ω ′ α,θ 0 ) + o(k min(α-1,-1) )
uniformly in k and with ω ′ α,θ 0 = φ α + φ ′ 0 where φ ′ 0 = arg

c 1,1
c1,1 (e iθ 0 ) and φ α = arg

χ 2 0 -1 χ2 0 -1 α .
First we have to prove the lemma Lemma 3 For -1 2 < α < 1 2 and θ 0 ∈]0, π[ we have, for all integer k sufficiently large

γ -k = 2 k + α sin(α) π cos (θ 0 k + φ α )) + o(k min(α-1,-1) ),
uniformly in k and where γ k is the coefficient of order k of the function (χχ 0 -1) α (χ χ0 -1) α ( χ χ0 -1) α ( χχ 0 -1) α .

Proof of Lemma 3:

In all this proof we denote respectively by γk , γ 1,k , γ 2,k the Fourier coefficient of order k of (χ-1) α ( χ-1) α , (χχ 0 -1) α ( χ χ0 -1) α , (χ χ0 -1) α ( χχ 0 -1) α . Clearly γk = sin(πα)

π 1 k+α γ 1,k = χ k 0 γk , γ 2,k = ( χ0 ) k γk . Assume also k ≥ 0. We have γ -k = v+u=-k γ 1,u γ 2,v . For an integer , k 0 and k 0 = k τ , 0 < τ < 1 we can split this sum into u<-k-k 0 γ 1,u γ 2,-k-u + -k+k 0 u=-k-k 0 γ 1,u γ 2,-k-u + -k 0 -1 u=-k+k 0 +1 γ 1,u γ 2,-k-u + k 0 u=-k 0 γ 1,u γ 2,k-u + u>k 0 γ 1,u γ 2,-k-u . Write k 0 u=-k 0 γ 1,u γ 2,-k-u = k 0 u=-k 0 γ 1,u ( χ0 ) k+u (γ -k-u -γ-k + γ-k ). Since k 0 u=-k 0 γ 1,u ( χ0 ) k+u (γ -k-u -γ-k ) = sin(πα) π k 0 u=-k 0 γ 1,u ( χ0 ) k+u -u (k + u + α)(k + α) (12)
it follows that

k 0 u=-k 0 γ 1,u γ 2,-k-u = γ-k k 0 u=-k 0 γ 1,u ( χ0 ) -k-u + O(k 0 k -2 ) = γ-k (χ 0 ) k χ 2 0 -1 ( χ0 ) 2 -1 α + γ-k (χ 0 ) k |u|≥k 0 γ 1,u χ u 0 + O(k 0 k -2 ) = γ-k (χ 0 ) k χ 2 0 -1 ( χ0 ) 2 -1 α + O (k 0 k) -1 + O(k 0 k -2 ) = γ-k (χ 0 ) k χ 2 0 -1 ( χ0 ) 2 -1 α + O(k τ -2 ).
In the same way we have

-k+k 0 u=-k-k 0 γ 1,u γ 2,k-u = γ-k (χ 0 ) -k ( χ0 ) 2 -1 χ 2 0 -1 α + O(k τ -2 ).
Now using Lemma 9 it is easy to see that

u<-k-k 0 γ 1,u γ 2,-k-u ≤ M 1 (k 0 k) -1 (13)
u>k 0 γ 1,u γ 2,-k-u ≤ M 2 (k 0 k) -1 (14)
with M 1 and M 2 no depending from k. For the sum S =

-k 0 -1 u=-k+k 0 +1
γ 1,u γ 2,-k-u we can remark, using an Abel summation, that

|S| ≤ M 3 (k 0 k) -1 + -k 0 -1 u=-k+k 0 +1 1 (u + α)(k -u + α) - 1 (u + 1 + α)(k -u -1 + α)
M 3 no depending from k. Consequently the main values theorem provides

|S| ≤ M 3 (k 0 k) -1 + -k 0 -1 u=-k+k 0 +1 k -2u (k -u) 2 u 2 . ( 15
)
with M 3 no depending from k. Then Euler and Mac-Laurin formula provides the upper bound

|S| ≤ O (k 0 k) -1 + -k 0 -1 -k+k 0 +1 k -2u (k -u) 2 u 2 du. Since -k 0 -1 -k+k 0 +1 k -2u (k -u) 2 u 2 du ≤ 3k (k + k 0 ) 2 -k 0 -1 -k+k 0 +1 1 u 2 du we get -k 0 -1 u<-k+k 0 +1 γ 1,u γ 2,k-u = O (k 0 k) -1 and γ k = 2 k + α sin(α) π cos (θ 0 k + φ α ) + O (k 0 k) -1 + O(k α-2 ).
Then with a good choice of τ we obtain the expected formula. The uniformity is a direct consequence of the equations ( 12), ( 13), ( 14), [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF]. ✷

The rest of the proof of Lemma 3 can be treated as the end of the proof of property 1.

Expression of

T -1 N (2 2α (cos θ -cos θ 0 ) 2α c 1 ) k+1,1 .
First we have to prove the next lemma

Lemma 4 For α ∈] -1 2 , 1 2 [ we have a function F N,α ∈ C 1 [0, δ] for all δ ∈]0, 1[, such that i) ∀z ∈ [0, δ[ |F N,α (z)| ≤ K 0 (1 + | ln(1 -z + 1 + α N )|)
where K 0 is a constant no depending from N .

ii) F N and F ′ N have a modulus of continuity no depending from N . iii)

T -1 N |χ -χ 0 | 2α |χ -χ0 | 2α c 1 k+1,1 = = β (α) k,θ 0 ,c 1 - 2 N k u=0 β (α) k-u,θ 0 ,c 1 F α,N ( u N ) cos(uθ 0 ) + R N,α uniformly in k, 0 ≤ k ≤ N , with R N,α = o N -1 k u=0 β (α) k-u,θ 0 ,c 1 F α,N ( u N ) if α > 0 and R N,α = o N α-1 k u=0 β (α) k-u,θ 0 ,c 1 F α,N ( u N ) if α < 0
Remark 4 This lemma and the continuity of F N,α in zero imply directly Theorem 5.

Remark 5 Lemma 4 and the continuity of the function F α imply that

T -1 N 2 2α (cos θ -cos θ 0 ) 2α c 1 1,1 = β (α) 0,θ 0 ,c 1 + 1 N β (α) 0,θ 0 ,c 1 F N,α (0) (1 + o(1)) .
Since F N,α (0) = α 2 + o(1) (see [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF]) the hypothesis β 0,θ 0 ,c 1 = 1 means that the coefficients of the predictor polynomial are

T -1 N 2 2α (cos θ -cos θ 0 ) 2α c 1 k+1,1 (1 + o(1)
) uniformly in k (it is a direct consequence of the equality [START_REF] Böttcher | Toeplitz matrices and determinants with Fisher-Hartwig symbols[END_REF].Indeed these of the orthogonal polynomial are 1)) (we can refer to the equations 3 and 2).

T -1 N (2 2α (cos θ -cos θ 0 ) 2α c 1 ) N -k+1,1 (1 + o(
Proof of the lemma 4: In the rest of the paper we slighty change of notation and denote by γ k the Fourier coefficient of order k of the function

(χχ 0 -1) α (χ χ0 -1) α c 1,1
( χ χ0 -1) α ( χχ 0 -1) α c1,1 by γ k . As for [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] and using the inversion formula and Corollary 6 we have to consider the sums

H m,N (u) = ∞ n 0 =0 γ -(N +1+n 0 ) ∞ n 1 =0 γ -(N +1+n 1 +n 0 ) ∞ n 2 =0 γ -(N +1+n 1 +n 2 ) × • • • × ∞ n 2m-1 =0 γ -(N +1+n 2m-2 +n 2m-1 ) ∞ n 2m =0 γ -(N +1+n 2m-1 +n 2m ) γ u-(N +1+n 2m )   . If S 2m = ∞ n 2m =0 γ -(N +1+n 2m-1 +n 2m ) γ u-(N +1+n 2m )
we can write, following the previous Lemma, S 2m = S 2m,0 + S 2m,1 with

S 2m,0 = 4 sin(πα) π 2 ∞ n 2m =0 cos (N + 1 + n 2m-1 + n 2m )θ 0 + 2ω ′ α,θ 0 cos (N + 1 + n 2m ) -u)θ 0 + 2ω ′ α,θ 0 × 1 N + 1 + n 2m-1 + n 2m + α 1 N + 1 + n 2m -u + α = 2 sin(πα) π 2 ∞ n 2m =0 cos (n 2m-1 + u)θ 0 ) N + 1 + n 2m-1 + n 2m + α 1 N + 1 + n 2m -u + α + ∞ n 2m =0 cos 2(N + 1 + n 2m + 4ω ′ α ) + n 2m-1 -u) θ 0 × 1 N + 1 + n 2m-1 + n 2m + α 1 N + 1 + wn 2m -u + α
Let us study the order of the second sum. To do this we can evaluate the order of the expression

M j=0 χ j 0 1 N + 1 + n 2m-1 + j + α 1 N + 1 + j -u + α
where M goes to the infinity and N = o(M ). As for the previous proofs it is clear that this sum is bounded by

M j=0 1 N + 2 + n 2m-1 + j 1 N + 2 + j -u - 1 N + 1 + n 2m-1 + j 1 N + 1 + j -u . Obviously 1 N + 2 + n 2m-1 + j 1 N + 2 + j -u - 1 N + 1 + n 2m-1 + j 1 N + 1 + j -u ≤ 2N + 2 + 2j + n 2m-1 -u (N + 1 + n 2m-1 + j) 2 (N + 1 + j -u) 2 and 2N + 2 + 2j + n 2m-1 -u (N + 1 + n 2m-1 + j) 2 (N + 1 + j -u) 2 = 1 N + 1 + j + n 2m-1 + 1 N + 1 + j -u 1 (N + 1 + j + n 2m-1 )(N + 1 + j -u) ≤ 1 N 1 (N + 1 + j + n 2m-1 )(N + 1 + j -u) .
In the other hand we have, for

α ∈]0, 1 2 [ S 2m,1 = o ∞ n 2m =0 1 N + 1 + n 2m-1 + n 2m + α 1 N + 1 + n 2m -u + α and for α ∈] -1 2 , 0[. S 2m,1 = o N α ∞ n 2m =0 1 N + 1 + n 2m-1 + n 2m + α 1 N + 1 + n 2m -u + α .
Hence we can write

S 2m = S ′ 2m (cos (θ 0 (n 2m-1 + u)) + r m,α ) , with S ′ 2m = +∞ n 2m =0 1 N + 1 + n 2m-1 + n 2m + α 1 N + 1 + n 2m -u + α . and r m,α = o(1) if α ∈]0, 1 2 [ r m,α = o(N α ) if α ∈] -1 2 , 0[. For z ∈ [0, 1] we define F m,N,α (z) by F m,N,α (z) = ∞ n 0 =0 1 N + 1 + n 0 ∞ n 1 =0 1 N + 1 + w 1 + w 0 × • • • × ∞ n 2m-1 =0 1 N + 1 + n 2m-2 + n 2m-1 + α × ∞ n 2m =0 1 N + 1 + n 2m-1 + n 2m + α 1 1 + 1+α N + n 2m N -z .
Repeating the same idea as previously for the sums on n 2m-1 , • • • , n 0 we finally obtain

H m,N (u) = 2 N sin(πα) π 2m+2 F m,N,α ( u N )(cos(uθ 0 ) + R N,α ).
with R N,α as announced previously. We established in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] the continuity of the function F m,N,α and the uniform convergence (see [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] Lemma 4). Moreover for all z ∈ [0, δ] , 0 < δ < 1 we have the inequality

1 1 + 1+α N + n 2m N -z ≤ 1 1 + 1+α N -δ . Hence 1 + 1+α N -δ 1 + 1+α N + n 2m N -z 2 ≤ 1 + 1+α N -δ 1 + 1+α N + n 2m N -z and 1 1 + 1+α N + n 2m N -z 2 ≤ 1 1 + 1+α N -δ 1 1 + 1+α N + n 2m N -z .
These last inequalities and the proof of Lemma 4 in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] prove that

F N,α is in C 1 [0, 1[.
Always in [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] we have obtained that, for all z in [0, 1],

F N,α (z) ≤ K 0 1+ ln(1 -z + 1 + α N ) (16)
where K 0 is a constant no depending from N . Now we have to prove the point ii) of the statement. For z, z

′ ∈ [0, δ] z -z ′ (1 + 1+α N + n 2m N -z)(1 + 1+α N + n 2m N -z ′ ) ≤ |z -z ′ | 1 -δ 1 1 + 1+α N + n 2m N -δ
that implies, with the inequality ( 16)

|F N,α (z) -F N,α (z ′ ) | ≤ |z -z ′ | K 0 1 + ln(1 -δ + 1+α N ) 1 -δ . (17)
In the same way we have

|z -z ′ | ((1 + 1+α N + n 2m N -z) + (1 + 1+α N + n 2m N -z ′ ) (1 + 1+α N + n 2m N -z) 2 (1 + 1+α N + n 2m N -z ′ ) 2 ≤ 2|z -z ′ | 1 (1 -δ) 2 1 1 + 1+α N + n 2m N -δ
and always with the inequality ( 16)

|F ′ N,α (z) -F ′ N,α (z ′ ) | ≤ 2|z -z ′ | K 0 1 + ln(1 -δ + 1+α N ) (1 -δ) 2 . ( 18
)
uniformly in k with the equation [START_REF] Brockwell | Times series: theory and methods[END_REF]. Then an Abel summation provides that the quantity

k-k 0 u=0 (k -u) α-1 cos((k -2u)θ 0 + ω α,φ 0 )F N,α ( u N
) is bounded by

M 1 k α-1 0 + k-k 0 u=0 |(k -u -1) α-1 F N,α ( u + 1 N ) -(k -u) α-1 F N,α ( u N )| with M 1 no depending from k. Moreover k-k 0 u=0 |(k -u -1) α-1 F N,α ( u + 1 N ) -(k -u) α-1 F N,α ( u N )| ≤ k-k 0 u=0 |(k -u -1) α-1 -(k -u) α-1 ||F N,α ( u N )| + k-k 0 u=0 |F N,α ( u + 1 N ) -F N,α ( u N )||(k -u -1) α-1 |
From the inequality 16 (we have assumed 0

< k N < δ) we infer k-k 0 u=0 |(k -u -1) α-1 -(k -u) α-1 ||F N,α (u)| ≤ M 2 k w=k 0 v α-2
with M 2 no depending from k. We finally get

k-k 0 u=0 |(k -u -1) α-1 -(k -u) α-1 ||F N,α (u)| = O   k w=k 0 v α-2   = O k α-1 0 = o(k α )
Identically Lemma 4 and the main value theorem provides

k-k 0 u=0 |F N,α ( u + 1 N ) -F N,α ( u N )||(k -u -1) α-1 | ≤ M 3 k α N = o(k α )
with M 3 no depending from N . By definition of k 0 and with Property 1 we have easily the existence of a constant M 4 , always no depending from k, such that for α > 0

k u=k-k 0 β (α) k-u,θ 0 ,c 1 F N,α ( u N ) cos(uθ 0 ) ≤ M 4 k α 0 .
Consequently for α > 0

2 k u=0 β (α) k-u,θ 0 ,c 1 F N,α ( u N ) cos(uθ 0 ) = K α,θ 0 ,c 1 cos((k -α)θ 0 + ω α,θ 0 ) k u=0 β(α) k-u F N,α ( u N ) + o(k α ).
uniformly in k with the definition of the constants M i , 1 ≤ i ≤ 4 and we get the Lemma for α > 0.

Since we have the result for the positive case we assume in the rest of the demonstration that α ∈] -1 2 , 0[. Recall that now γ ∈] max( α β , -α 1-α ), 1[.

First we have to evaluate the sum

k u=k-k 0 β (α) k-u,θ 0 ,c 1 cos(uθ 0 )F α (u). Since F N,α ∈ C 1 [0, δ] we have for k-k 0 N ≤ u N ≤ k N ≤ δ < the formula F α,N ( u N ) -F α,N ( k N ) + F α,N ( k N ) = F α,N ( k N ) + O( k 0 N ) = F α,N ( k N ) + o(k α ) uniformly in k (see once a more the definition of γ). Property 1 provides β (α) k-u,θ 0 ,c 1 = β (α) k-u,θ 0 ,c 1 + o(k β-1
). Hence we can write, uniformly in k,

2 k u=k-k 0 β (α) k-u,θ 0 ,c 1 cos(uθ 0 )F α ( u N ) = 2ℜ   χ k 0 k u=k-k 0 β (α) k-u,θ 0 ,c 1 ( χ0 ) k-u F α ( k N )   + o(k α ) = 2ℜ χ k 0 k 0 v=0 β (α) v,θ 0 ,c 1 ( χ0 ) v F α ( k N ) + o(k α ) = -2ℜ   χ k 0 ∞ v=k 0 +1 β (α) v,θ 0 ,c 1 ( χ0 ) v F α ( k N )   + o(k α ).
Moreover we have, uniformly with Property 1,

2 ∞ v=k 0 +1 β (α) v,θ 0 ,c 1 ( χ0 ) v = K α,θ 0 ,c 1 ∞ v=k 0 +1
β(α) v e i(vθ 0 +ω α,θ 0 ) + e -i(vθ 0 +ω α,θ 0 ) e -ivθ 0 + o(k β 0 ).

Consequently γ ∈] max( α β , -α 1-α ), 1[ infer that 2 ∞ v=k 0 +1 β (α) v,θ 0 ,c 1 ( χ0 ) v = K α,θ 0 ,c 1 ∞ v=k 0 +1 β(α)
v e i(vθ 0 +ω α,θ 0 ) + e -i(vθ 0 +ω α,θ 0 ) e -ivθ 0 + o(k α ).

We have

∞ v=k 0 +1 β(α) v (e i(vθ 0 +ω α,θ 0 ) + e -i(vθ 0 +ω α,θ 0 ) )e -ivθ 0 = ∞ v=k 0 +1 β(α) v (e i(ω α,θ 0 ) + e -i(2vθ 0 +ω α,θ 0 ) ) = ∞ v=k 0 +1 β(α) v e i(ω α,θ 0 ) + R. An Abdel summation provides |R| ≤ M 4 k α-1 0 = o(k α ) uniformly in k. Hence we have 2 k u=k-k 0 β (α) k-u,θ 0 ,c 1 cos(uθ 0 )F α ( u N ) = -K α,θ 0 ,c 1 cos (kθ 0 + ω α,θ 0 ) ∞ v=k 0 +1 β(α) v F N,α ( k N ) + o(k α ) = K α,θ 0 ,c 1 cos (kθ 0 + ω α,θ 0 ) k 0 v=0 β(α) v F N,α ( k N ) + o(k α ).
Then Remark 5 and the hypothesis β 0 0,θ 0 ,c 1 give the equalities

δ (α) u,θ 0 = T -1 N 2 2α (cos θ -cos θ 0 ) α c 1 u+1,1 (1 + o(1))
and for the same reasons

δ(α) u = T -1 N (1 -cos θ) α c 1 ) u+1,1 (1 + o(1)) .
According to Lemma 6 we have to treat the two sums (with the hypothesis

x < y) S 1,α = k u=0 δ (α) u,θ 0 δ (α) l-k+u,θ 0 and S 2,α = k u=0 δ (α) N -k+u,θ 0 δ (α)
N -l+u,θ 0 . For a sufficiently large integer k 0 we can split the sum S 1,α into

k 0 u=0 δ (α) u,θ 0 δ (α) l-k+u,θ 0 and k u=k 0 +1 δ (α) u,θ 0 δ (α)
l-k+u,θ 0 . We have

k 0 u=0 δ (α) u,θ 0 δ (α) l-k+u,θ 0 ≤ K α,θ 0 ,c 1 Γ(α) (l -k) α-1 (1 - l -k N ) α M k 0 . with M = max{δ (α) u,θ 0 }. Assume now k 0 = N γ with 0 < γ < α. We get k 0 u=0 δ (α) u,θ 0 δ (α) l-k+u,θ 0 = o(N 2α-1 ) (19)
In the other hand we have, following Theorem 4

k u=k 0 +1 δ (α) u,θ 0 δ (α) l-k+u,θ 0 = |K α,θ 0 | 2 k u=k 0 +1 cos (uθ 0 + ω α,θ 0 ) cos ((l -k)θ 0 + ω α,θ 0 ) u α-1 (1 - u N ) α (l -k + u) α-1 (1 - l -k + u N ) α + o(k 2α-1 )
As previously we obtain, with an Abel summation, that

k u=k 0 +1 δ (α) u,θ 0 δ (α) l-k+u,θ 0 = |K α,θ 0 ,c 1 | 2 Γ 2 (α) cos ((l -k)θ 0 ) k u=k 0 +1 u α-1 (1 - u N ) α (l -k + u) α-1 (1 - l -k + u N ) α + S ′ 1,α . with |S ′ 1,α | = O   k u=k 0 +1 |ρ N (u + 1) -ρ N (u)|   and ρ N (u) = u α-1 (1 - u N ) α (l -k + u) α-1 (1 - l -k + u N ) α .
With the main value theorem we can write

|S ′ 1,α | = O   k u=k 0 +1 |ρ ′ N (c)|   u < c < u + 1. Hence |S ′ 1,α | = O   4 j=0 Σ (j) 1,α   = o(N 2α-1 ).
Finally we obtain

k u=k 0 +1 δ (α) u,θ 0 δ (α) l-k+u,θ 0 = |K α,θ 0 ,c 1 | 2 Γ 2 (α) cos ((l -k)θ 0 ) k u=k 0 +1 u α-1 (1 - u N ) α (l -k + u) α-1 (1 - l -k + u N ) α + o(N 2α-1 ).
As for the equation ( 19) we get

k 0 u=0 δ(α) u δ(α) l-k+u . Consequently we can conclude S 1,α = |K α,θ 0 ,c 1 | 2 cos ((l -k)θ 0 ) k u=0 δ(α) u δ(α) l-k+u + o(N 2α-1 ). ( 20
)
As previously we can split the sum S 2,α into

k-k 1 -1 u=0 δ (α) N -k+u,θ 0 δ (α) N -l+u,θ 0 and k u=k-k 1 δ (α) N -k+u,θ 0 δ (α)
N -l+u,θ 0 .

Using Lemma 5 we obtain the bound

k u=k-k 1 δ (α) N -k+u,θ 0 δ (α) N -l+u,θ 0 ≤ k u=k-k 1 |β (α) N -k+u,θ 0 ,c 1 δ (α) N -l+u,θ 0 | + k u=k-k 1 |δ (α) N -l+u,θ 0 | 1 N N -k+u v=0 |β (α) N -k+u-v,θ 0 ,c 1 ||F N,α ( v N )| + o(N 2α-1 ). Assume now k 1 = o(N ). We have k u=k-k 1 |β (α) N -k+u,θ 0 ,c 1 δ (α) N -l+u,θ 0 | ≤ O   (N -l + k) α-1 ( l -k N ) α k u=k-k 1 (N -k + u) α-1   ≤ O N 2α-1 1 -(1 - k 1 N ) α = o(N 2α-1 ) and k u=k-k 1 |δ (α) N -l+u,θ 0 | 1 N N -k+u v=0 |β (α) N -k+u-v,θ 0 ,c 1 ||F N,α ( v N )| = O (N -l + k) α-1 ( l -k N ) α k 1 N α-1 1 0 ln(1 -t + α + 1 N )dt = o(N 2α-1 )
Lastly we obtain, still with an Abel summation

k-k 1 -1 u=0 δ (α) N -k+u,θ 0 δ (α) N -l+u,θ 0 = |K α,θ 0 ,c 1 | 2 Γ 2 (α) cos ((l -k)θ 0 ) k-k 1 -1 u=0 (N -k + u) α-1 ( k -u N ) α (N -l + u) α-1 ( l -u N ) α
+ o(N 2α-1 ).

Merging this last equality with (20) we obtain

S 2,α = |K α,θ 0 ,c 1 | 2 cos ((l -k)θ 0 ) k u=0 δ(α) N -k+u δ(α) N -l+u + o N 2α-1 . (21)
The equations ( 20) and ( 21) and Lemma 6 provide Theorem 6 for the case 1 2 > α > 0. The uniformity is a direct consequence of Theorem 4 and Lemmas 4 and 5.

6 Proof of Corollary 4 and 5

4 1 Remark 2

 12 Demonstration of Theorem 4 4.1 Asymptotic of β (α) k,θ 0 ,c In the rest of this paper we denote by c 1,1 the function in

Fsin πα π 2m F

 2m m,N,α (z). Let us denote by F N,α (z) the sum +∞ m=0 m,N,α (z). The function F N,α is defined, continuous and derivable on [0,1[

Lemma 7 .Lemma 8 1 N 1 T 1 - 1 N

 781111 For θ 0 ∈]0, π[ and α ∈]0,1 2 [ we haveT N (2(cos θ -cos θ 0 )c 1 ) -T N 2 2α (cos θ -cos θ 0 ) 2α c 1 ≤ K( 1 2 α)Nwhere K is a constant no depending from N .Proof : By the main value Theorem we have|2 2α (cos θ -cos θ 0 ) 2α -2(cos θ -cos θ 0 )| ≤ 4(1 -2α)2 cα(θ) (cos θ -cos θ 0 ) cα(θ) | with 0 < c α (θ) < 1 -2α. Hence the function ψ α → θ → 2 cα(θ) (cos θ -cos θ 0 ) cα(θ) c 1 (θ) is in L 1 (T) . For all integer k, 0 ≤ k ≤ N we consider the integral I k = 2π 0 2 2α (cos θ -cos θ 0 ) 2α -2(cos θ -cos θ 0 ) c 1 (θ)e -ikθ dθ. Assume 1 2α → 0 and put ǫ, 0 < ǫ < 1 -2α, for α sufficiently closed from 1 2 . Put I k = I k,1 + I k,2 + I k,3 with I k,1 = θ 0 -ǫ 0 2 2α (cos θ -cos θ 0 ) 2α -2(cos θ -cos θ 0 ) c 1 (θ)e -ikθ dθ, I k,2 = θ 0 +ǫ θ 0 -ǫ 2 2α (cos θ -cos θ 0 ) 2α -2(cos θ -cos θ 0 ) c 1 (θ)e -ikθ dθ, I k,3 = 2π θ 0 +ǫ 2 2α (cos θ -cos θ 0 ) 2α -2(cos θ -cos θ 0 ) c 1 (θ)e -ikθ dθ. It is easy to see that |I k,1 | and |I k,3 | are bounded by M (1 -2α) with M is a positive real no depending from k or N . Easily |I k,2 | ≤ 1 -2α ψ α 1 . Hence |I k | ≤ M 2 (1 -2α)where M 2 is a positive real no depending from k or N . In the other hand it is well known that for a N × N matrix A we have This last result achieves the proof.✷ Let δ > 0 a fixed real. For α < 1 2 such that1 2α sufficiently near of zero we have for all integer k, 0≤ k ≤ N T -1 N (|χχ 0 ||χ -χ0 |c 1 ) (χ k ) -T -1 N |χχ 0 | 2α |χ -χ0 | 2α c 1 (χ k ) ≤ o(N -δ ).Proof : Let us denote by T 1/2,N the matrix T -1 N (|χχ 0 ||χ -χ0 |c 1 ) and by T α,N the matrixT -|χχ 0 | 2α |χ -χ0 | 2α c 1 . Obviously T 1/2,N = T α,N Id + T -1 α,N T 1/2,N -T α,N .Corollary 2 and Lemma 6 imply the existence of a positve real C such that for all integers k, l, 0 ≤ k, l ≤ N , we have(T α,N ) k,l ≤ N C . (22) Since T α,N T 1/2,N -T α,N ≤ T -1 α,N T 1/2,N -T α,Nthe previous Lemma impliesT -1 α,N T 1/2,N -T α,N ≤ C ′ ( 1 2 α)N C+2 . (23) Put α = 1 2o(N -(C+3) ). From (23) the matrix Id + T -1 α,N (T 1/2,N -T α,N ) N = Id + T -1 α,N (T 1/2,N -T α,N ) --1α,N . Then we can write, for all integer k, 0≤ k ≤ N T -1 1/2,N (χ k ) -T -1 α,N (χ k ) ≤ Id + T -1 α,N (T 1/2,N -T α,N ) -Id T -1 α,N (χ k ) ≤ T -1 α,N (T 1/2,N -T α,N ) T -1 α,N (χ k ) 1 -T -1 α,N (T 1/2,N -T α,N ).As for the equation (22) we have obviously a constant J no depending from k or N such that T -1 α,N (χ k )| ≤ O(N J ). If α = 1 2 + o(N -(C+J+3+δ) ) we have T -1 N (|χχ 0 ||χ -χ0 |c 1 ) (χ k ) -T -|χχ 0 | 2α |χ -χ0 | 2α c 1 (χ k ) ≤ o(N -δ ) ✷

Using [START_REF] Simon | Orthogonal polynomials on the unit circle, Part 2: spectral theory[END_REF] and [START_REF] Szegö | Orthogonal polynomials[END_REF] we get the point ii).

To achieve the proof we have to remark that the uniformity in k in the point iii) is a direct consequence of Property 2. ✷

We have now to state the following lemma Lemma 5 for k N → x, 0 < x < 1 we have

This Lemma and Lemma 4 imply the equality

with (see [START_REF] Rambour | Inverse asymptotique des matrices de Toeplitz de symbole (1 -cos θ) α f 1 , -1 2 < α ≤ 1[END_REF] Lemma 3)

Proof of lemma 5: With our notation assume x ∈ [0, δ], 0 < δ < 1. Put k 0 = N γ with γ ∈ ] max( α β , -α 1-α ), 1[ if α < 0, and γ ∈]0, 1[ if α > 0. We can splite the sum 2

) cos(uθ 0 ). Property 1 and the assumption on β show that

uniformly in k. It is known that the second sum is also

With Lemma 4 we obtain, as previously

uniformly in k. Since we have seen that the sum

we can also conclude, as for α > 0

The uniformity is clearly provided by the uniformity in Lemma 4 and by the previous remarks. This last remark is sufficient to prove Lemma 5. ✷

Then Theorem 4 is a direct consequence of the inversion formula and of Lemma 5.

Proof of Theorem 6

Let us recall the following formula, which can be related with the Gobberg-Semencul formula.

Let P N,α,θ 0 and P N,α be the predictor polynomials of |χχ 0 | 2α |χ χ0 |c 1 and |1 -χ| 2α . We put

Proof : With an Abel summation we obtain, if


Kris Heylen
email: kris.heylen@kuleuven.be

Stephen Bond
email: stephen.bond@kuleuven.be

Dirk De Hertog
email: dirk.dehertog@kuleuven.be

Hendrik Kockaert
email: hendrik.kockaert@kuleuven.be

Frieda Steurs
email: frieda.steurs@kuleuven.be

Ivan Vulić
email: ivan.vulic@cs.kuleuven.be

TermWise: Leveraging Big Data for Terminological Support in Legal Translation ⋆

Keywords: Legal Terminology, Automatic Knowledge acquisition, Big Data, Context Sensitive Suggestion

Increasingly, large bilingual document collections are being made available online, especially in the legal domain. This type of Big Data is a valuable resource that specialized translators exploit to search for informative examples of how domain-specific expressions should be translated. However, general purpose search engines are not optimized to retrieve previous translations that are maximally relevant to a translator. In this paper, we report on the TermWise project, a cooperation of terminologists, corpus linguists and computer scientists, that aims to leverage big online translation data for terminological support to legal translators at the Belgian Federal Ministry of Justice. The project developed dedicated knowledge extraction algorithms and a server-based tool to provide translators with the most relevant previous translations of domain-specific expressions relative to the current translation assignment. In the paper, we give an overview of the system, give a demo of the user interface and then discuss, more in general, the possibilities of mining big data to support specialized translation.

Introduction

Translators in specialized domains are confronted with source texts that are teeming with highly specific terminology and other domain-specific expressions. Even the most experienced of translators regularly needs to check the translation of such expressions against a reliable resource. Although (online) specialized dictionaries and state-of-theart Computer Assisted Translation (CAT) tool offer some terminological support, the coverage of Translation Memories (TM), Term banks and Term Bases is often insufficient. Typically, translators turn to online collections of bilingual documents and search these with a general-purpose search engine (see [START_REF] Biel | Legal terminology in translation practice: dictionaries, googling or discussion forums[END_REF] for a discussion of typical search behavior). However, finding relevant examples is often hard and time-consuming and the reliability of online sources in not always guaranteed. In this paper we present the outcome of the TermWise project, which tries to leverage big online collections of bilingual documents to offer additional terminological support for domain-specific translation in a user-friendly way. The TermWise project adds an extension to existing CAT-tools in the form an extra cloud-based database, which we call a Term&Phrase Memory. It provides one-click access to translations for individual terms and domain-specific expressions that stem from known, trustworthy online sources and that are sorted for relevance to the translator's current translation assignment. The Term&Phrase Memory has been compiled by applying newly developed statistical knowledge acquisition algorithms to large parallel corpora harvested from official, public websites. Although these algorithms are language-and domain-independent, the tool was developed in a project with translators from the Belgian Federal Justice Department (FOD Justitie/SPF Justice) as end-user partners. Therefore the tool is demonstrated in a case study of bidirectional Dutch-French translation in the Belgian legal domain. In this paper, we first describe the specific needs that our end-user group expressed and how we translated them into the new functionality of the Term&Phrase Memory. Next, we summarize the term extraction and term alignment algorithms that were developed to compile the Term&Phrase Memory from large parallel corpora. Section 4 describes how the Term&Phrase Memory functions as server database that is now, in this proof-of-concept phase, accessed via a lightweight stand-alone tool, but that is designed to be fully integrated with a CAT user-interface so as to provide context-sensitive terminological support in the normal translation work-flow. In Section 5, we present the user-based evaluation of the tool that was carried out by students of Translation Studies and professional translators at the Belgian Federal Justice Department. Section 6 concludes with a discussion of how TermWise is an example of a dedicated linguistic search tool that allows translators to exploit Big Data that takes the form of large online bilingual document collections.

User needs of Legal Translators

Like other domain-specific translators, the translators at the Belgian Ministry of Justice are confronted with source texts full of domain-specific terminology which requires exact (as opposed to interpretative) translation and which even skilled translators need to check against a reference source once in a while. However, existing (online) Belgian legal dictionaries have limited coverage and are outdated. Also in the commercial CAT-tool used by the Ministry, the support for terminological look-up is quite limited. As with most CAT-tools, it does come with a Term Base functionality, but this type of terminological dictionary is initially empty and entries have to be added manually. Even a large organization like the Ministry cannot afford to invest much time in Term Base compilation. They acquired an externally compiled Term Base, but its coverage is limited and it contains no informative examples of the idiomatic usage of terms in contexts. Such proper phraseological usage of terms is especially important in legal language, where validity of a text depends on the usage of the appropriate formulae. Although the commercial tool's Translation Memory (TM) automatically gives translation suggestions, its retrieval on the level of entire sentences or even paragraphs is too coarse-grained for finding examples of individual words and phrases. A concor-dancer does allow for a manual look-up of a specific expression, but occurrences are not sorted for relevance, nor do they come with meta-data about the source document that could allow translators to assess its relevance and reliability. Additionally, the TM only keeps track of the Ministry's in-house translations, and does not include the vast body of relevant bilingual legal documents translated at other departments. The translators therefore often resort to doing Google searches for terms and phrases in open on-line legal document repositories to check previous translations in specific contexts. However, also here, the relevance of the search hits must be assessed manually. Based on this situation, we identified the following user needs: To our knowledge, this combination of functionalities is not implemented in any existing CAT-tool [START_REF] Reinke | State of the art in translation memory technology[END_REF]. In TermWise they are grouped in a separate module, which we will call a Term&Phrase Memory, so that in principle this module can be integrated in existing CAT-tools. However, commercial CAT-tool developers do not readily allow plug-ins by third parties. Also, the focus of the TermWise project was to deliver a proof-of-concept for the Term&Phrase Memory's functionality, not to develop a fully functional CAT-tool. Therefore, we opted to implement a stand-alone, lightweight tool to showcase the new functionality of the Term&Phrase Memory, but in such a way that it can easily interact with the current commercial CAT software of the Belgian Ministry of Justice. In the next section, we discuss which type of information is included in the Term&Phrase Memory and how it was compiled. Section 4 describes the user interface.

-

Corpus and Knowledge Acquisition

A number of official bilingual legal document collections are put online by the Belgian Federal Government (e.g. juridat 4 , De Kamer/La Chambre5 ) but for our case study, we focused on the largest collection, viz. the online version of the Belgian Official Journal (Belgisch Staatsblad/Moniteur Belge6 ), which publishes laws, decrees, and official communications of both the national state and the federal entities, in both French and Dutch. We implemented a web crawler in python to systematically download all the issues to our server. For the case-study, we only use the issues from 1997 to 2006 because they have been published as a curated, open-souce corpus (100M words) [START_REF] Vanallemeersch | Belgisch staatsblad corpus: Retrieving french-dutch sentences from official documents[END_REF] 7 . However, in a next stage, the aim is to continually update the corpus with new issues.

All issues were language-checked8 , tokenized and POS-tagged 9 , and sentence-aligned 10with publicly available tools. The webcrawler also retrieved the source department (e.g. ministry, agency) for all documents. Both the Dutch and French corpus were POStagged with TreeTagger. To extract domain-specific expressions and their translations, we followed the extract-then-align paradigm that is predominant in the literature on bilingual terminology extraction (e.g., see [START_REF] Daille | Towards automatic extraction of monolingual and bilingual terminology[END_REF]; [START_REF] Gaussier | Flow network models for word alignment and terminology extraction from bilingual corpora[END_REF]; [START_REF] Hervé Déjean | An approach based on multilingual thesauri and model combination for bilingual lexicon extraction[END_REF]; [START_REF] Le | Mutual bilingual terminology extraction[END_REF]; [START_REF] Lu | Towards bilingual term extraction in comparable patents[END_REF]). In this paradigm, terms are first extracted for the two languages separately and then in a second step aligned cross-lingually. Although both tasks are well known in NLP and have many existing implementations, most current tools are geared towards delivering intermediate results for a Machine Translation system or further manual lexicon compilation. In the Term&Phrase Memory, however, the output has to be usable directly by end-users. We therefore developed our own statistical algorithms for term extraction and term alignment to accommodate the specific user needs above. The knowledge acquisition proceeded in two steps.

STEP 1: Domain-Specific N-gram Extraction Following [START_REF] Lise | Phrasemes in legal texts[END_REF], we consider expressions of variable length as relevant for the legal domain. These do not only include single and multi-word terms that refer to legal concepts (typically NPs), but also phraseologies (e.g. typical verb-NP combinations), and formulaic expressions that can comprise entire clauses. The term extraction algorithm therefore considers n-grams of variable length without imposing predefined languagespecific POS patterns as is the case in most term extraction algorithms. Instead, the relevance of an n-gram is assessed based on its external independence and its internal coherence. Independence is the extent to which an n-gram can occur in different contexts. Following [START_REF] Ferreira Da Silva | Using localmaxs algorithm for the extraction of contiguous and non-contiguous multiword lexical units[END_REF], this is operationalized as a maximazation of frequency differences relative to the n-1 and n+1 grams in an n-gram expansion progression. Coherence is the extent to which the lexemes within an n-gram tend to co-occur in an informational unit. This is measured as the Mutual Information of the n-gram's POS-sequence. The algorithm is described in more detail in [START_REF] De | TermWise Xtract: Towards a contextualised term model[END_REF]. Note that the expressions extracted do not necessarily correspond to theoretically motivated, concept-based terminological units, but rather to domain-specific expressions in general that are of practical use to a translator. The extraction step resulted in a list of 649,602 n-grams for French and 639,865 n-grams for Dutch.

STEP 2: Bilingual N-gram Alignment

The goal of the alignment step was to provide for each Dutch n-gram a ranked subset of likely translations from the French n-grams list and vice versa. To build these ranked subsets, we developed a statistical algorithm for bilingual lexicon extraction (BLE) from parallel corpora, called SampLEX, and adapted it to handle n-grams of variable length. In a pre-processing step, the aligned sentences in the corpus are represented as a bag-ofterms taken from the French and Dutch input lists. SampLEX uses a strategey of data reduction and sub-corpora sampling for alignment. For more details about the algorithm and its properties, and benchmarking against other BLE models, we refer the reader to [START_REF] Vulić | Sub-corpora sampling with an application to bilingual lexicon extraction[END_REF]. Running SampLEX results for each Dutch n-gram in the list of French n-grams sorted by translation probability and vice versa. Also, the document and sentence ID of each occurrence of a candidate translation-pair in the corpus is returned. As a postprocessing step, a hard cut-off of the output ranked lists of translation candidates is performed. Some example output is displayed in Table 1

Context-sensitive Database Querying

The Term&Phrase Memory is conceived to function as an additional database accessible from within a CAT-tool's user-interface, next to the Translation Memory and Term Base. As with terms contained in a manually crafted Term Base, the terminological expressions included in the Term&Phrase Memory are highlighted in the source text of the translator's new assignment. By clicking on them, their previous translations-in-context are shown in a separate pane. Figure 1 illustrates this for the expression méthodes particulières de recherche in segment 5 of a Belgian-French legal document. The examples are ranked by relevance, defined as the similarity of their respective source documents to the current source text. The meta-data of the examples' source documents (e.g. issuing ministry or agency, state or federal level) and a link to the online version is also provided, both in html and pdf. This way, the user can assess the relevance and reliability of the translation's source. If the user agrees with a suggested translation, a button click copies it to the active segment in the target text pane.

Although the Term&Phrase Memory is meant to be integrated into a CAT tool, in the current test phase, it is implemented as a stand-alone tool. However, to make the tool easily usable next to a CAT tool, it is possible to upload the xliff file that CAT tools use to store translation projects in a segmented format. This makes sure that the segmentation of the source text in the TermWise tool is compatible with the one in CAT Figure 2 shows the architecture behind the TermWise tool. The system consists of a server, which handles translation requests, and a client, which issues the requests and displays the returned results in a GUI. When handling a translation request, the server takes as input a xliff-file or plain txt file and returns an XML file containing the segmentized document, translation suggestions for each segment, the n-grams found in the document, and translation suggestions for each n-gram together with context-sensitive annotated usage examples. The translation suggestions for segments correspond to the fuzzy matches from Translation Memories in traditional CAT-tools, but in this case the entire online document collection of the Belgisch Staatsblad/Moniteur Belge functions as a TM. The fuzzy matching algorithm is similar to that in existing software will not be further discussed here. Instead we will focus on handling of n-grams for the new Term&Phrase functionality.

The Term&Phrase Memory consists of (a) a list of paired, sentence-aligned documents from the Belgian Official Journal annotated with their source department, and (b) a dictionary of the n-grams found in those documents. In the latter, each n-gram is associated with a list of translation candidates of a given translation probability, and each n-gram translation pair is associated with the list of documents and line numbers in which that translation is found.

When the server receives an the input document in xliff format the segmentation is checked. If it is in plain txt , it is first segmentized using the Alpino tokeniser [START_REF] Van Noord | At Last Parsing Is Now Operational[END_REF]. N-grams are extracted from the segmentized input document by consulting the n-gram dictionary of the same language. A ranked list of similar corpus documents and their N-gram translations to be suggested are chosen on the basis of the given translation probabilities and on document similarity. The list of documents that are similar to the input document is compared with the list of documents for each n-gram translation pair. The relevance value for an n-gram translation pair is determined by a weighted interpolation of its given translation probability and the cosine similarity of the highest-ranking document on its list (based on a "set of n-grams" vector space model). If the relevance value exceeds a configurable threshold, that n-gram translation pair is displayed and suggested to the user. Example sentences are extracted from the highest-ranking document and from other high-ranking documents from the same source department.

Evaluation

The TermWise tool is evaluated by two end-user groups. In December 2014, 19 students of legal translation at the KU Leuven, campus Antwerp were made acquainted with the tool and then asked to translate a legal document from French into their native Dutch with the help of the TermWise tool alongside SDL Trados Studio 2011 that had the legal Translation Memory and Term Base of Belgian Federal Justice Department loaded. More specifically, the students were asked to record all the expressions in the source text that they normally would look up outside of the CAT tool and report whether they were present in the TermWise tool. The result are shown in Figure 3. Although not all desired expression were covered, students reported significant gains in look-up time.

Currently, seven professional translators at the Belgian Ministry of Justice are assessing the usability of the tool in their daily translation practice. First, legal translators are invited to make use of the tool to translate an unseen legal text and give comments and feed-back on the Term&Phrase Memory functionality and coverage as they are translating. Afterwards, they are also asked to fill in a survey on the general usability of the tool and the new functionality it offers. Results are expected by September 2014. The results of this qualitative evaluation will be used to improve the tool's userfriendliness and to fine-tune the parameters of the knowledge acquisition algorithms and the context-sensitive search function. Big Data is a buzz word in ICT in general and also in the translation industry. Discussions on the opportunities that big parallel corpora 11 offer for translation, usually focus on three aspects:

1. Sharing translation memories as open data (e.g. [START_REF] Sandrini | Open Translation Data. Die gesellschaftliche Funktion von bersetzungsdaten[END_REF]) 2. More data to improve (statistical) Machine Translation (e.g. [START_REF] Choudhury | TAUS Translation Technology Landscape Report[END_REF], p. 60) 3. More data to improve term extraction for the compilation of multilingual term bases or ontologies (e.g. [START_REF] De | Babouk: Focused Web Crawling for Corpus Compilation and Automatic Terminology Extraction[END_REF])

However, these approaches deal with derived products (TM's, MT systems or Term ontologies) and do not acknowledge that the translators themselves might want to exploit the data directly to help them in their translation process. Actually, professional translators are often very good at assessing applicability of a translation by comparison to previous examples and only resort to dictionaries when real conceptual confusion is at stake (for which good terminological work is still crucial). However, translators do need support to find translation examples that are informative and relevant to their current translation assignment in the deluge of available parallel data. Additionally, meta-data about the source of a previous translation is crucial to assess the reliability and appropriateness of the example. Clearly, general search engines like Google are not optimized for this type of linguistic search, but also parallel corpus search tools like Linguee 12only allow for the context-insensitive look-up of expressions that do not take into account the specific assignment that a translator is working on. The TermWise project aims precisely to combine access to large and constantly expanding online bilingual document collections with support for highly context-specific translation needs. More specifically, the Term&Phrase Memory functionality presented in this paper improves over current practices in the following ways:

-Highly domain-specific expressions are identified for the translator, whereas in concordance searches in current CAT and corpus search tools, translators have to select expressions for look-up themselves. Thanks to the dedicated term extraction algorithm, these expressions go beyond traditional noun phrases and include phrasemes and typical formulae -Moreover, the domain-specific expressions have already been looked up for the translator beforehand as the source text is submitted to a pre-search when it is uploaded to the tool. The translator just has to click the expression in the source text to get to the examples. -Like in state-of-the-art parallel corpus search tools, the domain-specific expressions are aligned to their translation and the translator does not have to locate the relevant passage in a bilingual document. -Unlike with corpus search tools, the examples are sorted for relevance to the current translation assignment: Searches for expressions are not executed in isolation, but the context of the source text is taken into account. -Unlike in a general search engine, the translator only gets translation examples from selected reliable sources and the meta-data of the source is readily provided.

We believe this type of functionality complements other resources that translators have available. Machine Translation can reduce translation time, but post-editing will remain necessary for the foreseeable future, and post-editors need easy access to online repositories to check translations. Also, high quality term banks and specialized (online) dictionaries remain a crucial resource for translators, but these are time-consuming and expensive to compile and maybe not necessary for all terminological needs of translators. Informative translation examples from qualitative and reliable sources can go a long way. In short, we argue that Term&Phrase Memory offers a novel functionality that is highly useful for specialized translation.

  Access to previous translations of domain-specific single and multi-word expressions -Examples of usage in context to infer correct phraseology -Information about the source documents of the translation examples -Examples from all relevant documents that are available online -Sorting the examples by relevance to the current translation assignment -Easy access to the examples from within the CAT-tool

Fig. 1 .

 1 Fig. 1. Screen cap of TermWise GUI with n-grams highlighted in the source text and translation examples displayed in the Term&Phrase Memory pane

Fig. 2 .

 2 Fig. 2. TermWise Client-Server Architecture

Fig. 3 .

 3 Fig. 3. Evaluation results with students

Table 1 .

 1 . Example output of the SampLEX algorithm for n-grams. Translation direction is French to Dutch.

	sur la proposition du conseil d' administration
	op voorstel van de raad van bestuur	Prob: 0.621
	op voordracht van de raad van bestuur	Prob: 0.379
	16 mai 1989 et 11 juillet 1991
	16 mei 1989 en 11 juli 1991	Prob: 1.0
	sur la proposition du ministre
	de voordracht van de minister	Prob: 0.481
	op voorstel van de minister	Prob: 0.111
	op voordracht van de minister	Prob: 0.074
	. . .	. . .

http://www.cass.be/

http://www.dekamer.be/

http://www.ejustice.just.fgov.be/cgi/welcome.pl

http://opus.lingfil.uu.se/MBS.php

TextCat: http://odur.let.rug.nl/ ˜vannoord/TextCat/

TreeTagger[START_REF] Schmid | Probabilistic part-of-speech tagging using decision trees[END_REF]

Geometric Mapping and Alignment system[START_REF] Melamed | Pattern recognition for mapping bitext correspondence In Jean Véronis[END_REF]

For an overview of the exploitation of comparable corpora, see[START_REF]Building and Using Comparable Corpora[END_REF].

see[START_REF] Volk | Elena Callegaro Innovations in Parallel Corpus Search Tools[END_REF] for an overview and comparison of current tools

⋆ TermWise was funded by KU Leuven IOF grant KP/09/001. Special thanks to Martine Perpet of the FOD Justitie.


Juan Carlos Diaz Vasquez1

Carolina Franco
email: cfranco3@eafit.edu.co

TERMINOLOGY MANAGEMENT SYSTEMS AND ITS ROLE IN THE DEVELOPMENT OF THE LANGUAGE INDUSTRY IN COLOMBIA

Keywords: Terminology Management Systems TMS, Language Industry, Linguistic Software, International Standards, Internationalization

This paper reflects the importance of Information and Communication Technologies (ICTs), especially Terminology Management Systems TMSs, as a key factor for the development of the language industry in Colombia and its economic global projection. Services offered within the scope of the Language Industry are mostly related to TMS, including tools for terminology extraction, knowledge management and communication, throughout the productive chain. The article concludes upon a survey conducted in the three biggest cities of Colombia about the appropriation levels of such tools in this industry and its impact on firm development.

Introduction

This research paper highlights the importance and dependence of the language industry on the newest ICT. Such technologies are the foundations for the effective providing of such services, which include, among others: translation, interpretation, subtitling, dubbing, software localization, web site globalization, development of language technologies, language education, consulting [LTC, 2009)]. And which make use of tools such as Terminology Management Systems, Translation Memories, Lexicons, Thesaurus, Taxonomies, Ontologies, Web-Dictionaries, E-mails, Videoconferencing Systems, which are intrinsic to linguistic activities. Additionally, these tools have propelled the internationalization of the Language Industry, through the reduction of communication costs and facilitating access to information. They are in many cases, even with unawareness of the producer, the entry modes to foreign markets and therefore essential part of the internationalization process of companies providing these services.

In general terms, Terminology Management Systems are probably the most common tool for technical and scientific translation activities, which represent a large proportion of the language industry activities. In this sense, the research pretends to answer the following question: Which are the most common technologies used in the language industry in these cities in Colombia and what is their role in the development of the companies?

Context

The digital revolution, since its early beginnings of Internet access, has been considered as a driver of economic growth, directly influencing GDP levels as pointed out by [START_REF] Katz | La contribución de la banda ancha al desarrollo económico[END_REF]. For the service sector, which has a high component of human capital, the proper management and use of information are of significant relevance for organizational operations, this explains why ICT have leveraged the growth of the sector worldwide [START_REF] Javalgi | Internationalization of services: Identifying the building-blocks for future research[END_REF]. ICT access and proper connectivity is considered imperative for the internationalization process, not only as a tool for communication with suppliers, customers and strategic partners but as an enabler of commercial processes to enter new markets through e-commerce.

In addition to the effects explained by [START_REF] Katz | La contribución de la banda ancha al desarrollo económico[END_REF] in the previous figure, the penetration of the broadband has some extra worthwhile mentioning effects, such as the improvement of education, increased possibilities for freedom of speech, increase of political conscience by accessing diverse sources of information, access to scarce products and services, etc.

In 2007, Latin America presented low levels of internet access compared to more developed countries (OCDE) and compared to the world average. The bright side is that such levels have presented patterns of growth since 1993 [López & Hilbert, 2010]. Nevertheless, it is important to keep in mind that the positive social and economic effects deriving from the access to broadband and ICT do not depend exclusively from the number of users in the region; it also depends on the quality of the service, and the education and training of the social actors for its appropriate use.

Colombia has been recognized for its production of primary goods. However, as the service sector and knowledge intensive industries gain importance in the economy also do the technologies for information and content management. Nonetheless, specific tools for Knowledge Management -KMhave developed at a slow pace and have received poor attention from the private, public and academic sectors [START_REF] Baquero | An Exploration of Knowledge Management in Colombia[END_REF]. In knowledge intensive industries such as the language industry, is of high relevance the documentation of practices, techniques and employees implicit knowledge, in order for such practices to be preserved and improved by the organization to establish more effective processes. The study conducted by [START_REF] Baquero | An Exploration of Knowledge Management in Colombia[END_REF], presented that only 22% of the large-and medium-size enterprises in Colombia had a KM program implemented (written policies and strategies of KM) in 2007. One limitation for this study is the lack of segregation by sector since the average results may vary from the primary, secondary and tertiary sector. We could however induce, that if the largest companies in Colombia have not prioritized the information management, the situation for micro-and small-size companies is less encouraging.

In 2009, the Ministry of Communications of the Republic of Colombia was transformed to the Ministry of Information and Communication Technologies. Its purpose is to increase the access of Colombians to ICT through the implementation of new policies and programs intended to improve social, economic and politic development [MinTIC, 2014a].

The different strategies developed by the Ministry and implemented with the support of local administrations, are centered in a policy named "Vive Digital" (Live Digital). Such policy looks for the creation and strengthening of a digital ecosystem based on four strategic axes: Infrastructure, services offered by operators, applications, and users; while strengthening the demand and supply of the digital market [MinTIC, 2014b]. One of the projects that captured our attention is "Gobierno en Línea" (On-line Government), which promotes the modernization of the State and of its interactions with its citizens through ICTs. It integrates sites of information, procedures and services of the different government entities (Gobierno en Línea, ND).

Despite the slow growth in the access to Internet and ICT, the country still presents lags in this subject. For diverse industries, both national and international, this represents a great barrier for the access to the Colombian market, where only 16% of the surveyed Colombians participated on e-commerce in 2012 [Ipsos MediaCT, 2012].

The service sector in Colombia is the sector that has presented the biggest growth between 1970and 2011, according to Fedesarrollo [2011]]. However, such growth has not been reflected in the Colombian exports. This presents a broad spectrum of opportunities, specially for products and services with high knowledge and technological content since in 2000 only 12.7% of Colombian exports belonged to these types [START_REF] Rojas | nueva economía y TICS: Colombia, un caso de rezago tecnológico[END_REF]. For 2010, the percentage of manufactures with high and medium technology content was approximately 34% [START_REF] Torres | Estructura tecnológica de las exportaciones industriales en América Latina 1990-2010[END_REF]. This sector along with services, represent then high potential to be exploited, with regional referents as Mexico 75%, Costa Rica 50% and Brazil 44%, in the same period [Ibid].

Terminology Management Systems

The technic side of translation has become one of two necessary conditions that determine the professional capabilities of translation activities and other linguistic services; the level of technical capabilities is often a mandatory requirement to be competitive in a market that is more and more driven by ICTs. Covering the topic of software tools for every linguistic aspect requires a broad analysis of many of them. A common denomination that covers several tools for such services is Computer Assisted translation, which is the main focus of this article. To be more specific in tracking technologies for translation purposes in the scientific and technical domain we will focus in tools for terminology management, whose use is expected to be more widely spread.

The Terminology Management Systems are the software intended to administrate terminology data [START_REF] Arntz | Einführung in die Terminologiearbeit[END_REF], which are necessary for computer assisted terminographic work and for terminology management. Since such systems were conceived at first as special data banks for translations, they allow, mostly, interactions with text processing systems [Ibíd].

Terminology Management Systems are computerized programs that facilitate collection, preservation and access to terminological data for reference of linguistic professionals, translators, terminologists and technical editors in specialized fields. [START_REF] Vásquez | Aplicación de Normas Terminológicas como apoyo al Sector Exportador en Colombia[END_REF]. By using such systems, it is possible to create terminology databases with entries containing term-related information.

Since Information Technologies have changed over time, so is terminology doing and moving forward to address more users' needs. Some of those IT's changes is the virtualization of the content so it is no longer storage in a given unit but accessible through Internet. These trends allow the use of different kind of mobile devices accessing the content in the Internet at the same time. There is then the cloud-based terminology [START_REF] Varga | Virtualization of research in terminology. Cloud-base terminology tools[END_REF]. It has turned TMS's into collaborative tools on the web. These Terminology Management tools have developed solutions to different user including creation of terminology management networks, educational tools (elearning) and commercial solutions (product data classifications).

TermCoord (www.termcoord.eu) is a very good example of a terminology network. This initiative fosters international collaboration within the framework of the EU in order to develop the mega terminology database IATE that is the biggest interactive terminology database from Europe. Termwiki (www.termwiki.com) is also a social network initiative offering Terminology Management tools for open communities and commercial purposes.

Other initiatives bringing Terminology Management solutions in an interactive way is the Euro Term Bank (www.eurotermbank.com). This platform enables users to participate in collaborative terminology projects editing and sharing the same information.

All these examples represent trends in the development of Terminology Management Systems that are necessary to carry on translation related tasks. Knowing these developments is a requirement for firms offering technical translation.

METHODOLOGY

This research has an exploratory purpose. Several studies have been developed in the language industry in Europe and the United States [START_REF] Kelly | The top 100 language service providers[END_REF][START_REF] Taus | Lack of Interoperability Costs the Translation Industry a Fortune[END_REF]LTC, 2009], but little research has been conducted in this topic in Latin America. This research is being conducted with a quantitative analysis of primary nominal variables obtained through a web-based survey, from a positivist epistemological consideration that allows the acquisition of new knowledge through data collection [START_REF] Bryman | Social Research Methods[END_REF]. Although considered as opposed positions, the constructivist epistemological consideration also plays an important role since the social actors according to cultural, social and economic conditions might influence the results. Although [START_REF] Bryman | Social Research Methods[END_REF] also proposes that, the logic of a quantitative research is deductive, our research, in line with its exploratory purpose is developed inductively, since there is no initial hypothesis and the theory was proposed once the results were analyzed.

The choice for this data collection method was based on the advantages and disadvantages proposed by [START_REF] Bryman | Social Research Methods[END_REF] from a variety of methods. The selfadministered surveys via e-mail, allow us to reach a high number of subjects in a short term and with low costs, additionally this allows to the respondent to answer conveniently accordingly to the availability of technical information required, which offers an advantage compared to methods such as interviews. However, the same characteristic presented us with the inconvenience of a low response rate, which we tried to avoid by contacting by phone the subjects of interest, looking for their previous acceptance to participate in the research.

The survey was conducted with firms in the language industry in the three main cities of Colombia [Bogota, Medellin and Cali]. The section of the survey that interests this article was focused on obtaining information about the use of Information and Communication Technologies (ICT), its relevance for the growth and expansion of the firms and the impact of the policies of the Ministry of Information and Communication Technologies in the development of a National "Digital Ecosystem" [MinTIC, 2013].

Finally, secondary data was obtained to complement the understanding and analysis of primary information. The main source for such data was the Colombian Ministry of Information and Communication Technologies. An analysis of existing literature was also conducted to provide context to the results of this research in comparison to previous research.

Results

We identified a total of 251 firms in the language industry in the three main cities of Colombia: Bogota, Medellin and Cali. These cities were selected under two crite-ria: the offer of academic programs related with the industry (bachelor, specialization, master and doctorate degrees) and the population. In such criteria, these three cities ranked highest in Colombia.

From this initial population, we discarded 29 firms in the first filter since they were mainly free-lancers; these were legally constituted firms where the owner is the only translator. From the remaining 222 firms we were only able to contact 106, from which only 30 answered our survey and only 29 answered it completely.

Such low response rate hinders our ability to make inferences about the whole industry; therefore, results need to be presented merely in a descriptive way rather than in a generalized one.

Discussion

Despite the great efforts of the Colombian government to increase the appropriation of ICT and the inclusion of new technologies in the productive processes of companies, a wide use of them is not perceived among the industry. This might be explained by the fact that the main target for the ICT Ministry is the education sector rather than the productive sectors. Therefore, their impact in the productive sector is expected in the long run.

The lack of widespread use of technologies encouraging the systematization of processes is also reflected in the lack of accession to international protocols and standards. Only 8 out of 30 firms, comply with at least 1 national/international regulation.

Equally important, is the poor use of technologies for Knowledge Management, since, in line with the lack of systematization of process, it makes the firm's opera-

Statistics

Total Responses tions highly dependent of its employees. Consequently, if there is a change of personnel, much of the knowledge acquired with experience would be lost; what makes the language industry in Colombia less productive and less efficient in the long run.

It is very interesting to note how, despite of the developments of software specific for terminology management, some of these firms are still attached to tools such as Web Dictionaries. These tools, although of easy access and availability, are not reliable enough to support the translation process of technical and specialized documents.

Such findings allow us to present these firms of the language industry in Colombia, as firms with very informal production. There is little standardization of processes, not enough application of international standards over terminology, knowledge and content management such as ISO 26162, EN 15038, quality standards such as ISO9001 and even national standards over service supply as the NTC5808, which reflect a lack of professionalization.

This last characteristic is also reflected in the low recognition translation professionals have in the country. The study conducted by Clavijo, Duque, Franco, Mendoza & Rodriguez in 2008 reflected that in Colombia, the translation activities are, to a great extent, performed by professionals of fields different of linguistics and have empirical knowledge of other languages, but few knowledge of terminology management and its impact in the translation process. This lack of knowledge of terminology specificities, might then explain the lack of appropriation of facilitating technologies.

Conclusions

The Language Industry in Colombia, as represented by this few firms, is still an industry in development, which shows low levels of professionalization and standardization regarding specifically the use of ICTs. These are issues of high importance for a higher impact of this industry in the economy. For such process is necessary the accession and compliance with international standards and regulations establishing quality parameters for the service and the processes of this particular industry.

In Colombia, there is low participation of technologies in this industry what led us to conclude that firms in the language industry are highly dependent on the knowledge and expertise of their employees. The lack of appropriation of these technologies can be explained under the light of the above-mentioned conditions: informality and lack of knowledge on international standards.

Although Colombia has made significant progress in terms of technology appropriation, the Ministry of Communications was even changed for a more updated and comprehensive entity: the Ministry of Information and Communication Technologies, such efforts have been mainly focused on the education sector but its impact in the productive sector is yet to be seen.

Consequently, in order to increase the appropriation of technology tools with the purpose of increasing the productivity of firms, it is required a bigger joint action. Such action should be conducted by the industry, the Ministry of ICT, software pro-viders, international standard and regulation organizations (such as ISO) to inform, promote and support the inclusion of such tools in the operation and management of firms in the language industry.

The appropriation of Terminology Management Systems not only increases firm's efficiency and productivity but allows companies within this industry to provide higher quality services assuring coherence and standardization of the language.

Table I .

 I Results.

	Value	Count Percent
			%
	Email	30	100%
	Web site	25	83.3%
	Shared web site	2	6.7%
	Social Networks	20	66.7%
	Translation software	15	50%
	Translation memories	14	46.7%
	Terminology Management Sys	8	26.7%
	tems
	Mobile Apps	7	23.3%
	Other	5	16.7%