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

Philippe Rambour

To cite this version:

Philippe Rambour. Asymptotic of the terms of the Gegenbauer polynomial on the unit circle and applications to the inverse of Toeplitz matrices. 2013. �HAL-00874074v2�

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

Philippe Rambour^∗

Abstract

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

The first part of this paper is devoted to the study of the orthogonal polynomials on the unit circle, with respect of a weight of typefα:θ7→2^2α(cosθ−cosθ0)^2αc1withθ0∈]0, π[,

−¹2 < α < ¹₂ andc1a sufficiently smooth function. In a second part of the paper we obtain an asymptotic of the entries (TNfα)⁻_k+1,l+1¹ forα > 0 and for sufficiently large values of k, l, withk6=l.

Mathematical Subject Classification (2000)

Primary 15B05, 33C45;Secondary 33D45, 42C05, 42C10.

Keywords: Orthogonal polynomials, Gegenbauer polynomials, inverse of Toeplitz ma- trices.

1 Introduction

The study of the orthogonal polynomials on the unit circle is an old and difficult problem (see [16], [17] or [18]). The Gegenbauer polynomials on the torus are the orthogonal polynomials on the circle with respect to a weight of type f_α : θ 7→ 2^2α(cosθ−cosθ₀)^2αc₁ with α > −¹2

andc₁ a positive integrable function. In this paper we assume−¹₂ < α≤ ¹₂ andc₁ sufficiently smooth regular function. It is said that a function k is regular if k(θ) >0 for all θ ∈T and k ∈ L¹(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 functionh inL¹(T) we denote by T_N(h) the Toeplitz matrix of orderN with symbolh the (N + 1)×(N + 1) matrix such that

(T_N(h))_i+1,j+1 = ˆh(j−i) ∀i, j 0≤i, j≤N

where ˆm(s) is the Fourier coefficient of order s of the function m (see, for instance [3] and [4]). There is a close connection between Toeplitz matrices and orthogonal polynomials on the complex unit circle. Indeed the coefficients of the orthogonal polynomial of degreeN with respect ofhare also the coefficients of the last column ofT_N⁻¹(h) except for a normalisation (see [11]). Here we give an asymptotic expansion of the entries (T_N(f_α))⁻¹_k+1,1 (Theorem 4). Using

∗Universit´e de Paris Sud, Bˆatiment 425; F-91405 Orsay Cedex; tel : 01 69 15 57 28 ; fax 01 69 15 60 19 e-mail : philippe.rambour@math.u-psud.fr

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 [15]. In this last work we have treated the case of the symbols h_αdefined by θ 7→ (1−cosθ)^αc whith −¹₂ < α ≤ ¹₂ and the same hypothesis on c as on c₁. We have stated the following Theorem which is an important tool in the demonstration of Theorem 4.

Theorem 1 ([15]) If −¹₂ < α≤ ¹₂,α 6= 0 we have for c∈A(T,³₂) and 0< x <1 c(1) (TN(hα))⁻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∈[δ₁, δ₂]with 0< δ₁ < δ₂ <1, with the definition

D´efinition 1 For all positive real τ we denote by A(T, τ) the set A(T, τ) ={h∈L²(T)|X

s∈Z

|s^τˆh(s)|<∞}

This theorem has also been proved for α∈N^∗ in [14] and for α∈]¹₂,+∞[\N^∗ in [13].

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 ∈]0,¹₂] and _N^k → x,

l

N → y and 0 < x 6= y < 1. First we obtain these entries as a function of cos(l−k)θ₀ and (TN(hα))⁻¹_k+1,l+1. It is Theorem 6. With the same hypothesis as for Theorem 1 we have stated in [15] the following Theorem

Theorem 2 ([15]) For0< α < ¹₂ we have

c(1) (T_N(h_α))⁻¹[N x]+1,[N y]+1=N^2α−1 1

Γ²(α)G_α(x, y) +o(N^2α−1) uniformly in (x, y) for 0< δ₁ ≤x6=y <1.

Theorem 2 has been proved for α ∈ N^∗ in [14], for α = ¹₂ in [15] and for α ∈]¹₂,+∞[\N^∗ in [13]. The quantities G_α(x, y) is the integral kernel onL²(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(l−k)θ₀ 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 [2, 1, 6] and to [7, 8, 9, 2, 5, 10, 12] 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 ([11]). To have the polynomial used for the prediction theory we put

Φ^∗_n(z) = Xn

k=0

(Tn(f))⁻¹_k+1,N+1

(T_n(f))⁻¹_N+1,N+1z^k, |z|= 1. (1) We define the polynomial Φ^∗_n (see [16]) as

Φ^∗_n(z) =zⁿΦ¯_n(1

z), (2)

that implies, with the symmetry of the Toeplitz matrix Φ^∗_n(z) =

Xn

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^∗_nq

(Tn(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 ([11]),

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 ssuch that −n≤s≤n,

d1

|P_n|²(s) = ˆh(s).

It implies

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

1

|P_n|²

=T_n(h).

2 Main results

2.1 Main notations

In all the paper we consider the symbol defined byθ7→2^2α(cosθ−cosθ₀)^2αc₁ wherec₁ = ^|P_|Q|^| withP, Q∈R[X], without zeros on the united circle and−¹₂ < α < ¹₂ and 0< θ₀ < π.We have c₁ =c_1,1¯c_1,1 withc_1,1 = ^P_Q. Obviously c_1,1 ∈H²⁺(T) since H²⁺(T) ={h∈L²(T)|u <0 =⇒ ˆh(u) = 0}. Ifχis the functionθ7→e^iθ and ifχ₀ =e^iθ0 we putg_α,θ0,c1 = (χ−χ₀)^α(χ−χ₀)^αc_1,1 and g_α,θ0 = (χ−χ₀)^α(χ−χ₀)^α since(2(cosθ−cosθ₀))^2α = |χ−χ₀|^2α|χ−χ₀|^2α. Clearly g_α,θ0,c1, g_α,θ0 ∈ H²⁺(T) and 2^2α(cosθ−cosθ₀)^2αc₁ = g_α,θ0,c1g_α,θ0,c1, 2^2α(cosθ−cosθ₀)^2α = g_α,θ0g_α,θ0. Then we denote by β_k,θ^(α)

0,c¹ the Fourier coefficient of g_α,θ⁻¹

0,c¹ and by β_k,θ^(α)

0 the one of g_α,θ⁻¹

0. Without loss of generality we assumeβ_0,θ^(α)

0,c1 = 1. We put also ˜β_k^(α) =gd˜_α⁻¹(k) with

˜

g_α= (1−χ)^α.

2.2 Orthogonal polynomials

Theorem 4 Assume θ₀ ∈]0, π[ and −¹₂ < α < ¹₂. Then we have for all integers k, _N^k → x, 0< x <1, the asymptotic

T_N⁻¹ |χ−χ₀ |^2α|χ−χ¯₀ |^2αc₁

k+1,1 =

=K_α,θ0,c1cos (kθ₀+ω_α,θ0) T_N⁻¹ |χ−1|^2α

k+1,1(1 +o(1))

uniformly in kfor x∈[δ₀, δ₁],0< δ₀ < δ₁ <1,and withω_α,θ0 =αθ₀+ arg (c_1,1(θ₀))−^πα₂ and K_α,θ0,c¹ = 2^−α+1(sinθ₀)^−α

q

c⁻¹₁ (χ₀) .

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_N⁻¹ |χ−χ₀ |^2α|χ−χ¯₀ |^2αc₁

k+1,1=

= K_α,θ0,c1

Γ(α) cos (kθ₀+ω_α,θ0)k^α−1(1− k

N)^α+o(N^α−1) uniformly in k for x∈[δ₀, δ₁] 0< δ₀ < δ₁ <1.

Moreover the equalities (2) and (3) provide Corollary 3 Let ΦN =

XN

j=0

δ_jχ^j be the orthogonal polynomial of degreeN (Gegenbauer poly- nomial) with respect to the weight θ 7→ 2^2α(cosθ−cosθ₀)c₁(θ), with −¹₂ < α < ¹₂. Then we have, for _N^j →x,0< x <1,

δ_j =N^α−1K_α,θ0,c1

Γ(α) cos (N −jθ₀+ω_α,θ0)j^α(1− j

N)^α−1+o(N^α−1).

uniformly in j for x∈[δ₀, δ₁], 0< δ₀ < δ₁ <1.

We can also point out the asymptotic of the coefficients of orderkof the predictor polynomial when _N^k →0.

Theorem 5 With the same hypotheses as in Theorem 4 we have, if k

N →0 when N goes to the infinity

T_N⁻¹ |χ−χ₀ |^2α|χ−χ¯₀ |^2αc₁

k+1,1 =β_k,θ^(α)_0,c1+O(1 N).

Lastly whenαapproaches ¹₂ we obtain the entries of the last column ofT_N(2(cosθ−cosθ₀)c1).

Corollary 4 Assume θ₀ ∈]0, π[. Then for all integersk for _N^k →x, 0< x <1, we have the asymptotic

T_N⁻¹(|χ−χ₀ ||χ−χ¯₀ |c₁)

k+1,1 =

=K_1/2,θ0,c1cos kθ₀+ω_1/2,θ0r 1 k− 1

N +o(√ N) uniformly in k for x∈[δ₀, δ₁], 0< δ₀ < δ₁ <1.

Remark 1 This corollary implies that the coefficient of order kof the orthogonal polynomial with respect of θ7→2(cosθ−cosθ₀)c₁(θ) isK_1/2,θ0,c1cos kθ₀+ω_{1/2,θ0 1}

k−_N¹−1

+o(√ N)

2.3 Application to Toeplitz matrices

Theorem 6 Assume θ₀ ∈]0, π[and 0 < α < ¹₂. For _N^k →x, _N^l →y and 0 < x6=y <1, we have asymptotic

T_N⁻¹ |χ−χ₀ |^2α|χ−χ¯₀ |^2αc₁

k+1,l+1 =

=|K_α,θ0,c1|²cos (θ₀(k−l)) T_N⁻¹ |χ−1|^2α

k+1,l+1+o(N^2α−1) uniformly for k, l such that 0< δ₁ < x6=y < δ₂ <1.

At it has been said in the introduction this statement and the results of [15] provides the next corollary.

Corollary 5 Assume α∈]0,¹₂]andθ₀∈]0, π[. Let G_α be the function defined on 0< x6=y <

1 by

G_α(x, y) = x^αy^α Γ²(α)

Z 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_N⁻¹ |χ−χ₀ |^2α|χ−χ₀ |^2αc₁

[N x]+1,[N y]+1=

=N^2α−1|K_α,θ0,c¹|²cos (θ₀([N x]−[N y]))G_α(x, y) +o(N^2α−1) uniformly in k, l for 0< δ₁ ≤x6=y≤δ₂ <1.

2.4 Jacobi polynomial (in a particular case)

We note that in Theorem 6 one passes from the zeroesχ₀ andχ₀ to two zeroesχ₁ =e^iθ1 and χ₂ =e^iθ2 with |θ₁−θ₂| ∈]0, π[. Namely it is easy to see that

T_N⁻¹(|χ−χ₁|^2α|χ−χ₂|^2αc₁) =

∆(χ^1/2₁ χ^1/2₂ )T_N⁻¹

|χ^1/2₁ χ^−1/2₂ −ψ|^2α|χ^−1/2₁ χ^1/2₂ −ψ|^2αc_1,ψ−1

∆⁻¹(χ^1/2₁ χ^1/2₂ ) with ∆(χ^1/2₁ χ^1/2) is the diagonal matrix defined by

∆(χ^1/2₁ χ^1/2₂ )

i,j = 0 if i 6= j and (∆(ψ))_j,j = (χ^1/2₁ χ^1/2₂ )^j.

From this and Equation (2) we deduce the following proposition Proposition 1 Let Φ_1,2=X

j=0

δ˜_jχ^j be the orthogonal polynomial (Jacobi polynomial) with re- spect to the weight|χ−χ₁|^2α|χ−χ₂|^2α, withα∈]−¹₂,¹₂]. LetK_α,θ1,θ2 be the real2^−α+1|sin(θ₁− θ₂)|^−α. Then we have, for _N^j →x, 0< x <1.

δ˜_j =N^α−1K_α,θ1,θ²((χ1χ₂)^1/2)^N−jcos

(θ₁−θ₂

2 )(N −j) +ω_α,θ1−θ²

j^α(1− _N^j )^α−1

Γ(α) +o(N^α−1), uniformly in j for x∈[δ₀, δ₁], 0< δ₀ < δ₁ <1.

3 Inversion formula

3.1 Definitions and notations

Let H²⁺(T) and H²⁻(T) the two subspaces of L²(T) defined by H²⁺(T) = {h ∈ L²(T)|u <

0 =⇒ h(u) = 0ˆ } and H²⁻(T) = {h ∈ L²(T)|u ≥ 0 =⇒ ˆh(u) = 0}. We denote by π₊ the orthogonal projector onH²⁺(T) andπ₋the orthogonal projector onH²⁻(T). It is known (see [9]) that if f ≥0 and lnf ∈L¹(T) we have f =g¯g with g∈H²⁺(T). Put Φ_N = ^g_¯gχ^N+1. Let H_ΦN and H_Φ^∗

N be the two Hankel operators defined respectively onH²⁺ and H²⁻ by H_ΦN : H²⁺(T)→H²⁻(T), H_ΦN(ψ) =π₋(Φ_Nψ),

and

H_Φ^∗_N : H²⁻(T)→H²⁺(T), H_Φ^∗_N(ψ) =π₊( ¯ΦNψ).

3.2 A generalised inversion formula

We have stated in [15] for a precise class of non regular functions which contains cos^α(θ−θ₀)c₁ and (cosθ−cosθ₀)^αc₁ the following lemma (see the appendix of [15] for the demonstration), Lemma 1 Let f be an almost everywhere positive function on the torus T such thatlnf, f, and ¹_f are in L¹(T). Then f =g¯g with g∈H²⁺(T). For all trigonometric polynomials P of degree at most N, we define G_N,f(P) by

G_N,f(P) = 1 gπ₊

P

¯ g

− 1

gπ₊ Φ_N X∞ s=0

H_Φ^∗

NH_ΦNs

π₊Φ¯_Nπ₊ P

¯ g

! . For all P we have

• The serie X∞ s=0

H_Φ^∗

NH_ΦNs

π₊Φ¯_Nπ₊ P

¯ g

converges in L²(T).

• det (T_N(f))6= 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=D

π₊ χ^k

¯ g

χ^l

¯ g

E

−DX^∞

s=0

H_Φ^∗

NH_ΦNs

π₊Φ¯_Nπ₊ χ^k

¯ g

Φ¯_N χ^l

¯ g

E . Lastly if γ_u,α,θ= ^gd^α,θ0

gα,θ0(u) we obtain as in [15] the formal result H_Φ^∗

NH_ΦNm

π₊Φ¯_Nπ₊ χ^k

¯ g

= Xk

u=0

β_u,θ^(α)

0,c¹

X∞ n⁰=0

X∞ n¹=1

¯

γ_{−(N+1+n1+n0),α,θ0} X∞

n²=0

γ_{−(N+1+n1+n2),α,θ0}· · · X∞ n²m−1=1

¯

γ_{−(N+1+n2m−1+n2m−2),α,θ0} X∞

n2m=0

γ_{−(N+1+n2m−1+n2m),α,θ0}γ¯_{−(u−(N+1+n2m),α,θ0)}

! χⁿ⁰

3.3 Application to the orthogonal polynomials

With the corollary 6 and the hypothesis on β_0,θ^(α)_0,c1 the equality in the corollary 6 becomes, forl= 1, and forf =|χ−χ¯₀|^2α|χ−χ¯₀|^2αc₁

(T_N(f))⁻¹_1,k+1=β_k,θ^(α)

0,c1− Xk

u=0

β_k−u,θ^(α)

0,c1H_N(u) (4)

with

H_N(u) = X+∞

m=0

X∞ n0=0

γ_N+1+n0,α,θ⁰

X∞ n1=0

¯

γ_{−(N+1+n1+n0),α,θ0} X∞

n2=0

γ_{−(N+1+n1+n2),α,θ0}· · · X∞ n2m−1=0

¯

γ_{−(N+1+n2m−1+n2m−2),α,θ0} X∞

n2m=0

γ_−(N+1+n2m−1+n²m),α,θ⁰γ¯_(u−(N+1+n2m),α,θ⁰

!!

Our proof consists in the computation of the coefficients β_u,θ^(α)_0,c1, 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−cosθ)c1 multiplied by a trigonometric coefficient. That provides the expected link with the formulas in Theorems 1, 2.

4 Demonstration of Theorem 4

4.1 Asymptotic of β_k,θ^(α)

0,c1

Remark 2 In the rest of this paper we denote by c_1,1 the function in H²⁺(T) such that c₁ =c_1,1c_1,1. In all this proof we put φ₀ = arg (c_1,1(θ₀))

Property 1 For −¹₂ < α < ¹₂ and θ₀ ∈]0, π[we have, for sufficiently large k and for the real β defined by β=α−¹₂ if α <0 and β =α if α >0,

β_k,θ^(α)

0,c¹ =K_α,θ0,c1cos(kθ₀+ω_α,θ0)k^α−1

Γ(α) +o(k^β−1) uniformly in k. With K_α,θ0,c1 = √ ¹

c¹(χ⁰)2^−α+1(sinθ₀)^−α and ω_α,θ0 as in the statement of Theorem 4.

First we have to prove the lemma

Lemma 2 For −¹₂ < α < ¹₂ andθ₀ ∈]0, π[ we have, for a sufficiently large k.

β_k,θ^(α)₀ =K_α,θ0cos((k+α)θ₀+ω_α)k^α−1

Γ(α) +o(k^β−1).

uniformly in k, withK_α,θ0 = 2^−α+1(sinθ₀)^−α, ω_α=−^πα2 , andβ 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,θ0 −K_α,θ0cos((k+α)θ₀+ω_α)k^α−1 Γ(α)

< ǫk^β−1 and β_k,θ^(α)

0,c¹−K_α,θ0,c¹cos((k+α)θ₀+ω_α)k^α−1 Γ(α)

< ǫk^β−1. Proof : With our notations we can write

β_k,θ^(α)

0 = Xk

u=0

β˜_u^(α)(χ0)^uβ˜_k−u^(α) (χ0)^k−u. Putk₀=k^γ with 0< γ <1 such that foru > k₀ we have

β˜^(α)_u = u^α−1

Γ(α) +O(k^α−2) (5)

uniformly in u (see [19]). Writting fork≥k₀ Xk

u=0

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u=

k0

X

u=0

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u

+

k−kX0−1

u=k0+1

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u

+ Xk

u=k−k⁰

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u. The first sum is also

k0

X

u=0

β˜_u^(α)(χ₀)^u

β˜_k−u^(α) −β˜_k^(α)+ ˜β_k^(α)

(χ₀)^k−u. We observe that

k⁰

X

u=0

β˜_u^(α)(χ₀)^u

β˜_k−u^(α) −β˜_k^(α)≤ 1 Γ(α)

k⁰

X

u=0

|(k−u)^α−1−k^α−1||β˜_u^(α)|. (6) Consequently

k0

X

u=0

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u=

k0

X

u=0

β_u^(α)(χ₀)^2u

!

¯ χ^k₀k^α−1

Γ(α) +R₁

=



 X+∞

u=0

β˜^(α)_u (χ₀)^2u− X+∞

u=k0

β˜_u^(α)(χ₀)^2u



χ¯^k₀k^α−1 Γ(α) +R₁

withR₁=O(k^α−2+γ) if α <0 and R₁ =O(k^α−2+γα) if α >0. Then Lemma 9 implies

|

+∞X

u=k0

β˜_u^(α)(χ₀)^2u| ≤ |β˜_k^(α)

0 χ^2u₀ |+ X∞ u=k0

|β˜_u+1^(α) −β˜_u^(α)|

Γ(α) , (7)

that is

+∞X

u=k⁰

β˜_u^(α)(χ₀)^2u=O(k₀^α−1).

Finally we get

k⁰

X

u=0

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u= k^α−1

Γ(α)χ₀^k(1−χ²₀)^−α+O

k^{(α−1)(γ+1)} +R₁. Analogously we obtain

Xk

u=k−k0

β˜_u^(α)(χ₀)^uβ˜_k−u^(α) (χ₀)^k−u =χ^k₀k^α−1

Γ(α)(1−χ¯²₀)^−α+O

k^{(α−1)(γ+1)} +R₂, withR₂ asR₁.

For the third sum an Abel summation provides

k−kX0−1

u=k⁰+1

β˜_u^(α)(χ0)^uβ˜_k−u^(α) (χ0)^k−u =χ₀^k β˜_k^(α)

0

β˜_k−k^(α)

0σ_k0−1

+

k−kX0−2

u=k⁰

( ˜β^α_uβ˜_k−u^(α) −β˜_u+1^(α)β˜^(α)_k−u−1)σ_u



+ ˜β_k−k^(α)

0−1β˜_k^(α)

0 σ_k−k0

withσ_v= 1 +χ²₀+·+χ^2v₀ . This last sum is also equal toχ₀^k(A+B),with

|A|=O

β˜_k^(α)₀ β˜_k^(α)

=O(k^α−1₀ k^α−1) =o(k^{(α−1)(γ+1)}) and

B =−

k−kX⁰−2

u=k0

1

Γ²(α) u^α−1(k−u)^α−1−(u+ 1)^α−1(k−u−1)^α−1 χ^2u+2₀ 1−χ²₀. The main value Theorem implies

|B| ≤M k





k−kX0

v=k0

v^α−2(k−v)^α−2



, (8)

withM no depending from k. With the Euler and Mac-Laurin formula it is easyly seen that

k−kX⁰

v=k0

v^α−2(k−v)^α−1∼k₀^α−2(k−k₀)^α−1+k^α−1₀ (k−k₀)^α−2+ Z _k−k0

k0

t^α−2(k−t)^α−2dt.

The decomposition Z k−k⁰

k0

t^α−2(k−t)^α−2dt= Z k/2

k0

t^α−2(k−t)^α−2dt+ Z k−k⁰

k/2

t^α−2(k−t)^α−2dt

provides the estimation |B| = O(k₀^α−1k^α−1) = O(k^{(α−1)(γ+1)}). If α > 0 and 0 < γ < 1 we have

β_k,θ^(α)

0 = k^α−1 Γ(α)

χ^k₀(1−χ²₀)^−α+χ^k₀(1−χ²₀)^−α

+o(k^α−1) Ifα <0 andγ = ¹₂ we get

β^(α)_k,θ0 = k^α−1 Γ(α)

χ^k₀(1−χ²₀)^−α+χ^k₀(1−χ²₀)^−α

+o(k^β−1) withβ =α− ¹₂. On the another hand we have

β_k,θ^(α)

0 = 2k^α−1 Γ(α)ℜ

e^−ikθ0(1−cos(2θ₀)−isin(2θ₀))^−α

+o(k^β−1)

= 2^1−αk^α−1 Γ(α)ℜ

e^−ikθ0(sin(θ₀) (sinθ₀−icosθ₀))^−α

+o(k^β−1) Since θ₀ ∈]0, π[ we have (sinθ₀(sinθ₀−icosθ₀))^−α= (sinθ₀)^−αe^{iα(π2−θ0)}

This last remark gives the definition of ω_α. The equations (6), (7), (8), imply the uniformity

that completes the proof of the lemma. ✷

To ends the proof of the property we need to obtainβ^(α)_k,θ

0,c1 fromβ_k,α,θ^(α)

0 for a sufficiently large k. We can remark that a similar case has been treated in [13] for the function (1−χ)^αc₁. 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 ∈ L²(T)|P

u∈Zu^p|ˆh(u)|<∞}for all positive integerp. We have,β_m,θ^(α)_0,c1 = Xm

s=0

β_m,θ^(α)₀c_m−s. For 0< ν <1 we can write

Xm

s=0

β_s,θ^(α)

0c_m−s=

m−mX^ν

s=0

β^(α)_s,θ

0c_m−s+

Xm

s=m−m^ν+1

β_s,θ^(α)

0c_m−s. Lemma 2 provides

Xm

s=m−m^ν+1

β_s,θ^(α)₀c_m−s = K_α,θ0

Xm

s=m−m^ν

s^α−1

Γ(α)(cos ((s+α)θ₀+ω_α)c_m−s

!

+o(m^β−1)

Xm

s=m−m^ν+1

|c_m−s| and, since P

s∈Z|c_s|<∞, we have Xm

s=m−m^ν+1

β_s,θ^(α)₀c_m−s=K_α,θ0

s^α−1 Γ(α)

Xm

s=m−m^ν

s^α−1

Γ(α)(cos ((s+α)θ₀+ω_α)c_m−s+o(m^β−1).

We have always

Xm

s=m−m^ν

(s^α−1−m^α−1)cm−s

≤(1−α)m^ν+α−2 Xm

s=m−m^ν

|c_m−s|. (9) The convergence of (c_s) implies

K_α,θ0

Xm

s=m−m^ν

s^α−1−m^α−1+m^α−1

Γ(α) (cos ((s+α)θ₀+ω_α)c_m−s

=K_α,θ0

m^α−1 Γ(α)

Xm

s=m−m^ν

(cos ((s+α)θ₀+ω_α)c_m−s+O(m^α−2+ν).

For all positive integer p the functionc_1,1 A(p,T)). Hence one can prove first

X∞ v=m^ν+1

e^−ivθc_v≤(m^−pν)X

s∈Z

|c_s| (10)

and secondly Xm

s=m−m^ν

(cos ((s+α)θ₀+ω_α)c_m−s = 1 2

Xm

s=m−m^ν

e^isθ0c_m−s

!

e^{i(θ0α+ωα)}

+1 2

Xm

s=m−m^ν

e^−isθ0c_m−s

!

e^{−i(θ0α+ωα)}

= 1 2

c⁻¹_1,1(e^−iθ0)e^{i(mθ0+θ0α+ωα)}+c⁻¹_1,1(e^iθ0)e^{−i(mθ0+θ0α+ωα)} +O(m^−pν).

Since c⁻¹_1,1(e^iθ0) =c⁻¹_1,1(e^−iθ0) that last formula provides Xm

s=m−m^ν

(cos ((s+α)θ₀+ω_α)c_m−s= q

c⁻¹₁ (χ₀) cos ((m+α)θ₀+ω_α+φ₀) +O(m^−pν) (11) and

Xm

s=m−m^ν+1

β^(α)_s,θ

0c_m−s=K_α,θ0

m^α−1 Γ(α)

q

c⁻¹₁ (χ₀) cos ((m+α)θ₀+ω_α+φ₀) +O(m^α−1−pν) +O(m^α−2+ν) +o(m^β−1).

On the other hand we have (becausec⁻¹_1,1 inA(T, p))

m−mX^ν

s=0

β_s,αc_m−s

≤ 1 m^2ν

X

v∈Z

v^p|c_v|max

s∈N(|β_s,θ^(α)₀|).

For a good choice of p and ν we obtain the expected formula for β_α,θ0,c¹. The uniformity is provided by Lemma 2 and the equation (9) and (10).

4.2 Estimation of the Fourier coefficients of ^g_g^α,θ0

α,θ0

Property 2 Assume−¹₂ < α < ¹₂ andθ₀ ∈]0, π[then we have for all integerk≥0sufficiently large

d g_α,θ0

g_α,θ0

(−k) = 2 k+α

sin(πα)

π cos(θ₀k+ 2ω_α,θ^′ ₀) +o(kmin(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^iθ0) and φ_α= arg_χ2 0−1

¯ χ²0−1

α

. First we have to prove the lemma

Lemma 3 For −¹₂ < α < ¹₂ andθ₀ ∈]0, π[ we have, for all integerk sufficiently large γ_−k= 2

k+α sin(α)

π cos (θ0k+φ_α)) +o(kmin(α−1,−1)),

uniformly in k and where γ_k is the coefficient of orderk of the function ^(χχ_{( ¯χ¯χ}^{0−1)α(χ¯χ0−1)α}

0−1)^α( ¯χχ0−1)^α. Proof of Lemma 3: In all this proof we denote respectively by ˜γ_k, γ_1,k, γ_2,k the Fourier co- efficient of order k of ^(χ−1)_{( ¯χ−1)}^αα,^(χχ_{( ¯χχ¯}^0−1)α

0−1)^α,^(χ¯_{( ¯χχ}^χ0−1)α

0−1)^α. Clearly ˜γ_k = ^sin(πα)_{π k+α}¹ γ_1,k = χ^k₀˜γ_k, γ_2,k = ( ¯χ₀)^k˜γ_k.Assume also k≥0. We have γ_−k= X

v+u=−k

γ_1,uγ_2,v.For an integer , k₀ and k₀ =k^τ, 0< τ <1 we can split this sum into

X

u<−k−k⁰

γ_1,uγ_2,−k−u+

−k+kX0

u=−k−k⁰

γ_1,uγ_2,−k−u+

−kX0−1 u=−k+k⁰+1

γ_1,uγ_2,−k−u

+

k0

X

u=−k⁰

γ_1,uγ_2,k−u+ X

u>k⁰

γ_1,uγ_2,−k−u. Write

k0

X

u=−k0

γ_1,uγ_2,−k−u=

k0

X

u=−k0

γ_1,u( ¯χ₀)^k+u(˜γ_−k−u−γ˜_−k+ ˜γ_−k).

Since

k⁰

X

u=−k0

γ_1,u( ¯χ₀)^k+u(˜γ_−k−u−γ˜_−k) = sin(πα) π

k⁰

X

u=−k0

γ_1,u( ¯χ₀)^k+u −u

(k+u+α)(k+α) (12)

it follows that

k⁰

X

u=−k0

γ_1,uγ_2,−k−u= ˜γ_−k

k⁰

X

u=−k0

γ_1,u( ¯χ₀)^−k−u+O(k₀k⁻²)

= ˜γ_−k(χ₀)^k

χ²₀−1 ( ¯χ₀)²−1

α

+ ˜γ_−k(χ₀)^k X

|u|≥k0

γ_1,uχ^u₀+O(k₀k⁻²)

= ˜γ_−k(χ₀)^k

χ²₀−1 ( ¯χ₀)²−1

α

+O (k₀k)⁻¹

+O(k₀k⁻²)

= ˜γ_−k(χ₀)^k

χ²₀−1 ( ¯χ₀)²−1

α

+O(k^τ−2).

In the same way we have

−k+kX0

u=−k−k⁰

γ_1,uγ_2,k−u= ˜γ_−k(χ₀)^−k

( ¯χ₀)²−1 χ²₀−1

α

+O(k^τ−2).

Now using Lemma 9 it is easy to see that X

u<−k−k0

γ_1,uγ_2,−k−u≤M₁(k₀k)⁻¹ (13)

X

u>k⁰

γ_1,uγ_2,−k−u ≤M₂(k0k)⁻¹ (14)

withM₁andM₂no depending fromk. For the sumS =

−kX0−1 u=−k+k⁰+1

γ_1,uγ_2,−k−uwe can remark, using an Abel summation, that

|S| ≤M₃(k₀k)⁻¹+

−kX0−1 u=−k+k⁰+1

1

(u+α)(k−u+α) − 1

(u+ 1 +α)(k−u−1 +α)

M₃ no depending fromk. Consequently the main values theorem provides

|S| ≤M₃(k₀k)⁻¹+

−kX0−1 u=−k+k⁰+1

k−2u

(k−u)²u². (15)

withM₃ no depending fromk. Then Euler and Mac-Laurin formula provides the upper bound

|S| ≤O (k₀k)⁻¹ +

Z −k0−1

−k+k⁰+1

k−2u (k−u)²u²du.

Since Z _−k0−1

−k+k0+1

k−2u

(k−u)²u²du≤ 3k (k+k₀)²

Z _−k0−1

−k+k0+1

1 u²du