Jean-François Coulombel
email: jean-francois.coulombel@math.univ-toulouse.fr

Grégory Faye
email: gregory.faye@math.univ-toulouse.fr

Generalized Gaussian bounds for discrete convolution powers

Keywords: AMS classification: 42A85, 35K25, 60F99, 65M12 convolution, difference approximation, stability, local limit theorem

We prove a uniform generalized Gaussian bound for the powers of a discrete convolution operator in one space dimension. Our bound is derived under the assumption that the Fourier transform of the coefficients of the convolution operator is a trigonometric rational function, which generalizes previous results that were restricted to trigonometric polynomials. We also allow the modulus of the Fourier transform to attain its maximum at finitely many points over a period.

For 1 ≤ q < +∞, we let q (Z; C) denote the Banach space of complex valued sequences indexed by Z and such that the q norm, defined for u : Z → C by

u q :=   j∈Z |u j | q   1/q
, is finite. We also let ∞ (Z; C) denote the Banach space of bounded complex valued sequences indexed by Z and equipped with the norm:

u ∞ := sup j∈Z |u j | .
Throughout this article, we use the notations

U := {ζ ∈ C | |ζ| > 1} , D := {ζ ∈ C | |ζ| < 1} , S 1 := {ζ ∈ C | |ζ| = 1} , U := U ∪ S 1 , D := D ∪ S 1 .
If w is a complex number and ρ a positive real number, the notation B ρ (w) stands for the open ball in C centered at w and with radius ρ, that is B ρ (w) := {z ∈ C | |z -w| < ρ}.

The notation σ(T ) stands for the spectrum of a bounded operator T acting on a Banach space E. We also use the notation • E→E for the operator norm on a Banach space E.

Introduction and main result 1.A reminder on Laurent operators

Let us recall a few facts about Laurent operators on q (Z; C). If a ∈ 1 (Z; C), we let L a denote the socalled Laurent (or convolution) operator associated with the sequence a [TE05, Nik02], which is defined by:

L a : u j j∈Z -→ ∈Z a u j- j∈Z = a u , (1)
whenever the defining formula (1) for the sequence L a u makes sense. Here and below, always stands for the convolution product of two sequences indexed by Z. In particular, Young's inequality shows that L a acts boundedly on q (Z; C) for any q ∈ [1, +∞]:

∀ u ∈ q (Z; C) , L a u q ≤ a 1 u q .
The spectrum of L a is also well-understood since the celebrated Wiener-Levy theorem, see [START_REF] Newman | A simple proof of Wiener's 1/f theorem[END_REF], characterizes the invertible elements of 1 (Z; C) for the convolution product (and we have the morphism property L a • L b = L a b ). Namely, the spectrum of L a as an operator acting on q (Z; C) does not depend on q and is nothing but the image of the Fourier transform of the sequence a:

σ (L a ) = ∈Z a e i ξ | ξ ∈ R .
Since a belongs to 1 (Z; C), its Fourier transform is continuous on R. It actually belongs to the so-called Wiener algebra. Following, among other works, [Tho65, DSC14, RSC15, RSC17], we are interested here in giving uniform pointwise bounds for the n-th iterated convolution product a • • • a = a n as the number n gets large. We use the convention a 1 := a and a n := a (n-1) a for n ≥ 2. Note that, by the morphism property L a • L b = L a b , we have (L a ) n = L a n for any n ∈ N. Beyond their own analytical interest, sharp bounds for the coefficients (a n ) j or the precise description of their asymptotic behavior are useful in probability theory and in numerical analysis. In probability theory, the coefficients a correspond to the probability P(X = ) where X is a random variable with values in Z. Considering the random walk:

Y n := X 1 + • • • + X n ,
where the X m 's are identically distributed, independent and follow the same law as X, then (a n ) j = P(Y n = j) , for any n ∈ N * and j ∈ Z. In this context, asymptotic expansions for (a n ) j are referred to as local limit theorems and may be found in [START_REF] Petrov | Sums of independent random variables[END_REF]Chapter VII]. These expansions involve the Gaussian function and Hermite polynomials. In numerical analysis, the study of the iterates (L a ) n , n ∈ N, arises when one discretizes an evolutionary linear partial differential equation (set on the real line R) by means of a finite difference scheme. The transport equation or the heat equation are typical examples. We refer for instance to [START_REF] Richtmyer | Difference methods for initial value problems[END_REF][START_REF] Gustafsson | Time dependent problems and difference methods[END_REF]. From Young's inequality:

(L a ) n u ∞ = L a n u ∞ ≤ a n 1 u ∞ ,
one observes that boundedness of the sequence ( a n 1 ) n∈N is a sufficient condition1 for what is referred to, in this context, as stability in the maximum norm, that is:

sup n∈N (L a ) n ∞ → ∞ < +∞ .
The fundamental result in [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF] characterizes, under suitable assumptions, the elements a ∈ 1 (Z; C) such that the geometric sequence (a n ) n∈N is bounded in 1 (Z; C), see also [START_REF] Hedstrom | Norms of powers of absolutely convergent Fourier series[END_REF][START_REF] Després | Finite volume transport schemes[END_REF][START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF][START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] and references therein for further developments. The sufficient part of the characterization in [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF] (see also [START_REF] Strang | Polynomial approximation of Bernstein type[END_REF][START_REF] Després | Finite volume transport schemes[END_REF]) is performed by deriving a suitable "algebraic" pointwise bound for the coefficient (a n ) j (see [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF]Lemma 2.4]). This bound is obtained by integrating by parts the Fourier transform of a n , and this manipulation requires the Fourier transform of a to be C 2 . For the derivation of these algebraic bounds, the support of the sequence a may be arbitrary. Refining and optimizing this approach, the algebraic bound in [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF] was turned in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] into a generalized Gaussian bound, for finitely supported sequences a, thanks to a suitable contour deformation. The contours chosen in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] can go arbitrarily far away from the real line (where the Fourier transform of a is defined at first), which is the reason why the authors in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] assume a to have finite support, so that its Fourier transform extends to a holomorphic function on the whole complex plane. The necessity of the conditions of [Tho65, Theorem 1] for a to be power bounded in 1 (Z; C) is proved under the assumption that a is finitely supported. This assumption on the support was removed in [START_REF] Hedstrom | Norms of powers of absolutely convergent Fourier series[END_REF].

Our goal in this article is to extend the results in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] in two directions: we first wish to consider sequences a with infinite support, since such sequences arise when one considers implicit discretizations of partial differential equations. We also wish to relax the assumption made in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] that the modulus of the Fourier transform of a attains its maximum at only one point over each period (say, at 0, in the interval [-π, π]). When the modulus of the Fourier transform attains its maximum at more than one point over a period, the arguments in either [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF] or [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF], rely on (C ∞ ) partitions of unity which destroy the holomorphy of the Fourier transform. This is the reason why the bounds obtained in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] in that situation are only of "sub-Gaussian" type (compare for instance [DSC14, Theorem 3.1] with [DSC14, Theorem 3.5]). Actually, the results in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] were first refined and extended in [START_REF] Randles | On the convolution powers of complex functions on Z[END_REF], where local limit theorems are proved for complex valued sequences a, and then further extended in [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] to deal with multidimensional situations. Another extension that is achieved in [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] is the proof of generalized Gaussian bounds for finitely supported sequences a whose modulus of the Fourier transforms attains its maximum at several points (under a technical assumption that is discussed below).

Our approach in this article is quite different from the one in [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF][START_REF] Després | Finite volume transport schemes[END_REF][START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF][START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] where the coefficient (a n ) j is represented by an integral involving the Fourier transform of a (to the n-th power).

Here we rather follow an approach which is commonly referred to in the partial differential equation community as "spatial dynamics", which amounts to representing (a n ) j in terms of the resolvent of the operator L a . The link between the two comes from the so-called functional calculus [START_REF] Conway | A course in functional analysis[END_REF] Chapter VII] which expresses the temporal Green's function (here the coefficient (a n ) j ) in terms of the spatial Green's function, which is the solution to the resolvent equation:

z I -L a u = δ , z ∈ σ(L a ) ,
where δ stands for the "discrete" Dirac mass (δ j = 1 if j = 0, and 0 otherwise). A detailed analysis of the spatial Green's function with sharp holomorphic extensions and bounds is provided in Section 2 of the present article under conditions that are similar to but, to some extent, less restrictive than those in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF][START_REF] Randles | Convolution powers of complex functions on Z d[END_REF]. Our main technical assumption is the fact that we consider nonzero drift velocities, which enables us to pass smoothly (that is, holomorphically) from "temporal" to "spatial" representations. Some cases with vanishing group velocities (in the terminology of [START_REF] Trefethen | Group velocity in finite difference schemes[END_REF]) are discussed at the end of this article.

Once we have sharp holomorphic extensions and bounds for the spatial Green's function, our final argument relies on a suitable choice of contours in the defining expression of the temporal Green's function. The choice of contours can be interpreted as an application of the saddle point method [dB81]. A fundamental contribution in this direction is [START_REF] Zumbrun | Pointwise semigroup methods and stability of viscous shock waves[END_REF] (for the stability analysis of viscous shock profiles) and we also refer to [START_REF] Godillon | Green's function pointwise estimates for the modified Lax-Friedrichs scheme[END_REF] for an application of this method to the stability analysis of discrete shock profiles. As a matter of fact, our motivations for deriving generalized Gaussian bounds in the broadest possible context stems from the stability analysis of discrete shock profiles but also from the theory of numerical boundary conditions for hyperbolic equations. An application of the techniques developed in this article to finite rank perturbations of Toeplitz operators (on2 (N; C) rather than 2 (Z; C)) is given in [START_REF] Coulombel | Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions[END_REF]. Discrete shock profiles will be considered in a forthcoming work. We now make several assumptions and state our main result.

Assumptions and main result

This work is much inspired by the theory of partial differential equations and its numerical approximations. Hence, instead of sticking to the convolution operators L a of the introduction, we shall rather use operators in the form: We thus consider from now on two "convolution" operators Q 0 and Q 1 on Z with finite support:

∀ σ = 0, 1 , ∀ j ∈ Z , (Q σ u) j := p =-r a ,σ u j+ , (2)
where r, p ∈ N and the a ,σ 's are complex numbers 2 . In what follows, we always write:

Q 0 = L φ 0 , and Q 1 = L φ 1 , (3)
where φ 0 and φ 1 are finitely supported elements of 1 (Z; C). Both operators Q 0 and Q 1 act boundedly on any q (Z; C), 1 ≤ q ≤ +∞. Our main focus below is on the three cases q = 1, q = 2 and q = +∞. The integers r, p in (2) define the common stencil of the operators Q 0 , Q 1 . They are fixed by enforcing the conditions:

| a -r,1 | + | a -r,0 | > 0 , | a p,1 | + | a p,0 | > 0 .
Our first assumption is the following.

Assumption 1. The operator Q 1 is an isomorphism on 2 (Z; C), that is:

∀ κ ∈ S 1 , Q 1 (κ) := p =-r a ,1 κ = 0 , (4)
and it satisfies furthermore the index condition:

1 2 i π S 1 Q 1 (κ) Q 1 (κ) dκ = 0 . (5)
The function Q 1 in (4) is referred to below as the symbol of the convolution operator Q 1 . We can similarly define the symbol Q 0 associated with Q 0 :

∀ κ ∈ S 1 , Q 0 (κ) := p =-r a ,0 κ .
Recalling the definition (3) of the sequence φ 1 , the condition (4) implies that φ 1 is invertible in 1 (Z; C) (thanks to the Wiener-Levy theorem [START_REF] Newman | A simple proof of Wiener's 1/f theorem[END_REF]). We are then interested in the operator L := Q -1 1 Q 0 and more specifically in its powers L n as n becomes large. Since φ 1 is invertible in 1 (Z; C), we can write

L = L φ with φ := φ -1 1 φ 0 . Since we are interested in L = Q -1
1 Q 0 , we can always multiply Q 0 and Q 1 by the same nonzero complex number, which does not modify L . In view of Assumption 1, we thus always assume Q 1 (1) = 1 from now on.

We briefly discuss the support of the sequence φ in order to compare our framework with that of [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] or [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF]. The generalized Gaussian bounds in [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] or [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] are obtained by assuming that φ has finite support (which makes its Fourier transform an entire function). In our case, two situations occur:

• The support of φ 1 is a singleton, that is Q 1 = a ,1 S for some integer between -r and p (recall the notation S for the shift operator). Because of the condition (5), we have = 0 so Q 1 is a nonzero multiple of the identity, and our normalization convention Q 1 (1) = 1 makes Q 1 be the identity. In that case, φ = φ 0 is finitely supported. In numerical analysis, this situation corresponds to explicit schemes. In probability theory, this situation corresponds to random walks with finite range.

• The support of φ 1 contains at least two elements. Then the inverse φ -1 1 of φ 1 for the convolution product has an infinite support. Apart from "trivial" cases where a factorization is possible, φ = φ -1 1 φ 0 will also have infinite support. An example of this situation is provided in Section 4. In numerical analysis, this situation corresponds to implicit schemes.

As can be expected, a crucial role is played below by the symbol of L which is defined by:

∀ κ ∈ S 1 , F (κ) := Q 0 (κ) Q 1 (κ) . (6)
The main difference between [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] or [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] and the present work is that we allow F (exp(i ξ)) to be a trigonometric rational function of ξ rather than just a trigonometric polynomial in ξ. (Other results, such as local limit theorems, are derived in [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF] under the assumption that F is of class C ∞ on S 1 , but we focus here on the derivation of generalized Gaussian bounds.) In other words, we deal here with the class of sequences in 1 (Z; C) whose Fourier transforms are trigonometric rational functions. The following assumption on F is inspired by the fundamental contribution [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF]. The link between our Assumption 2 below and the classification obtained in [Tho65, page 280] in the case of trigonometric polynomials is discussed in Appendix A at the end of this article (see Lemma 16).

Assumption 2. The function F defined in (6) satisfies max κ∈S 1 |F (κ)| = 1. Furthermore, there exists a finite set of points {κ 1 , . . . , κ K }, K ≥ 1, in S 1 such that:

∀ κ ∈ S 1 \ κ 1 , . . . , κ K , F (κ) < 1 , (7)
and for all index k = 1, . . . , K, F (κ k ) belongs to S 1 . Moreover, for any k = 1, . . . , K, there exist a nonzero real number α k , an even integer 2 µ k ≥ 2 and a complex number β k with positive real part such that:

F κ k e i ξ F (κ k ) = exp -i α k ξ -β k ξ 2 µ k + O ξ 2 µ k +1 , (8)
as ξ tends to 0.

As in [Tho65, DSC14, RSC17], the maximum of F on the unit circle S 1 is normalized to be 1. Thanks to Beurling's result (see [START_REF] Riesz | Functional analysis[END_REF]page 428]):

∀ a ∈ 1 (Z; C) , lim n→∞ a n 1/n 1 = max θ∈R ∈Z a e i θ ,
the case where the maximum equals 1 is the limit case where the question of stability is not straightforward.

In view of the result of Lemma 16 in Appendix A, we just wish to exclude in Assumption 2 the case where F has constant modulus on S 1 , and we then only consider the so-called points of type γ in the terminology of [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF]. The number α k in (8) is necessarily real since F (κ) belongs to D for all κ ∈ S 1 . The fact that all real numbers α k are nonzero is a major assumption that we make. It is fundamental below in the description of the so-called spatial Green's function. Examples of operators Q 0 , Q 1 for which Assumptions 1 and 2 are satisfied are provided in Section 4 at the end of this article. Since F (κ) belongs to D for all κ ∈ S 1 , the operator L is a contraction on 2 (Z; C), that is:

∀ u ∈ 2 (Z ; C) , L u 2 ≤ u 2 ,
since the 2 norms on both sides can be computed by the Parseval-Bessel identity. Of course, this implies that every power of L is also a contraction on 2 (Z; C). In the field of numerical analysis, this property is referred to as 2 -stability3 , or strong stability [START_REF] Strang | On the construction and comparison of difference schemes[END_REF][START_REF] Tadmor | Complex symmetric matrices with strongly stable iterates[END_REF], for the "numerical scheme":

Q 1 u n+1 = Q 0 u n , n ∈ N , u 0 ∈ 2 (Z) .
Let us now define the quantities:

∀ z ∈ C , ∀ = -r, . . . , p , A (z) := z a ,1 -a ,0 . (9)
The following assumption already appears in several works devoted to the stability analysis of numerical boundary conditions for discretized hyperbolic equations, see, e.g., [Kre68, Osh69, GKS72, GT81, Cou13] and references therein. Not only does it determine the minimal integers r and p in (2) (by prohibiting to add artificial zero coefficients), but it is also crucially used below to analyze the so-called resolvent equation ( 14). It might be relaxed though, but a more elaborate analysis would be required.

Assumption 3. The functions A -r and A p defined in (9) do not vanish on U .

For instance, if a -r,1 and a p,1 are nonzero, Assumption 3 means that a -r,0 /a -r,1 and a p,0 /a p,1 belong to D. If a -r,1 and a p,1 are both zero, then a -r,0 and a p,0 should both be nonzero.

Thanks to Assumption 3, we can define the following companion matrix:

M(z) :=        - A p-1 (z) A p (z) . . . . . . - A -r (z) A p (z) 1 0 . . . 0 0 . . . . . . . . . 0 0 1 0        ∈ M p+r (C) , (10)
which is holomorphic on the set {z ∈ C | |z| > exp(-η)} for some parameter η > 0 which only depends on the location of the root of A p (if it exists). A crucial observation is that the upper right coefficient of M(z) is always nonzero, because of Assumption 3 and up to restricting η, so the matrix M(z) is invertible for all relevant values of z. We shall repeatedly use the inverse matrix M(z) -1 in what follows.

The analysis in this article heavily relies on a precise description of the spectrum of M(z) as z runs through U (and even sometimes slightly through D). This description is given in Lemma 1 below, and uses the following two assumptions.

Assumption 4. Either Q 1 is the identity, or a -r,1 and a p,1 are nonzero.

If Q 1 is the identity, Assumption 1 is trivially satisfied since Q 1 ≡ 1. In the other case, the complex numbers a -r,1 and a p,1 are both nonzero. In that case, Q 1 is a meromorphic function on C with a single pole (that is located at 0) of order r. (When r equals zero, there is no pole.) By the residue theorem [START_REF] Rudin | Real and complex analysis[END_REF], the index condition (5) is equivalent to Q 1 having r zeros (counted with multiplicity) in D \ {0}. Because κ r Q 1 (κ) is a polynomial of degree p + r, Q 1 then has p zeros in U .

Assumption 5. For all index k = 1, . . . , K, let us define z k := F (κ k ) ∈ S 1 . Then for any k = 1, . . . , K, the set:

I k := ν ∈ {1, . . . , K} | z ν = z k (11)
has either one or two elements4 . Furthermore, in case it has two elements, which we denote ν k,1 , ν k,2 , then α ν k,1 α ν k,2 < 0. (Let us recall that the drift parameters α k are given in Assumption 2.)

From now on, we always make Assumptions 1, 2, 3, 4 and 5. Our main result is a partial extension of [DSC14, Theorem 3.1] and [RSC17, Theorem 1.8]. It gives a uniform, generalized Gaussian bound for the convolution coefficients of the powers L n . A precise statement is the following.

Theorem 1. Let the operators Q 0 , Q 1 in (2) satisfy Assumptions 1, 2, 3, 4 and 5 and the normalization condition Q 1 (1) = 1. According to the above two cases in Assumption 4, we have:

• Explicit case. If Q 1 is
the identity, then there exist two constants C > 0 and c > 0 such that the operator L = Q 0 satisfies the uniform generalized Gaussian bound:

∀ n ∈ N * , ∀ j ∈ Z , (L n δ) j ≤ C K k=1 1 n 1/(2 µ k ) exp   -c |j -α k n| n 1/(2 µ k ) 2 µ k 2 µ k -1   , ( 12
)
where δ denotes the discrete Dirac mass defined by δ j = 1 if j = 0 and δ j = 0 otherwise.

• Implicit case. If Q 1 is not the identity, then there exist constants C > 0, L > 0 and c > 0 such that the operator L = Q -1 1 Q 0 satisfies the bounds:

∀ n ∈ N * , ∀ |j| ≤ L n , (L n δ) j ≤ C K k=1 1 n 1/(2 µ k ) exp   -c |j -α k n| n 1/(2 µ k ) 2 µ k 2 µ k -1   , (13a) ∀ n ∈ N * , ∀ |j| > L n , (L n δ) j ≤ C exp (-c n -c |j|) . (13b)
A comparison between Theorem 1 and the analogous results in [Tho65, DSC14, RSC17] is provided in the next Subsection. Otherwise, the rest of this article is organized as follows. In Section 2, we prove sharp bounds on the so-called spatial Green's function. This is where Assumptions 3, 4 and 5 are used. Then we use these preliminary bounds in Section 3 to obtain the uniform bounds (12) and ( 13) for what we call the temporal Green's function. Examples and possible extensions are given in Section 4. The proofs of some intermediate and related results are gathered in Appendix A.

What is new ? and what is not ?

Let us first observe that the sequence G := L δ corresponds to the Laurent series expansion of F near

S 1 : ∀ κ ∈ S 1 , F (κ) = j∈Z G j κ j .
In particular, if G satisfies a generalized Gaussian bound of the form:

∀ j ∈ Z , | G j | ≤ C exp(-c |j| s ) ,
for some positive constants C and c and some exponent s > 1, then F extends to a holomorphic function on C \ {0}. In our framework, we have F (κ) = Q 0 (κ)/ Q 1 (κ). Assuming that Q 0 (κ) and Q 1 (κ) have no common factor, the only possible case where F extends to a holomorphic function on C \ {0} is when Q 1 does not vanish on C \ {0}. Because of the form of Q 1 and the index condition (5), the only situation in which F extends to a holomorphic function on C \ {0} is when Q 1 is the identity5 . This argument explains why, in (13), the bound for (L n δ) j "degenerates" to exp(-c |j|) for any fixed n (e.g., n = 1) and for large j's.

We now compare Theorem 1 with [DSC14, Theorem 3.1] and [RSC17, Theorem 1.8] which, to our knowledge, are the two prior references on generalized Gaussian bounds for convolution powers of complex sequences. In [DSC14, Theorem 3.1], the authors consider (in our notation) the explicit case (Q 1 is the identity) with K = 1, but they make no assumption on the drift parameter α 1 (while we assume α 1 = 0 in Theorem 1). When specifying [RSC17, Theorem 1.8] to one space dimension, the result in [RSC17, Theorem 1.8] covers the explicit case (Q 1 is the identity) with K ≥ 1, but it is then further assumed:

α 1 = • • • = α K , µ 1 = • • • = µ K , β 1 = • • • = β K .
As in [DSC14, Theorem 3.1], it is not assumed in [RSC17, Theorem 1.8] that the common value of the α k 's should be nonzero.

In our opinion, the main novelty here consists in considering sequences a such that the α k 's, µ k 's and β k 's are, to some extent, "arbitrary". This is not entirely true since we assume that the z k 's are not "too much" equal (Assumption 5) and we further assume that all α k 's are nonzero. We believe that this restriction on the α k 's in Theorem 1 is purely technical. For instance, in Corollary 2, we give an example of a general situation in the explicit case where some α k can be zero and where the result of Theorem 1 can be used to obtain the same bound as in (12). We also note that the theory of numerical boundary conditions in [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF] (see [START_REF] Coulombel | Stability of finite difference schemes for hyperbolic initial boundary value problems[END_REF] for a thorough exposition) covers cases with vanishing group velocities (by making use in several occurrences of Puiseux expansions). In the continuous setting, reaction diffusion equations are another occurence where spatial dynamics and pointwise Green's function bounds cover some problems with zero drift velocities, see, e.g., [START_REF] Faye | Asymptotic stability of the critical Fisher-KPP front using pointwise estimates[END_REF]. We thus hope that we shall be able to fully remove the restriction α k = 0 as well as Assumption 5 in the future.

Our second improvement is to consider sequences in 1 (Z; C) whose Fourier transforms are trigonometric rational functions, which is relevant for implicit numerical schemes. Possible extensions of our work are listed in Section 4.

The spatial Green's function

The spectrum of L as an operator on 2 (Z; C) is the parametrized curve F (S 1 ). We know from Assumption 2 that this curve touches the unit circle S 1 at the points z k , k = 1, . . . , K, and that it is located inside the open unit disk D otherwise. Hence the resolvent set of L contains at least U \ {z 1 , . . . , z K }. For such values of z, we can thus define the sequence G(z) ∈ 2 (Z; C) (the capital G letter stands for Green, as in Green's function) by the formula:

z I -L G(z) = δ , (14)
where we recall that δ stands for the Dirac mass (δ j = 1 if j = 0 and δ j = 0 if j ∈ Z \ {0}).

From the definition L = Q -1 1 Q 0 , the equation ( 14) can be equivalently rewritten:

z Q 1 -Q 0 G(z) = Q 1 δ ,
and the definitions (2), (9) give the final form:

∀ j ∈ Z , p =-r A (z) G j+ (z) = (Q 1 δ) j , (15)
together with the integrability conditions at infinity G(z) ∈ 2 (Z; C).

Spectral properties

We introduce the augmented vectors:

∀ j ∈ Z , W j (z) :=    G j+p-1 (z) . . . G j-r (z)    ∈ C p+r , e :=      1 0 . . . 0      ∈ C p+r ,
and rewrite equivalently (15) as:

∀ j ∈ Z , W j+1 (z) -M(z) W j (z) = (Q 1 δ) j A p (z) e . (16)
The construction and analysis of the solution to the recurrence relation (16) relies on the following spectral splitting lemma, which is originally due to Kreiss [START_REF] Kreiss | Stability theory for difference approximations of mixed initial boundary value problems[END_REF] in the context of finite difference approximations.

Lemma 1 (Spectral splitting). Let z ∈ U \ {z 1 , . . . , z K } and let the matrix M(z) be defined as in (10). Then M(z) has:

• no eigenvalue on S 1 , • r eigenvalues in D \ {0},
• p eigenvalues in U (eigenvalues are counted with multiplicity).

Let now k ∈ {1, . . . , K} be such that the set I k in (11) is the singleton {k}. Then if α k > 0, the matrix M(z k ) has κ k ∈ S 1 as a simple eigenvalue, it has r -1 eigenvalues in D and p eigenvalues in U . If α k < 0, the matrix M(z k ) has κ k ∈ S 1 as a simple eigenvalue, it has r eigenvalues in D and p -1 eigenvalues in U .

Eventually, let now k ∈ {1, . . . , K} be such that the set I k in (11) has two elements ν k,1 , ν k,2 . Then the matrix M(z k ) has κ ν k,1 and κ ν k,2 as simple eigenvalues on S 1 , it has r -1 eigenvalues in D and p -1 eigenvalues in U .

The arguments are basically the same as in [START_REF] Kreiss | Stability theory for difference approximations of mixed initial boundary value problems[END_REF] but we give them here for the sake of completeness.

Proof of Lemma 1. We first recall that the matrix M(z) is given by (10) and that it is invertible for all z satisfying |z| > exp(-η) (thanks to Assumption 3). Hence 0 will never be an eigenvalue of M(z) for the relevant values of z. Let us then observe that κ ∈ C \ {0} is an eigenvalue of M(z) for z ∈ U if and only if z and κ satisfy the so-called dispersion relation:

p =-r A (z) κ = 0 ,
and the definition (9) of the functions A yields the equivalent form:

Q 1 (κ) z = Q 0 (κ) . (17)
In particular, for any z in the connected set U \ {z 1 , . . . , z K }, M(z) has no eigenvalue on the unit circle S 1 for otherwise we would have z = F (κ) for some κ ∈ S 1 and z ∈ {z 1 , . . . , z K }, which is precluded by Assumption 2. To obtain the first statement of Lemma 1, it thus remains to count the number of eigenvalues of M(z) in D \ {0} (we shall call such eigenvalues the stable ones). By the connectedness of U \ {z 1 , . . . , z K }, the number of stable eigenvalues of M(z) does not depend on z ∈ U \ {z 1 , . . . , z K }. In order to compute the precise number of such eigenvalues, we shall let z tend to infinity and determine the asymptotic behavior of these eigenvalues. This asymptotic behavior differs completely between the explicit and implicit cases (though the number of stable eigenvalues will be the same in both cases), which is the reason why we now deal with those two cases separately.

The explicit case (Q 1 = I). The dispersion relation (17) then reduces to:

z = p =-r a ,0 κ . (18)
If r = 0, then there are no eigenvalues in D \ {0} for any z for otherwise there would be at least one eigenvalue in D \ {0} for all z ∈ U and the triangle inequality in (18) would imply:

| z | ≤ p =0 | a ,0 | .
which is impossible because |z| can be arbitrarily large. The result is thus proved in the case r = 0 so we assume r ≥ 1 from now on (Assumption 3 then yields a -r,0 = 0). Following [START_REF] Kreiss | Stability theory for difference approximations of mixed initial boundary value problems[END_REF] (see also [START_REF] Coulombel | Stability of finite difference schemes for hyperbolic initial boundary value problems[END_REF] for the complete details), the number of eigenvalues of M(z) in D \ {0} is computed by letting z tend to infinity for in that case, all such (stable) eigenvalues of M(z) collapse to zero. Indeed, an eigenvalue of M(z) in D \ {0} cannot remain uniformly away from the origin for otherwise the right hand side of (18) would remain bounded while the left hand side tends to infinity. The final argument is the following (see [HP05, Theorem 4.2.1] for a general statement). For any z ∈ U \ {z 1 , . . . , z K }, the eigenvalues of M(z) are those κ = 0 such that:

κ r = 1 z p =-r a ,0 κ r+ ,
which is just an equivalent way of writing (18). Hence for z large, the small eigenvalues of M(z) behave at the leading order like the roots of the reduced equation:

κ r = a -r,0 z ,
and there are exactly r distinct roots close to 0 of that equation. Hence M(z) has r eigenvalues in D \ {0} for any z ∈ U \ {z 1 , . . . , z K }.

The implicit case (Q 1 = I). We then know that a -r,1 = 0 and a p,1 = 0. Moreover, the function Q 1 satisfies the index condition (5). By the residue theorem [START_REF] Rudin | Real and complex analysis[END_REF], this means that Q 1 has as many poles as roots in D and since it only has a pole of order r at 0, we can conclude that Q 1 has r roots in D \ {0}. Since κ r Q 1 (κ) is a polynomial of degree p + r, we also conclude that Q 1 has p roots in U , as already explained in the introduction.

From the definition (9), we compute:

lim z→∞ M(z) =       - a p-1,1 a p,1 . . . . . . - a -r,1 a p,1 1 0 . . . 0 0 . . . . . . . . . 0 0 1 0      
, and the eigenvalues of that (invertible) matrix are exactly those κ = 0 that satisfy Q 1 (κ) = 0. Hence for any sufficiently large z, M(z) has r eigenvalues in D \ {0} and p eigenvalues in U (which are close to the roots of Q 1 ). This completes the proof of the first statement in Lemma 1. It now remains to examine the situation at the points z k , k = 1, . . . , K. The arguments below are the same for the explicit and implicit cases so we stop distinguishing between the two from now on. We thus consider a point z k for 1 ≤ k ≤ K.

Case I. We assume that the index k ∈ {1, . . . , K} is such that the set I k in (11) is the singleton {k} and we assume for now α k > 0 in (8). Since the eigenvalues of M(z k ) are the roots of the dispersion relation:

Q 1 (κ) z k = Q 0 (κ) ,
we first observe that the only eigenvalue of M(z k ) on S 1 is κ k and we are now going to show that this eigenvalue is algebraically (and therefore geometrically) simple. The relation (8) gives:

F (κ k ) = - z k α k κ k = 0 .
Moreover, the characteristic polynomial of M(z) at κ equals z -F (κ) up to a nonvanishing holomorphic function of (z, κ) close to (z k , κ k ). This means that κ k is an algebraically simple eigenvalue of M(z k ) and can therefore be extended holomorphically with respect to z in a sufficiently small neighborhood of z k . We let κ k (z) denote this holomorphic extension, which satisfies z = F (κ k (z)) for any z close to z k . Performing a Taylor expansion, we compute:

κ k z k (1 + ) = κ k 1 -α k + O( 2 ) .
In particular, κ k z k (1 + ) belongs to D for > 0 small enough.

To conclude, we observe that the p + r -1 eigenvalues of M(z k ) which differ from κ k lie in D ∪ U . Those eigenvalues remain in D ∪ U as z k is perturbed into z k (1 + ) for a sufficiently small > 0. Using the previous step of the analysis, we know that M z k (1+ ) has r eigenvalues in D and p eigenvalues in U so the reader will easily get convinced that the only possible situation for the location of the eigenvalues of M(z k ) is the one stated in Lemma 1.

Cases II and III. It remains to deal with the case where I k is the singleton {k} and α k < 0 (Case II), and the final case where I k has two elements (Case III). The argument for Case II is the same as for Case I except that now the Taylor expansion of κ k shows that κ k z k (1 + ) belongs to U for > 0 small enough. The remaining details for that case are easily filled in. For Case III, M(z k ) has two eigenvalues on S 1 , which are, in our usual notation, κ ν k,1 and κ ν k,2 . The same argument as in Case I or Case II shows that one of these eigenvalues moves into D as z is perturbed from z k to z k (1 + ), and the other eigenvalue moves into U . This situation thus mixes Cases I and II. The conclusion follows and the proof of Lemma 1 is now complete.

Estimates for the spatial Green's function

This section is devoted to the analysis of the solution to the recurrence relation ( 16), which we recall is an equivalent formulation of (15). More precisely, our aim is to derive pointwise estimates on the spatial Green's function G j (z). We will divide the analysis depending on the position of z in the complex plane.

Away from the tangency points z 1 , . . . , z K , we expect to obtain a uniform exponential decay while near the tangency points only some kind of local boundedness is expected. We will rely on the spectral splitting given by Lemma 1 to compute pointwise estimates of the augmented vector W j (z).

We start with the estimates away from the tangency points.

Lemma 2 (Bounds away from the tangency points). Let z ∈ U \ {z 1 , . . . , z K }. Then there exists an open ball B δ (z), δ > 0, centered at z and there exist two constants C > 0, c > 0 such that:

∀ z ∈ B δ (z) , ∀ j ∈ Z , G j (z) ≤ C exp -c |j| .
Proof. We first introduce some notation. Let z ∈ U \ {z 1 , . . . , z K } be fixed. We know that M(z) in ( 10) is well-defined and holomorphic in a sufficiently small neighborhood of z (including in the case where z belongs to S 1 ). Moreover, because of the spectral splitting shown in Lemma 1, the matrix M(z) has no eigenvalue on S 1 for z close to z, and it has r, resp. p, eigenvalues in D, resp. U , for z close to z. Consequently, for z close to z, the so-called stable subspace, which is spanned by the generalized eigenvectors of M(z) associated with eigenvalues in D, has constant dimension r. Similarly, the unstable subspace, which is spanned by the generalized eigenvectors of M(z) associated with eigenvalues in U , has constant dimension p. We let E s (z), resp. E u (z), denote the stable, resp. unstable, subspace of M(z) for z close to z. We have the decomposition:

∀ z ∈ B δ (z) , C p+r = E s (z) ⊕ E u (z) ,
for some sufficiently small radius δ > 0. The associated projectors are denoted π s (z) and π u (z). The dynamics of ( 16) is therefore of hyperbolic type for any z ∈ B δ (z). The projectors π s (z) and π u (z) are given by contour integrals. For instance, we have:

π s (z) = 1 2 i π γ (w I -M(z)) -1 dw ,
where γ is a contour that encloses the stable eigenvalues (those in D) of M(z) (for instance, S 1 is such a contour). A similar formula holds for π u (z) with a contour that encloses the unstable eigenvalues. This formula shows that π s (z) depends holomorphically on z in the ball B δ (z) and consequently, the stable and unstable subspaces E s (z) and E u (z) depend holomorphically6 on z. Up to restricting δ, any complex number z in the open ball B δ (z) lies in the resolvent set of the operator L , hence there exists a unique solution (W j (z)) j∈Z ∈ 2 (Z; C p+r ) to (16). Since the dynamics of the iteration (16) for such z enjoys a hyperbolic dichotomy, the solution to (16) is given by integrating either from j to +∞, or from -∞ to j -1, depending on whether we compute the unstable or stable components of the vector W j (z). This leads to the expression:

∀ j ∈ Z , π u (z) W j (z) = - 1 A p (z) ≥0 (Q 1 δ) j+ M(z) -1-π u (z) e , (19)
for the unstable components, and to the expression:

∀ j ∈ Z , π s (z) W j (z) = 1 A p (z) j-1 =-∞ (Q 1 δ) M(z) j-1-π s (z) e , (20)
for the stable components. At this stage, we observe that the sequence Q 1 δ only has finitely many nonzero coefficients, which are given by:

∀ j ∈ Z , (Q 1 δ) j = a -j,1 , if j ∈ {-p, . . . , r}, 0 , otherwise.
Hence we see from (19) that π u (z) W j (z) vanishes for j ≥ r + 1, and we see from (20) that π s (z) W j (z) vanishes for j ≤ -p. For j ≤ r, we get:

π u (z) W j (z) = - 1 A p (z) r-j =max(-p-j,0) a -j-,1 M(z) -1-π u (z) e ,
and since the sequence (M(z) -π u (z)) ≥1 is exponentially decreasing, uniformly with respect to z ∈ B δ (z), we get the uniform bound7 :

∀ z ∈ B δ (z) , ∀ j ∈ Z , π u (z) W j (z) ≤ 0 , if j ≥ r + 1, C exp(-c |j|) , if j ≤ r. ( 21
)
Similar arguments, using the uniform exponential decay of the sequence (M(z) π s (z)) ≥1 , yield the bound:

∀ z ∈ B δ (z) , ∀ j ≥ 1 , π s (z) W j (z) ≤ 0 , if j ≤ -p -1, C exp(-c |j|) , if j ≥ -p. (22)
Adding ( 21) and ( 22) gives the claim of Lemma 2 since the spatial Green's function G j (z) is just one coordinate of the vector W j (z) ∈ C p+r .

We are now going to examine the behavior of the spatial Green's function G(z) close to any of the points z k , k = 1, . . . , K, where the spectrum of L is tangent to the unit circle. Let us first recall that the exterior U of the unit disk belongs to the resolvent set of L hence the spatial Green's function G(z) is well-defined in the "half-ball" B δ (z k ) ∩ U for any radius δ > 0. Our goal below is to extend holomorphically G(z) to a whole neighborhood of z k for each k, which amounts to passing through the (essential) spectrum of L . Our results are the following two lemmas.

Lemma 3 (Bounds close to the tangency points -Cases I and II). Let k ∈ {1, . . . , K} be such that the set I k in (11) is the singleton {k}. Then there exists an open ball B (z k ) and there exist two constants C > 0 and c > 0 such that, for any integer j ∈ Z, the component G j (z) defined on B (z k ) ∩ U extends holomorphically to the whole ball B (z k ) with respect to z, and the holomorphic extension satisfies the bound:

∀ z ∈ B (z k ) , ∀ j ∈ Z , G j (z) ≤    C exp -c |j| , if j ≤ 0, C κ k (z) j , if j ≥ 1, if α k > 0 (Case I), and
∀ z ∈ B (z k ) , ∀ j ∈ Z , G j (z) ≤    C κ k (z) |j| , if j ≤ 0, C exp -c j , if j ≥ 1, if α k < 0 (Case II),
where, in either case, κ k (z) denotes the (unique) holomorphic eigenvalue of M(z) defined close to z k and that satisfies κ k (z k ) = κ k .

Lemma 4 (Bounds close to the tangency points -Case III). Let now k ∈ {1, . . . , K} be such that the set I k in (11) has two elements {ν k,1 , ν k,2 } which are fixed by the convention α ν k,1 < 0 < α ν k,2 . Then there exists an open ball B (z k ) centered at z k and there exists a constant C > 0 such that, for any integer j ∈ Z, the component G j (z) defined on B (z k ) ∩ U extends holomorphically to the whole ball B (z k ) with respect to z, and the holomorphic extension satisfies the bound:

∀ z ∈ B (z k ) , ∀ j ∈ Z , G j (z) ≤      C κ ν k,1 (z) |j| , if j ≤ 0, C κ ν k,2 (z) j , if j ≥ 1, (Case III),
where κ ν k,1 (z), resp. κ ν k,2 (z), denotes the (unique) holomorphic eigenvalue of M(z) defined close to z k and that satisfies

κ ν k,1 (z k ) = κ ν k,1 , resp. κ ν k,2 (z k ) = κ ν k,2 .
The proofs of Lemma 3 and Lemma 4 are mostly identical so we just give the proof of Lemma 3 and indicate the minor refinements for the proof of Lemma 4.

Proof of Lemma 3. Most ingredients of the proof are similar to what we have already done in the proof of Lemma 2. We assume from now on α k > 0, the case α k < 0 being left to the interested reader. We just need to slightly adapt the notation used in the proof of Lemma 2 since the hyperbolic dichotomy of M(z) does not hold any longer in a whole neighborhood of z k . Since κ k is a simple eigenvalue of M(z k ), we can extend it holomorphically to a simple eigenvalue κ k (z) of M(z) in a neighborhood of z k . This eigenvalue is associated with the eigenvector:

E k (z) :=      κ k (z) p+r-1 . . . κ k (z) 1      ∈ C p+r ,
which also depends holomorphically on z in a neighborhood of z k . The vector E k (z) contributes to the stable subspace of M(z) for z ∈ U close to z k but the situation is unclear as z goes inside D (it actually depends on the position of z with respect to the spectrum of L ). The remaining p + r -1 eigenvalues of M(z) enjoy the now familiar hyperbolic dichotomy, uniformly with respect to z close to z k . We let below E ss (z), resp. E u (z), denote the strongly stable, resp. unstable, subspace of M(z) associated with those eigenvalues that remain uniformly inside D, resp. U , as z belongs to a neighborhood of z k . In particular, E ss (z), resp. E u (z), has dimension r -1, resp. p, thanks to Lemma 1, and we have the decomposition:

∀ z ∈ B (z k ) , C p+r = E ss (z) ⊕ Span E k (z) ⊕ E u (z) , (23)
for a sufficiently small radius > 0. We let below π ss (z), π k (z) and π u (z) denote the holomorphic projectors associated with the decomposition (23).

We first consider a point z ∈ B (z k )∩U so that the decomposition (23) holds and the Green's function G(z) ∈ 2 (Z; C) is well-defined as the only solution to (15). We use the equivalent formulation (16) and derive the following expressions that are entirely similar to those found in the proof of Lemma 2:

∀ j ∈ Z , π u (z) W j (z) = - 1 A p (z) ≥0 (Q 1 δ) j+ M(z) -1-π u (z) e , (24a)
π ss (z) W j (z) = 1 A p (z) j-1 =-∞ (Q 1 δ) M(z) j-1-π ss (z) e , (24b)
π k (z) W j (z) = 1 A p (z) j-1 =-∞ (Q 1 δ) κ k (z) j-1-π k (z) e . ( 24c
)
The strongly stable (π ss (z)W j (z)) and unstable (π u (z)W j (z)) components obviously extend holomorphically to the whole neighborhood B (z k ) of z k since the projectors π ss (z) and π u (z) depend holomorphically on z on that set and the sums on the right hand side of (24a) and (24b) are, at most, finite. Furthermore, by using the same type of bounds as in the proof of Lemma 2, we obtain:

∀ z ∈ B (z k ) , ∀ j ∈ Z , π u (z) W j (z) + π ss (z) W j (z) ≤ C exp(-c |j|) , (25)
for some appropriate constants C > 0 and c > 0. We now focus on the vector π k (z) W j (z) in (24c) which is aligned with the eigenvector E k (z). We see from (24c) that π k (z) W j (z) vanishes for j ≤ -p. For j in the finite set {-p + 1, . . . , r}, we have:

π k (z) W j (z) = 1 A p (z) j-1 =-p a -,1 κ k (z) j-1-π k (z) e ,
and for j ≥ r + 1, we have:

π k (z) W j (z) = 1 A p (z) r =-p a -,1 κ k (z) j-1-π k (z) e .
In either case, we see that the component π k (z) W j (z) extends holomorphically to the whole neighborhood B (z k ) of z k and we have a bound of the form8 :

∀ z ∈ B (z k ) , ∀ j ∈ Z , π k (z) W j (z) ≤ 0 , if j ≤ -p, C |κ k (z)| j , if j ≥ 1 -p. (26)
In order to conclude, we can always assume that the ball B (z k ) is so small that the modulus |κ k (z)| belongs to the interval9 [exp(-c), exp c] (it equals 1 at z k ). It then remains to add the bounds in (25) and (26) and to compare which is the largest. This completes the proof of Lemma 3 in the case α k > 0. The remaining Case II (α k < 0) is handled similarly except that now the eigenvector E k (z) contributes to the unstable subspace of M(z) for |z| > 1. The minor modifications are left to the reader.

Proof of Lemma 4. The proof of Lemma 4 is a mixture between Cases I and II in which now M(z) has two (holomorphic) eigenvalues whose modulus equals 1 at z k . One contributes to the stable subspace of M(z) and the other one contributes to the unstable subspace of M(z) for |z| > 1. The same ingredients as in the proof of Lemma 3 can then be applied with minor modifications.

Let us remark that all the claims in Lemma 2, Lemma 3 and Lemma 4 do not distinguish between the explicit and implicit case since they only rely on Lemma 1. In the implicit case, for the forthcoming estimates of the temporal Green's function of Section 3, we will also need to obtain bounds of the spatial Green's function G(z) for large values of z. These bounds are provided by the following result.

Lemma 5 (Bounds at infinity -Implicit case). If Q 1 is not the identity, then there exist a radius R ≥ 2 and two constants C > 0, c > 0 such that there holds:

∀ z ∈ B R (0) , ∀ j ∈ Z , G j (z) ≤ C exp -c |j| .
Proof. The proof is basically the same as that of Lemma 2. Indeed, we recall that in the implicit case, the matrix M(z) has a limit at infinity, given by:

      - a p-1,1 a p,1 . . . . . . - a -r,1 a p,1 1 0 . . . 0 0 . . . . . . . . . 0 0 1 0      
, and this matrix has a hyperbolic dichotomy because of Assumption 1. We can thus apply the same argument as in the proof of Lemma 2 for z in a neighborhood of infinity. In particular, we can use the fact that the sequences (M(z) -π u (z)) ≥1 and (M(z) π s (z)) ≥1 are exponentially decreasing, uniformly with respect to z in a neighborhood B R (0) c of infinity. The conclusion of Lemma 5 follows.

Let us observe that we can extend holomorphically each scalar component G j (z) to a neighborhood B (z k ) of z k , but that does not necessarily mean that the extended sequence G(z) lies in 2 (Z; C). For instance, in Case I of Lemma 3, the eigenvalue κ k (z) contributes to the stable subspace of M(z) for |z| > 1 but it starts contributing to the unstable subspace of M(z) as z crosses the spectrum of L (which coincides with the curve F (S 1 )). Hence the holomorphic extension G(z) ceases to be in 2 (Z; C) as z crosses the spectrum of L for it then has an exponentially growing mode in j at +∞.

We finally end this section with the following corollary, which is a direct consequence of Lemma 2, Lemma 3, Lemma 4 and Lemma 5 above by applying a standard compactness argument. We refer to Figure 1 for a geometrical representation of this result.

Corollary 1. There exists some > 0 such that for each ∈ (0, ) there exists some δ > 0 such that the Green's function G(z), defined initially as the unique solution G(z) ∈ 2 (Z; C) to (15) for each z in the resolvent set of L , has a unique holomorphic extension (also denoted G(z)) to the set

S := ζ ∈ C | e -δ < |ζ| ≤ e π ∪ K k=1 B (z k ).
Moreover there are constants C > 0 and c > 0 such that

• Whenever z ∈ S \ K k=1 B (z k ), ∀ j ∈ Z , G j (z) ≤ C exp -c |j| . ( 27
) • Whenever z ∈ B (z k ) for k = 1, • • • , K, the

The temporal Green's function

The starting point of the analysis is to use the inverse Laplace transform formula to express the so-called Green's function G n := L n δ as the following contour integral

∀ n ∈ N * , ∀ j ∈ Z , G n j = (L n δ) j = 1 2 i π Γ z n G j (z) dz , ( 28
)
where Γ is a closed curve in the complex plane surrounding the unit disk D and lying in the resolvent set of L . The idea will be to deform Γ in order to obtain sharp pointwise estimates on the temporal Green's function using our pointwise estimates on the spatial Green's function given in Lemma 2, Lemma 3, Lemma 4 and Lemma 5 above. To do so, we first change variable in (28), by setting z = exp(τ ), such that we get

G n j = 1 2 i π Γ e n τ G j (τ ) dτ , (29)
where without loss of generality Γ = {s + i | ∈ [-π, π]} for some s > 0 (and actually any s > 0 thanks to Cauchy's formula), and G j (τ ) is given by

∀ j ∈ Z , G j (τ ) := G j (e τ ) e τ .
The remaining of this section is devoted to the proof of Theorem 1. For the sake of clarity, we first treat the explicit case with K = 1, and then deal with the implicit case still with K = 1. And finally, we explain how the results generalize to K > 1 in both cases. (Let us recall that K denotes the number of tangency points of the spectrum of L within the unit circle S 1 ).

In the explicit case (when Q 1 is the identity), the analysis below uses the fact that each velocity α k lies in the open interval (-p, r). This fact is stated in the following lemma, which can be seen as a variation on the so-called Courant-Friedrichs-Lewy condition [START_REF] Courant | Über die partiellen Differenzengleichungen der mathematischen Physik[END_REF] and/or the Bernstein inequality for trigonometric polynomials. A proof of Lemma 6 is provided in Appendix A. Lemma 6. Under Assumptions 1 and 2, if Q 1 is the identity, then there holds:

∀ k = 1, . . . , K , -p < α k < r ,
where the α k 's are the drift velocities arising in (8).

We now deal with the proof of Theorem 1.

The explicit case with K = 1

We first remark that, since L = Q 0 is a convolution operator with finite stencil, for each n ≥ 1, there holds G n j = 0 , for j > r n or j < -p n . As a consequence, throughout this section, we assume that j and n satisfy

n ≥ 1 , -p n ≤ j ≤ r n .
We also assume without loss of generality that κ 1 = z 1 = 1 together with α 1 > 0 (the case α 1 < 0 being handled similarly). In that case, we have from (8) that

F e i ξ = exp -i α ξ -β ξ 2 µ + O ξ 2 µ+1 as ξ → 0, ( 30
)
where we dropped the index 1 to simplify our notations. Now, using Lemma 3, bounds close to the tangency point z = 1 for G j (z) translate into bounds near the origin τ = 0 for G j (τ ). More precisely, we have the following lemma which combines Lemma 3 and Corollary 1.

Lemma 7. There exist * > 0 and two constants 0 < β * < Re(β) < β * such that for each ∈ (0, * ) there exist some width η > 0 together with two constants, still denoted C > 0, c > 0, such that, for any integer j ∈ Z, the component G j (τ ) extends holomorphically on B (0) with bounds:

∀ τ ∈ B (0) , ∀ j ∈ Z , G j (τ ) ≤ C exp -c |j| , if j ≤ 0, C exp j Re( (τ )) , if j ≥ 1, ( 31
)
where is holomorphic on B (0) and has the Taylor expansion:

(τ ) = - 1 α τ + (-1) µ+1 β α 2 µ+1 τ 2 µ + O |τ | 2 µ+1 , ∀ τ ∈ B (0) , (32)
together with

Re( (τ )) ≤ - 1 α Re(τ ) + β * α 2 µ+1 Re(τ ) 2 µ - β * α 2 µ+1 Im(τ ) 2 µ , ∀ τ ∈ B (0) . ( 33
)
Furthermore, we have

∀ τ ∈ Ω := { -η < Re(τ ) ≤ π } \ B (0) , ∀ j ∈ Z , G j (τ ) ≤ C exp -c |j| . ( 34
)
Proof. The first part of the proof simply relies on writing κ(z) = exp(ω(z)) near z = 1 and using z = exp(τ ), such that after identification we have (τ ) = ω(exp(τ )). Indeed, the function κ is holomorphic in the ball B 0 (1) for some 0 > 0. Upon reducing the size of 0 , we can define a holomorphic function ω : B 0 (1) → C such that κ(z) = exp(ω(z)) for each z ∈ B 0 (1), and ω(1) = 0. We now let * > 0 small enough be such that for each ∈ (0, * ) and τ ∈ B (0), we have exp(τ ) ∈ B 0 (1). We can now define : B (0) → C as (τ ) := ω(exp(τ )) which is holomorphic in B (0) by construction. Finally, we remark that G j (τ ) extends holomorphically on B (0) for any j ∈ Z since G j (z) extends holomorphically on B 0 (1).

Next, we explain how to use the expansion (30) to obtain the desired Taylor expansion (32) for (τ ) near τ = 0. We first remark that for each ∈ (0, * ) and τ ∈ B (0) we have the identity e τ = F (κ(e τ )) = F (e (τ ) ).

As a consequence, we get the expansion

τ = -α (τ ) + β(-1) µ+1 (τ ) 2 µ + O | (τ )| 2 µ+1 ,
as τ → 0. Since is holomorphic in B (0) with (0) = 0, we can use the above equality to obtain, by identification, each term of its Taylor expansion and recover (32). Note that we can always reduce the size of * such that the expansion is valid for each ∈ (0, * ) and τ in B (0).

To complete the proof we now prove the existence of two positive real numbers β * and β * verifying 0 < β * < Re(β) < β * such that inequality (33) holds true in B (0) for each ∈ (0, * ). First we compute

(-1) µ+1 Re βτ 2 µ = (-1) µ+1 Re(β) Re τ 2 µ -(-1) µ+1 Im(β) Im τ 2 µ = -Re(β) Im(τ ) 2 µ -(-1) µ Re(β) Re(τ ) 2 µ -Re(β) µ-1 m=1 (-1) m 2 µ 2 m Re(τ ) 2 m Im(τ ) 2 (µ-m) -Im(β) µ-1 m=0 (-1) m+1 2 µ 2 m + 1 Re(τ ) 2 m+1 Im(τ ) 2 (µ-m)-1 .
Next using Young's inequality with some δ > 0, we get

Re(τ ) k Im(τ ) 2 µ-k ≤ k 2 µ δ 2 µ k Re(τ ) 2 µ + 2 µ -k 2 µ δ 2 µ 2 µ-k Im(τ ) 2 µ , k = 1, • • • , 2 µ -1 .
And, we also note that the remainder term can be bounded as

O |τ | 2 µ+1 ≤ C * |Re(τ )| 2 µ + |Im(τ )| 2 µ , τ ∈ B (0) ,
for each ∈ (0, * ) for some constant C > 0 independent of . We finally remark that δ > 0 can be taken arbitrarily small and that the leading order term in Im(τ ) 2 µ comes with a negative sign since we assume Re(β) > 0. As a consequence, upon eventually reducing the size of * , we can find 0 < β * < Re(β) < β * (depending only on * ) such that inequality (33) holds true in B (0) for each ∈ (0, * ).

Using Lemma 7, we readily see that when -np ≤ j ≤ 0, our estimates (31)-(34) from Lemma 7 can be combined to

∀ τ ∈ Ω ∪ B (0) , ∀ j ≤ 0 , G j (τ ) ≤ C e -c |j| ,
from which we automatically obtain the following estimate, using the contour Γ = {-η 29) for any 0 < η < η . Modifying the contour in (29) is legitimate thanks to Cauchy's formula and also because the integrals on the segments {-υ ± i π | υ ∈ [-η, π]} compensate one another.

+ i | ∈ [-π, π]} ⊂ Ω ∪ B (0) in (
Lemma 8. For each ∈ (0, * ) there exists constants C > 0 and c > 0 such that for all -n p ≤ j ≤ 0 with n ≥ 1, there holds

G n j ≤ C e -n η -c |j| ,
for any η ∈ (0, η ) with η > 0 the width given in Lemma 7.

From now on, we assume that 1 ≤ j ≤ nr. It turns out that we will need again to divide the analysis in two pieces. We will consider first the medium range where 1 ≤ j ≤ nδ where δ := α 2 . In that case we can prove the following lemma.

Lemma 9. For each ∈ 0, min * , α 2 µ 2β * 1 2 µ-1
there exists a constant C > 0 such that for n ≥ 1 and 1 ≤ j ≤ n δ, the following estimate holds:

G n j ≤ C e -n η 4 ,
for each η ∈ (0, η ) with η > 0 the width given in Lemma 7.

Proof. For each ∈ 0, min * , α 2 µ 2β * 1 2 µ-1 and for η ∈ (0, η ) with η > 0 given in Lemma 7, we use again the segment Γ = {-η 29). We denote by Γ in and Γ out the portions of the segment Re(τ ) = -η which lie either inside B (0) or outside B (0) with |Im(τ )| ≤ π. Standard computations (using Lemma 7) lead to 1 2 i π Γ out e n τ G j (τ ) dτ ≤ C e -n η -c j , and

+ i | ∈ [-π, π]} ⊂ Ω ∪ B (0) in (
1 2 i π Γ in e n τ G j (τ ) dτ ≤ C e -n η
Γ in e j Re( (τ )) |dτ | 2 π .

Next, we recall the estimate (33) on Re( (τ )) from Lemma 7, that is

Re( (τ )) ≤ - 1 α Re(τ ) + β * α 2 µ+1 Re(τ ) 2 µ - β * α 2 µ+1 Im(τ ) 2 µ , ∀ τ ∈ B (0) . As a consequence, for all τ ∈ Γ in ⊂ B (0), we have Re( (τ )) ≤ η α + β * α 2 µ+1 η 2 µ .
Here, we crucially used the fact that the term in Im(τ ) 2 µ in the estimate for Re( (τ )) comes with a negative sign. Summarizing, we have obtained that

-n η + j Re( (τ )) ≤ n η - 1 2 + β * 2α 2 µ η 2 µ-1
for each τ ∈ Γ in and 1 ≤ j ≤ nα 2 . Finally, since 0 < η < < α 2 µ 2β * 1 2 µ-1 , there holds that β * 2α 2 µ η 2 µ-1 < 1 4 , and we get the final estimate e -n η

Γ in e j Re( (τ )) |dτ | 2 π ≤ e -n η 4 .
This concludes the proof of the lemma.

log ( (L ))

Figure 2: Illustration of the geometry of the family of parametrized curved Γ p within the ball B (0) for different values of τ p ∈ [-η/2, 0 ]. The extremal curves are given when τ p = -η/2 to the left (magenta curve) and when τ p = 0 to the right (light blue curve) where 0 < 0 < is precisely defined such that Γ p with τ p = 0 intersects the segment {-η + i | ∈ [-π, π]} on the boundary of B (0). The dashed dark red curve represents the logarithm of the spectrum σ(L ). Note that with our careful choice of parametrization, we have that Γ p with τ p = 0 (dark blue curve) lies to the right of the spectral curve with tangency at the origin.

We now turn to the last case where n ≥ 1 and n δ ≤ j ≤ n r (recall δ = α/2 and α < r). Our generalized Gaussian estimates will precisely come from this part of the analysis. In order to proceed, we follow the strategy developed in [START_REF] Zumbrun | Pointwise semigroup methods and stability of viscous shock waves[END_REF] in the fully continuous case (see also [START_REF] Godillon | Green's function pointwise estimates for the modified Lax-Friedrichs scheme[END_REF] in a fully discrete case that corresponds to µ = 1 in our notation), and introduce a family of parametrized curves given by

Γ p := Re(τ ) - β * α 2 µ Re(τ ) 2 µ + β * α 2 µ Im(τ ) 2 µ = Ψ (τ p ) | -η ≤ Re(τ ) ≤ τ p (35)
with Ψ (τ p ) := τ p -β * α 2 µ τ 2 µ p . Note that the curves Γ p intersect the real axis at τ p . We now explain how we choose η > 0 and τ p > -η in the above definition of Γ p .

First, for each ∈ (0, * ), we fix η ∈ (0, η ) with η > 0 given in Lemma 7 such that the curve Γ p with τ p = 0 intersects {-η + i | ∈ [-π, π]} inside the open ball B (0). Then, we let 0 ∈ (0, ) which is uniquely defined as the value of τ p for which Γ p with τ p = 0 intersects the segment {-η + i | ∈ [-π, π]} precisely on the boundary10 of B (0) with η fixed previously. And finally, the specific value of τ p is fixed depending on the ratio ζ γ as follows

τ p :=                  ρ ζ γ if - η 2 ≤ ρ ζ γ ≤ 0 , 0 if ρ ζ γ > 0 , - η 2 if ρ ζ γ < - η 2 .
We refer to Figure 2 for an illustration of the geometry of Γ p for different values of τ p . There only remains to define ζ, γ and the function ρ. We let

ζ := j -n α 2 µ n , and γ := j n β * α 2 µ > 0 ,
and ρ ζ γ is the unique real root to the equation

-ζ + γ x 2 µ-1 = 0 , that is ρ ζ γ := sgn ζ γ |ζ| γ 1 2 µ-1 .
The motivation for introducing such quantities comes from the estimate (33) from Lemma 7. More precisely, for all τ ∈ Γ p ⊂ B (0), we have

j Re( (τ )) ≤ j - 1 α Re(τ ) + β * α 2 µ+1 Re(τ ) 2 µ - β * α 2 µ+1 Im(τ ) 2 µ = j - τ p α + β * α 2 µ+1 τ 2 µ p = -n τ p + n α -2 µ ζ τ p + γ τ 2 µ p ,
and our careful choice of τ p will always allow us to handle the terms inside the final parenthesis. We remark thatα 4µ ≤ ζ ≤ r-α 2 µ , and our generalized Gaussian estimates will come from those values of ζ ≈ 0. Before proceeding with the analysis, we note that for all τ ∈ Γ p , we have

Re(τ ) ≤ τ p -c * Im(τ ) 2 µ , (36)
for some constant c * > 0. Indeed, we remark that the function Ψ, defined as Ψ(t) = tβ * α 2 µ t 2 µ , satisfies Ψ (0) = 1 such that for each t ∈ [-η, ] one has Ψ (t) ≤ c 0 for some c 0 > 0. As a consequence, for each τ ∈ Γ p one has

- β * α 2 µ Im(τ ) 2 µ = Ψ(Re(τ )) -Ψ(τ p ) = - τp Re(τ ) Ψ (t) dt ≥ c 0 (Re(τ ) -τ p ) ,
which gives the desired estimate (36) with c * = β * c 0 α 2 µ . Furthermore, a straightforward application of the implicit function theorem gives the following result on the parametrization of the curves Γ p that we will be using in our estimates below.

Lemma 10. There exist * * ∈ (0, * ), some analytic function Φ : (- * * , * * ) × (- * * , * * ) → R and some constant C > 0 such that for any ∈ (0, * * ) and τ p ∈ (-, ) the curve Γ p can be parametrized as

Γ p = {τ ∈ B (0) | Re(τ ) = Φ(Im(τ ), τ p )} , with Re(τ ) = τ p - β * α 2 µ Im(τ ) 2 µ + O |Im(τ )| 2 µ+1 + |τ p | 2 µ+1
η 2 ≤ ρ ζ γ ≤ 0 when n δ ≤ j ≤ n r. The contour is composed of Γ out -∪ Γ in -∪ Γ p ∪ Γ in + ∪ Γ out + .
The contours Γ in ± and Γ out ± are the portions of the segment Re(τ ) = -η which lie either inside B (0) or outside B (0) with |Im(τ )| ≤ π while Γ p is defined in (35) and intersects the real axis at τ p .

Finally, in what follows, we will use the notation f g whenever f ≤ C g for some constant C > 0 independent of j and n.

• We first treat the caseη 2 ≤ ρ ζ γ ≤ 0 . For all τ ∈ Γ p ⊂ B (0) we obtain

n Re(τ ) + j Re( (τ )) ≤ n (Re(τ ) -τ p ) + n α -2 µ ζ τ p + γ τ 2 µ p ≤ -n c * Im(τ ) 2 µ + n α -2 µ ζ τ p + γ τ 2 µ p .
For the second term, we will use the specific form of

τ p = ρ ζ γ = sgn(ζ) |ζ| γ 1 2 µ-1 to get that -2 µ ζ τ p + γ τ 2 µ p = -γ (2 µ -1) |ζ| γ
As a consequence, we can derive the following bound

1 2 i π Γp e n τ G j (τ ) dτ Γp e nRe(τ )+jRe( (τ )) |dτ | e -n α (2 µ-1)γ |ζ| γ 2 µ 2 µ-1 Γp e -nc * Im(τ ) 2 µ |dτ | e -n α (2 µ-1)γ |ζ| γ 2 µ 2 µ-1 n 1 2 µ .
In the last inequality, assuming that ∈ (0, * * ), we have used Lemma 10 to get

Γp e -nc * Im(τ ) 2 µ |dτ | * - * e -nc * x 2 µ dx 1 n 1 2 µ
, where * ∈ (0, ) is defined as

* := α 2 µ β * (Ψ(τ p ) -Ψ(-η)) 1 2 µ , which is the positive root of -η - β * α 2 µ η 2 µ + β * α 2 µ 2 µ * = Ψ(τ p ) = τ p - β * α 2 µ τ 2 µ p .
Next we denote by Γ in ± and Γ out ± the portions of the segment Re(τ ) = -η which lie either inside B (0) or outside B (0) with |Im(τ )| ≤ π. We refer to Figure 3 for an illustration. Usual computations lead to

1 2 i π Γ out ± e n τ G j (τ ) dτ ≤ C e -n η -c j .
For all τ ∈ Γ in ± , we use that Im(τ )2 ≥ Im(τ * ) 2 where τ * = -η+i * with * = α 2 µ β * (Ψ(τ p ) -Ψ(-η))

1 2 µ .
That is τ * = -η + i * lies at the intersection of Γ p and the segment {-η

+ i | ∈ [-π, π]} with τ * ∈ B (0). As a consequence, for all τ ∈ Γ in ± we have Re( (τ )) ≤ - 1 α Re(τ ) + β * α 2 µ+1 Re(τ ) 2 µ - β * α 2 µ+1 Im(τ ) 2 µ = - τ p α + β * α 2 µ+1 τ 2 µ p - β * α 2 µ+1 Im(τ ) 2 µ -2 µ * ≥0 ≤ - τ p α + β * α 2 µ+1 τ 2 µ p .
Thus, we have

n Re(τ ) + j Re( (τ )) ≤ -n η + j - τ p α + β * α 2 µ+1 τ 2 µ p = n α -η α + j n -τ p + β * α 2 µ τ 2 µ p ≤ n α -(η + τ p ) α -2 µ ζ τ p + γ τ 2 µ p = n α -(η + τ p ) α -(2 µ -1) γ |ζ| γ for all τ ∈ Γ in ± . Finally, as -η 2 ≤ ρ( ζ γ ) = τ p we have η + τ p ≥ η 2 ,
we obtain an estimate of the form

1 2 i π Γ in ± e n τ G j (τ ) dτ ≤ C e -n η 2 -n α (2 µ-1) γ |ζ| γ 2 µ 2 µ-1
. Summarizing, we have obtained

G n j ≤ 1 2 i π Γp e n τ G j (τ ) dτ + 1 2 i π Γ out ± e n τ G j (τ ) dτ + 1 2 i π Γ in ± e n τ G j (τ ) dτ ≤ C    e -n α (2 µ-1)γ |ζ| γ 2 µ 2 µ-1 n 1 2 µ + e -n η -c j + e -n η 2 -n α (2 µ-1) γ |ζ| γ 2 µ 2 µ-1    .
• Next, we consider the case ρ(ζ/γ) > 0 for which we choose τ p = 0 . The contour Γ is decomposed into Γ p ∪ Γ out ± where Γ out ± are the portions of the segment Re(τ ) = -η which lie outside B (0) with |Im(τ )| ≤ π. In that case, we have that for all τ

∈ Γ p n Re(τ ) + j Re( (τ )) ≤ -n c * Im(τ ) 2 µ + n α -2 µ ζ τ p + γ τ 2 µ p = -n c * Im(τ ) 2 µ + n α -2 µ ζ 0 + γ 2 µ 0 since τ p = 0 in this case. But as ρ(ζ/γ) > 0 we get that ζ > 0 and ζ > 2 µ-1 0
γ, the last term in the previous inequality is estimated via

-2 µ ζ 0 + γ 2 µ 0 < -(2 µ -1) γ 2 µ 0 < 0 .
As a consequence, we can derive the following bound

1 2 i π Γp e n τ G j (τ ) dτ e -n α (2 µ-1)γ 2 µ 0 n 1 2 µ .
With our careful choice of 0 > 0, the remaining contribution along segments Γ out ± with Re(τ ) = -η can be estimated as usual as

1 2 i π Γ out ± e n τ G τ (j)dτ ≤ C e -n η -c j , as |τ | ≥ for τ ∈ Γ out ± . Summarizing, we have obtained G n j ≤ 1 2 i π Γp e n τ G j (τ ) dτ + 1 2 i π Γ out ± e n τ G j (τ ) dτ ≤ C e -n α (2 µ-1)γ 2 µ 0 n 1 2 µ + e -n η -c j .
• It remains to handle the last case ρ(ζ/γ) < -η/2 for which we choose τ p = -η/2, and we readily note that in this setting ζ < 0. The contour Γ is decomposed into Γ p ∪ Γ out ± ∪ Γ in ± where once again Γ in ± and Γ out ± are the portions of the segment Re(τ ) = -η which lie either inside B (0) or outside B (0) with |Im(τ )| ≤ π. For all τ ∈ Γ p ⊂ B (0), we find that

n Re(τ ) + j Re( (τ )) ≤ -n c * Im(τ ) 2 µ + n α µ ζ η + γ η 2 2 µ . Using that ρ(ζ/γ) < -η/2 which is equivalent to ζ/γ < -η 2 2 µ-1 , we get that µ ζ η + γ η 2 2 µ < -(2 µ -1) γ η 2 2 µ .
As a consequence, we can derive the following bound

1 2 i π Γp e n τ G j (τ ) dτ e -n α (2 µ-1)γ( η 2 ) 2 µ n 1 2 µ .
As usual, we have that

1 2 i π Γ out ± e n τ G j (τ ) dτ ≤ C e -n η -c j .
It only remains to estimate the contribution on Γ in ± . We proceed as before, and we have that for all

τ ∈ Γ in ± n Re(τ ) + j Re( (τ )) ≤ n α -(η + τ p ) α -2 µ ζ τ p + γ τ 2 µ p ≤ n α - η 2 α -(2 µ -1) γ η 2 2 µ = -n η 2 + (2 µ -1) γ α η 2 2 µ
, and this time we obtain

1 2 i π Γ in ± e n τ G j (τ ) dτ ≤ C e -n η 2 + (2 µ-1)γ α ( η 2 ) 2 µ .
In conclusion, we have obtained

G n j ≤ 1 2 i π Γp e n τ G j (τ ) dτ + 1 2 i π Γ out ± e n τ G j (τ ) dτ + 1 2 i π Γ in ± e n τ G j (τ ) dτ ≤ C e -n α (2 µ-1)γ( η 2 ) 2 µ n 1 2 µ + e -n η -c j + e -n η 2 + (2 µ-1)γ α ( η 2 ) 2 µ .
As a summary, gathering the above estimates, we can deduce the following result.

Lemma 11. For each ∈ (0, * * ) there exist constants C > 0 and c > 0 such that for n ≥ 1 and n δ ≤ j ≤ n r the following estimate holds:

G n j ≤ C n 1 2 µ exp -c |j -α n| n 1 2 µ 2 µ 2 µ-1
, where * * > 0 is given in Lemma 10. Generalized Gaussian estimates are derived near each j ≈ n α k , k = {1, 2, 3} in the interior of the sectors delimited by the magenta lines. In the explicit case, below the lines j = -n p and j = n r (blue), the Green's function G n j vanishes.

Proof. One only needs to check that the purely exponentially decaying in n contributions obtained when

ρ ζ γ > 0 or ρ ζ γ < -η
2 can be subsumed into generalized Gaussian estimates. For example, in the case ρ ζ γ > 0 , there exists some small constant c > 0 such that

-n ≤ -c |j -α n| n 1 2 µ 2 µ 2 µ-1 , as β * 2 α 2 µ-1 ≤ γ ≤ β * α 2 µ r , and j n α -1 = 2 µ ζ α > 2 µ α γ 2 µ-1 0 ≥ µ β * α 2 µ 2 µ-1 0 .
All other cases can be dealt with in a similar way.

Proof of Theorem 1. Combining Lemma 8, Lemma 9 and Lemma 11 proves our main Theorem 1 in the explicit case with K = 1. Indeed, we first fix ∈ 0, min * * ,

α 2 µ 2β * 1 2 µ-1 with 0 < * * < * from
Lemma 10, and then we fix η ∈ (0, η ) with η > 0 given in Lemma 7 such that the curve Γ p with τ p = 0 intersects {-η + i | ∈ [-π, π]} inside the open ball B (0). As a consequence, we can apply estimates from Lemma 8, Lemma 9 and Lemma 11 with this specific choice of and η, which gives the proof since purely exponentially decaying in n bounds obtained in Lemma 8 and Lemma 9 can be subsumed into generalized Gaussian estimates.

The implicit case with K = 1

The main difference compared with the explicit case is that now it is no longer true that G n j vanishes for j > n r or j < -p n. Nevertheless we observe that the results of Lemma 8, Lemma 9 and Lemma 11 still hold true. Actually, the proofs of Lemma 8, Lemma 9 and Lemma 11 naturally extend to -n L ≤ j ≤ 0 (Lemma 8), 1 ≤ j ≤ n δ (Lemma 9, unchanged) and n δ ≤ j ≤ n L (Lemma 11) for any large constant L ≥ max(p, r) that is fixed a priori. As a consequence, one only needs to consider the case n ≥ 1 and |j| > n L for some large constant L > 0 to be determined. To obtain the desired estimate in that case, we use the bound at infinity obtained in Lemma 5. More precisely, there exists R ≥ π/2 and two constants C > 0, c > 0 such that there holds

∀ τ ∈ {ζ ∈ C | Re(ζ) ≥ log R} , ∀ j ∈ Z , G j (τ ) ≤ C exp -c |j| . (37)
We then have the following result.

Lemma 12. Let L ≥ max(p, r) > 0 be large enough such that L > 2 log R / c with R and c as in (37).

Then, there exists C > 0, such that for n ≥ 1 and |j| > nL, we have

G n j ≤ C exp -n c L 4 - c 2 |j| .
Proof. In (29), we now use the contour

Γ = c |j| 2 n + i | ∈ [-π, π] . With our choice of L, we have that for all τ ∈ Γ, Re(τ ) = c |j| 2 n ≥ cL 2 > log R and so 1 2 i π Γ e n τ G j (τ ) dτ ≤ C e -c 2 |j| .
Finally, we notice that

- c 2 |j| ≤ -n c L 4 - c 4 |j| , for |j| > n L .
This completes the proof of the lemma.

The proof of Theorem 1 in the implicit case (for K = 1) then follows from the combination of the slight extensions of Lemma 8, Lemma 9 and Lemma 11 (once the large constant L is fixed as in Lemma 12).

The explicit and implicit cases with K > 1

We now briefly explain how to handle the general case with K > 1 and refer to Figures 4-5 for illustrations. From Assumption 2, we have the existence of K tangency points κ k with associated nonzero real numbers α k . We will distinguish two cases:

A. All α k are distinct from one and another.

B. There exist two or more α k which are equal.

We only discuss the explicit case here, as the implicit case does not distinguish between Cases A and B. First, we let be

τ k = i θ k := log(z k ) for θ k ∈ [-π, π] and θ k ∈] -π, π] be such that κ k = e i θ k for each k = 1, • • • , K.
In order to proceed, we need the following lemma which is a direct consequence of Corollary 1 and whose proof is identical to Lemma 7.

Lemma 13. There exist some * > 0 and two constants 0 < β * < Re(β k ) < β * for k = 1, • • • , K such that for each ∈ (0, * ) there exist some width η > 0 together with two constants, still denoted C > 0, c > 0, such that, for any integer j ∈ Z, the component G j (τ ) extends holomorphically on each B (τ k ) with bounds:

∀ τ ∈ B (τ k ) , ∀ j ∈ Z , G j (τ ) ≤ C exp -c |j| , if j ≤ 0, C exp j Re( k (τ )) , if j ≥ 1, (Case I) or ∀ τ ∈ B (τ k ) , ∀ j ∈ Z , G j (τ ) ≤ C exp j Re( k (τ )) , if j ≤ 0, C exp -c |j| , if j ≥ 1, (Case II) or ∀ τ ∈ B (τ k ) , ∀ j ∈ Z , G j (τ ) ≤ C exp j Re( ν k,1 (τ )) , if j ≤ 0, C exp j Re( νk,2 (τ )) , if j ≥ 1, (Case III)
where each k is holomorphic on B (τ k ) and has the Taylor expansion:

k (τ ) = i θ k - 1 α k (τ -τ k ) + (-1) µ k +1 β k α 2 µ k +1 k (τ -τ k ) 2 µ k + O |τ -τ k | 2 µ k +1 , ∀ τ ∈ B (τ k ) , together with Re( k (τ )) ≤ - Re(τ ) α k + β * α 2 µ k +1 k Re(τ ) 2 µ k - β * α 2 µ k +1 k (Im(τ ) -θ k ) 2 µ k , ∀ τ ∈ B (τ k ) . (38)
Furthermore, we have

∀ τ ∈ Ω := { -η < Re(τ ) ≤ π } \ K k=1 B (τ k ) , ∀ j ∈ Z , G j (τ ) ≤ C exp -c |j| .
Case A. This is precisely the case depicted in Figure 4 with K = 3. Without loss of generality we label the α k by increasing order such that

-p < α 1 < • • • < α k < • • • < α K < r.
For each k = 1, • • • , K we define two real numbers δ k < δ k such that we have the ordering

-p < δ 1 < α 1 < δ 1 < • • • < δ k < α k < δ k < • • • < δ K < α K < δ K < r , with sgn(δ k ) = sgn(δ k ) = sgn(α k ). For each k = 1, • • • , K,
we define the following sectors in the (j, n)plane:

D k := (j, n) ∈ Z × N * | n δ k ≤ j ≤ n δ k ,
together with

D * := {(j, n) ∈ Z × N * | -n p ≤ j ≤ n r} \ K k=1 D k .
Our first lemma pertains at obtaining exponential bounds in the region D * . We introduce two quantities Proof. We only sketch the proof as it is almost identical to the proofs of Lemma 8 and Lemma 9. Let ∈ (0, min( * , Λ * 1 , Λ * 2 )) and consider η ∈ (0, η ). We select the contour Γ = {-η + i | ∈ [-π, π]} and we denote by Γ in k the portion of Γ which lie within B (τ k ) and Γ out the union of the remaining portions. As a consequence, we have Γ = Γ in 1 ∪ • • • ∪ Γ in K ∪ Γ out , and we get

Λ * 1 := min k=1,••• ,K α 2 µ k k 2 β * 1 2 µ k -1 > 0 , and Λ * 2 := min k=1,••• ,K (δ k -α k ) α 2 µ k k 2 r β * 1 2 µ k -1 > 0 .
1 2 i π Γ e n τ G j (τ ) dτ ≤ K k=1 1 2 i π Γ in k e n τ G j (τ ) dτ + 1 2 i π Γ out e n τ G j (τ ) dτ .
Our objective is to bound each above term separately. Along Γ out , we get an estimate of the form

1 2 i π Γ out e n τ G j (τ ) dτ ≤ C e -n η -c |j| , ∀ (j, n) ∈ D * ,
as along Γ out the Green's function G j (τ ) enjoys the pointwise exponential bound from Lemma 13. We now derive pointwise bound for each contour integral along Γ in k , k = 1, • • • , K. We first handle the case where I k = {k}, and assume without loss of generality that α k > 0. Then Case I of Lemma 13 reads

∀ τ ∈ B (τ k ) , ∀ j ∈ Z , G j (τ ) ≤ C exp -c |j| , if j ≤ 0, C exp j Re( k (τ )) , if j ≥ 1, (39)
with

Re( k (τ )) ≤ - Re(τ ) α k + β * α 2 µ k +1 k Re(τ ) 2 µ k - β * α 2 µ k +1 k (Im(τ ) -θ k ) 2 µ k , ∀ τ ∈ B (τ k ) .
If (j, n) ∈ D * is such that j ≤ 0, then we directly get

1 2 i π Γ in k e n τ G j (τ ) dτ ≤ C e -n η -c |j| .
From now on, we therefore consider (j, n) ∈ D * with j ≥ 1. As in the proof of Lemma 9, we use the above estimate (38) to get that

Re( k (τ )) ≤ η α k + β * α 2 µ k +1 k η 2 µ k for each τ ∈ Γ in k ⊂ B (τ k ). As a consequence, for all (j, n) ∈ D * with 1 ≤ j ≤ n δ k , we have -n η + j Re( k (τ )) ≤ n η -1 + j n α k + j n β * α 2 µ k +1 k η 2 µ k -1 ≤ -n η     1 - δ k α k >0 - β * α 2 µ k k η 2 µ k -1     ≤ - n η 2 1 - δ k α k , since η is chosen such that 0 < η < η < < Λ * 1 .
And we have obtained the estimate

1 2 i π Γ in k e n τ G j (τ ) dτ ≤ C exp - n η 2 1 - δ k α k .
For the remaining cases (j, n) ∈ D * and n δ k ≤ j ≤ n r, we use a different contour near the ball B (τ k ). We refer to Figure 6 for an illustration. We introduce the contour

Γ k k := Re(τ ) - β * α 2 µ k k Re(τ ) 2 µ k + β * α 2 µ k k (Im(τ ) -θ k ) 2 µ k = Ψ k ( k ) | -η ≤ Re(τ ) ≤ k , with Ψ k ( k ) := k -β * α 2 µ k k 2 µ k k
and where 0 < k < is chosen such that Γ k k intersects Γ in k precisely on the boundary of B (τ k ). We note that there exists some constant c k > 0 such that for any τ

∈ Γ k k ⊂ B (τ k ), one has Re(τ ) ≤ k -c k (Im(τ ) -θ k ) 2 µ k , which yields n Re(τ ) + j Re( k (τ )) ≤ n Re(τ ) + j - Re(τ ) α k + β * α 2 µ k +1 k Re(τ ) 2 µ k - β * α 2 µ k +1 k (Im(τ ) -θ k ) 2 µ k = n Re(τ ) - j α k Ψ( k ) ≤ -n c k (Im(τ ) -θ k ) 2 µ k + n k α k α k - j n + j n β * α 2 µ k k 2 µ k -1 k ≤ - n k α k   δ k -α k >0 -r β * α 2 µ k k 2 µ k -1 k  
for each τ ∈ Γ k k . Now, since δ k -α k > 0 and 0 < k < < Λ * 2 , we have 1 2 i π Γ in k e n τ G j (τ ) dτ ≤ C exp -n k (δ k -α k ) 2 α k ,
which gives the desired estimate in the region (j, n) ∈ D * and n δ k ≤ j ≤ n r.

Let finally comment on the case where

I k = {ν k,1 , ν k,2 } with α ν k,1 < 0 < α ν k,2
. This time, Lemma 13 gives

∀ τ ∈ B (τ k ) , ∀ j ∈ Z , G j (τ ) ≤ C exp j Re( ν k,1 (τ )) , if j ≤ 0, C exp j Re( ν k,2 (τ )) , if j ≥ 1.
The analysis for (j, n) ∈ D * and j ≥ 1 is unchanged, and we apply the same strategy for (j, n) ∈ D * and j ≤ 0 without any difficulty. As a conclusion, we have obtained that there exist C > 0 and δ > 0 such that for each k = 1, • • • , K we have

1 2 i π Γ in k e n τ G j (τ ) dτ ≤ C e -n δ , ∀(j, n) ∈ D * ,
which ends the proof.

We prove in the next Lemma that we obtain generalized Gaussian estimates in each sector

D k , k = 1, • • • , K.
Lemma 15. There exists * ∈ (0, * ) such that for each ∈ (0, * ) there are constants C > 0 and c > 0 such that for any k = 1, • • • , K and (j, n) ∈ D k , the following estimate holds:

G n j ≤ C n 1 2 µ k exp   -c |j -α k n| n 1 2 µ k 2 µ k 2 µ k -1   .
Proof. Let (j, n) ∈ D k , that is n ≥ 1 and n δ k ≤ j ≤ n δ k . Assume without loss of generality that α k > 0. We first consider the case where I k = {k}. Once again, we introduce a family of parametrized curves Γ k p given by Γ k p := Re(τ ) -

β * α 2 µ k k Re(τ ) 2 µ k + β * α 2 µ k k (Im(τ ) -θ k ) 2 µ k = Ψ k (τ p,k ) | -η ≤ Re(τ ) ≤ τ p,k with Ψ k (τ p,k ) = τ p,k -β * α 2 µ k k
τ 2 µ k p,k and η > 0, τ p,k > -η are chosen as follows. For each ∈ (0, * ) we fix η ∈ (0, η ) such that the curve Γ k p with τ p,k = 0 intersects the ray {-η + i | ∈ [-π, π]} inside the ball B (τ k ). Furthermore, we let 0 < 0,k < be defined as the value of τ p,k for which Γ k p with τ p,k = 0,k intersects the ray {-η + i | ∈ [-π, π]} on ∂B (τ k ) with η fixed previously. Finally, τ p,k is now defined as

τ p,k :=                  ρ k ζ k γ k if - η 2 ≤ ρ k ζ k γ k ≤ 0,k , 0,k if ρ k ζ k γ k > 0,k , - η 2 if ρ k ζ k γ k < - η 2 ,
where ζ k , γ k and the function ρ k are set to

ζ k := j -n α k 2 µ k n , and
γ k := j n β * α 2 µ k k > 0 , with ρ k ζ k γ k
given by

ρ k ζ k γ k := sgn ζ k γ k |ζ k | γ k 1 2 µ k -1 .
In our estimate, we use a contour Γ k which consists of Γ k p in B (τ k ) and the ray {-η + i | ∈ [-π, π]} otherwise (see Figure 5, left panel for an illustration in the case K = 3). Depending on the ratio ζ k γ k , there exists (or not) a portion of the ray {-η

+ i | ∈ [-π, π]} within the ball B (τ k ) that we denote Γ in k . Note that when ρ k ζ k γ k > 0,k we have Γ in k = ∅.
The analysis along Γ k p ∪ Γ in k is exactly the same as the one conducted in the proof of Lemma 11 and we get that there exists * * ∈ (0, * ) such that for all ∈ (0, * * ) one has

1 2 i π Γ k p ∪Γ in k e n τ G j (τ ) dτ ≤ C n 1 2 µ k exp   -c |j -α k n| n 1 2 µ k 2 µ k 2 µ k -1   , (j, n) ∈ D k .
The fact that one needs to eventually decrease the size of comes from Lemma 10 which is needed to obtain the generalized Gaussian bound and prove that

Γ k p e -nc * k (Im(τ )-θ k ) 2 µ k |dτ | ≤ C n 1 2 µ k .
Along the ray {-η + i | ∈ [-π, π]}, we denote by Γ out all portions that lie outside the balls B (τ υ ) with υ = k, and we get 1 2 i π Γ out e n τ G j (τ ) dτ ≤ e -n η -c |j| , (j, n) ∈ D k .

Thus, it only remains to estimate the contour integral along the ray {-η + i | ∈ [-π, π]} within a ball B (τ υ ) with υ = k, that we denote Γ in υ . Let assume first that I υ = {υ}. We split the analysis in two cases.

(i) If α υ < 0, then we have the estimate

| G j (τ ) | ≤ C e -c j , τ ∈ Γ in υ ,
as 0 < n δ k ≤ j ≤ n δ k , and we obtain

1 2 i π Γ in υ e n τ G j (τ ) dτ ≤ C e -n η -c j , (j, n) ∈ D k .
(ii) If α υ > 0, then we have the following estimates for each τ ∈ Γ in υ ⊂ B (τ υ )

| G j (τ ) | ≤ C e j Re( υ (τ )) , τ ∈ Γ in υ , and
Re( υ (τ )) ≤ - Re(τ ) α υ + β * α 2 µυ+1 υ Re(τ ) 2 µυ - β * α 2 µυ+1 υ (Im(τ ) -θ υ ) 2 µυ .
As a consequence, we readily obtain that

-n η + j Re( υ (τ )) ≤ n η -1 + j n α υ + j n β * α 2 µυ+1 υ η 2 µυ-1 .
Thus if α υ > α k , we get that

-n η + j Re( υ (τ )) ≤ -n η     1 - δ k α υ >0 - δ k β * α 2 µυ+1 υ η 2 µυ-1     ≤ - n η 2 1 - δ k α υ ,
provided that η is chosen small enough, which is always possible by eventually reducing the size of * * . This gives

1 2 i π Γ in υ e n τ G j (τ ) dτ ≤ C e -n η 2 1- δ k αυ .
Finally, if α υ < α k we use a different contour inside the ball B (τ υ ). Namely, we use the contour

Γ υ υ := Re(τ ) - β * α 2 µυ υ Re(τ ) 2 µυ + β * α 2 µυ υ (Im(τ ) -θ υ ) 2 µυ = Ψ υ ( υ ) | -η ≤ Re(τ ) ≤ υ , with Ψ υ ( υ ) := υ -β * α 2 µυ υ 2 µυ υ
and where 0 < υ < is chosen such that Γ υ υ intersects the segment Γ in υ precisely on the boundary of B (τ υ ). We note that there exists some constant c υ > 0 such that for any τ ∈ Γ υ υ ⊂ B (τ υ ), one has

Re(τ ) ≤ υ -c υ (Im(τ ) -θ υ ) 2 µυ , which yields n Re(τ ) + j Re( υ (τ )) ≤ -n c υ (Im(τ ) -θ υ ) 2 µυ - n υ α υ δ k -α υ -δ k β * α 2 µυ υ 2 µυ-1 υ ,
for each τ ∈ Γ υ υ . Now, since δ k -α υ > α k -α υ > 0 and 0 < υ < , we can always further reduce the size of * * such that

1 2 i π Γ in υ e n τ G j (τ ) dτ ≤ C n 1 2 µυ exp -n υ (δ k -α υ ) 2 α υ ,
which gives the desired estimate.

If now I υ = {ν υ,1 , ν υ,2 }, then we have α ν υ,1 < 0 < α ν υ,2 and for 0 < n δ k ≤ j ≤ n δ k , we get

| G j (τ ) | ≤ C exp j Re ν υ,2 (τ ) , τ ∈ Γ in ν ,
such that the analysis is similar to the above case (ii). Finally, when I k = {ν k,1 , ν k,2 }, we necessarily have that α ν k,1 < 0 < α ν k,2 = α k and the analysis remains unchanged. As there exists some * ∈ (0, * * ) such that for all ∈ (0, * ) we have proved the desired generalized Gaussian bound.

Case B. In that case, two or more α k are equal. Note that for (j, n) ∈ D * the analysis remains unchanged and Lemma 14 still holds true in that case. Let us assume for simplicity that α ν 1 = α ν 2 for some couple of integers ν 1 = ν 2 , and all other α k 's are distinct. The estimate from Lemma 15 is still valid for (j, n) ∈ D k for each k ∈ {ν 1 , ν 2 }. For (j, n) ∈ D ν 1 = D ν 2 , in the ball B (τ ν 1 ) we use the contour Γ ν 1 p and in the ball B (τ ν 2 ) we use the contour Γ ν 2 p . And we refer to Figure 5 for an illustration of such contours. Reproducing the analysis of Lemma 15, we obtain the existence of C > 0 and c > 0 such that for (j, n) ∈ D ν 1 = D ν 2 the following estimate holds:

G n j ≤ C ν∈{ν 1 ,ν 2 } 1 n 1 2 µν exp -c |j -α ν n| n 1 2 µν 2 µν 2 µν -1 .
Finally, we remark that Lemma 12 naturally extends to the case K > 1 in the implicit setting. This concludes the proof of Theorem 1.

Examples and extensions

We first give several examples of operators (2) that fit into the framework of Theorem 1, and that arise when discretizing the transport equation:

∂ t u + ∂ x u = 0 , (t, x) ∈ R + × R , (40)
with Cauchy data at t = 0. We refer to [START_REF] Gustafsson | Time dependent problems and difference methods[END_REF][START_REF] Gustafsson | High order difference methods for time dependent PDE[END_REF] for a detailed analysis and more examples of finite difference schemes in that context.

Example 1: the Lax-Friedrichs scheme

The Lax-Friedrichs scheme is an explicit finite difference approximation of (40), which corresponds to the operators:

Q 1 := I , Q 0 := 1 + λ 2 S -1 + 1 -λ 2 S , (41)
where here and below, λ is a real parameter11 and S still denotes the shift operator defined by: S : u j j∈Z -→ u j+1 j∈Z .

We now restrict to λ ∈ (0, 1) so that both coefficients in the definition (41) are positive and they sum to 1. In probability theory, this corresponds to a random walk with probability (1 + λ)/2 to jump of +1 and probability (1 -λ)/2 to jump of -1 at each time iteration (recall our convention on the coefficients a which differs from the standard convolution product).

In the notation of (2), we have r = p = 1. Since we are dealing here with an explicit scheme, Assumptions 1 and 4 are trivially satisfied. The definition (6) reduces here to: F e i ξ = cos(ξ) -i λ sin(ξ) .

Computing:

F e i ξ 2
= cos 2 (ξ) + λ 2 sin 2 (ξ) , we find that F (κ) belongs to D for all κ ∈ S 1 , and F (κ) belongs to S 1 for such κ if and only if κ = ±1. We thus have (7) with κ 1 := 1 and κ 2 := -1, and the reader can check that (8) is satisfied with:

α 1 = α 2 = λ , β 1 = β 2 = 1 -λ 2 2 , µ 1 = µ 2 = 1 ,
which means that Assumption 2 is satisfied. Since the modulus of F (κ) attains its maximum at two points of S 1 , we cannot apply the uniform Gaussian bound from [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF]. The improvements of Theorem 1 and [RSC17, Theorem 1.8] are relevant in that case. Note that [RSC17, Theorem 1.8] can be used for the Lax-Friedrichs scheme (41) since we are in the case where α 1 = α 2 , β 1 = β 2 and µ 1 = µ 2 . The spectral curve F (S 1 ) (an ellipse) is illustrated in Figure 7 in the case λ = 1/2. We now turn to Assumption 3 and compute (see the general definition (9)):

A -1 (z) = - 1 + λ 2 , A 1 (z) = - 1 -λ 2 .
Figure 7: Left: Spectrum (blue curve) σ(L ) = F (S 1 ) for the Lax-Friedrichs scheme (41) with λ = 1/2. Right: The Green's function (marked points) at different time iterations for the Lax-Friedrichs scheme (41) compared with a fixed Gaussian profile centered at j = λn (solid lines). We started with an initial condition given by the Dirac mass δ.

Hence Assumption 3 is satisfied too. At last, Assumption 5 is satisfied since we have here K = 2, z 1 = 1 and z 2 = -1 = z 1 , which means that both sets I 1 and I 2 in (11) are singletons. Overall, the conclusion of Theorem 1 for the Lax-Friedrichs scheme in (41) is the uniform bound12 :

G n j ≤ C √ n exp -c (j -λ n) 2 n .
This behavior is illustrated in Figure 7 in the case λ = 1/2. For readers who are familiar with the theory of the transport equation (see [START_REF] Gustafsson | Time dependent problems and difference methods[END_REF] otherwise), the parameter λ stands for the ratio ∆t/∆x of the time and space steps. Hence the bound of Theorem 1 equivalently reads (with new constants that are still denoted C and c):

G n j ≤ C √ n exp -c (j ∆x -n ∆t) 2 ∆x (n ∆t) ,
which corresponds to the heat kernel at point j ∆x, time n ∆t with a diffusion coefficient proportional to ∆x.

Example 2: an implicit scheme

Our second example is based on the so-called method of lines for discretizing (40) (see [START_REF] Gustafsson | Time dependent problems and difference methods[END_REF][START_REF] Gustafsson | High order difference methods for time dependent PDE[END_REF] for a detailed exposition of the method and its outcome). Here we first apply the centered finite difference for the spatial derivative and we then apply the implicit Euler scheme for the time integration. As in the case of the Lax-Friedrichs scheme (41), we introduce a positive parameter λ > 0 (which plays the role of the ratio ∆t/∆x but its origin is meaningless here), and we use the operators:

Q 1 := I + λ 2 S -S -1 , Q 0 := I . (42)
In the notation of (2), this corresponds again to r = p = 1, but the scheme is now implicit because Q 1 is not the identity. We compute:

Q 1 e i ξ = 1 + i λ sin(ξ) = 0 , which means that Q 1 is an isomorphism on 2 (Z; C). The index condition (5) is also satisfied since the complex number Q 1 (κ) has positive real part for all κ ∈ S 1 so we can write:

Q 1 (κ) = exp q(κ) ,
thanks to the standard determination of the logarithm (which implies the validity of ( 5)). The operator L is given by:

L = 1 √ 1 + λ 2    ≥0 x S -+ ≥1 (-1) x S    ,
where x ∈ (0, 1) is given by:

x := √ 1 + λ 2 -1 λ .
We are thus dealing with a convolution operator with infinite support. We compute:

F e i ξ = 1 1 + i λ sin(ξ) ,
which means that F (κ) belongs to D for all κ ∈ S 1 and, again, F (κ) belongs to S 1 if and only if κ = ±1 (K = 2 in the notation of Assumption 2). Setting κ 1 = 1 and κ 2 = -1, we find that the relation (8) is satisfied with:

α 1 = λ , β 1 = β 2 = λ 2 2 , α 2 = -λ .
Assumption 2 is thus satisfied but we now have z 1 = z 2 = 1, and we immediately see that Assumption 5 is also satisfied: both sets I 1 and I 2 equal {1, 2} and α 1 α 2 = -λ 2 < 0. The spectral curve F (S 1 ) is illustrated in Figure 8 in the case λ = 1/2. As far as Assumption 3 is concerned, we compute:

A -1 (z) = - λ 2 z , A 1 (z) = λ 2 z ,
so Assumption 3 is satisfied again. We also note that Assumption 4 is satisfied since we have a -1,1 = -λ/2 and a 1,1 = λ/2. We can therefore apply Theorem 1 which, in the case of (42), yields the uniform Gaussian bound:

G n j ≤ C √ n exp -c (j + λ n) 2 n + exp -c (j -λ n) 2 n , | j | ≤ L n .
This behavior is illustrated in Figure 8 in the case λ = 1/2. Since we have an explicit formula for G 1 j , it is clear that the bound:

G 1 j ≤ C exp(-c | j |) ,
for large j's cannot be improved to some generalized Gaussian bound. This justifies why we need to distinguish the cases |j|/n 1 and |j|/n = O(1) in (13).

Figure 8: Left: Spectrum (blue curve) σ(L ) = F (S 1 ) for the implicit scheme (42) with λ = 1/2. Right:

The absolute value of the Green's function (marked points) at different time iterations for the implicit scheme (42) compared with two fixed Gaussian profiles centered at j = λ n and j = -λ n (solid lines). We started with an initial condition given by the Dirac mass δ.

Example 3: the O3 scheme

Next, as a third example, we consider the O3 scheme [START_REF] Després | Finite volume transport schemes[END_REF][START_REF] Després | Uniform asymptotic stability of Strang's explicit compact schemes for linear advection[END_REF] which is an explicit scheme of order 3 obtained as the convex combination of the Lax-Wendroff scheme [START_REF] Lax | On the stability of difference schemes[END_REF] and the Beam-Warming scheme [START_REF] Warming | Recent advances in the development of implicit schemes for the equations of fluid dynamics[END_REF]. The Lax-Wendroff scheme is an explicit finite approximation of (40) corresponding to the operators:

Q LW 1 := I, Q LW 0 := (1 -λ 2 )I + λ + λ 2 2 S -1 + -λ + λ 2 2 S ,
for some λ ∈ (0, 1). On the other hand, the Beam-Warming scheme is an explicit scheme given by

Q BW 1 := I, Q BW 0 := 1 - 3 2 λ + 1 2 λ 2 I + 2λ -λ 2 S -1 + -λ + λ 2 2 S -2 ,
for some λ ∈ (0, 1). In both cases, λ stands for the ratio ∆t/∆x. The O3 scheme is then defined as the following convex combination of the above two schemes

Q O3 1 := I, Q O3 0 := (1 -δ) Q LW 0 + δQ BW 0 , with δ = 1 + λ 3 . (43)
The expression for Q O3 0 can be simplified and we have

Q O3 0 = (2 -λ)(1 -λ 2 ) 2 I + λ(2 -λ)(1 + λ) 2 S -1 - λ(1 -λ 2 ) 6 S -2 + λ(2 -λ)(λ -1) 6 S.
In the notation of (2), this corresponds to r = 2 and p = 1. Once again, since we are dealing here with an explicit scheme, Assumptions 1 and 4 are trivially satisfied. The definition (6) gives in that case

F O3 e i ξ = (1 -δ) F LW e i ξ + δF BW e i ξ ,
Figure 9: Left: Spectrum (blue curve) σ(L ) = F (S 1 ) for the O3 scheme (43) with λ = 1/2. Right: The absolute value of the Green's function (marked points) at different time iterations for the O3 scheme (43) compared with a fixed generalized Gaussian profile H n j (44) centered at j = λ n (solide lines). We started with an initial condition given by the Dirac mass δ.

with

F LW e i ξ = 1 -λ 2 + λ 2 cos(ξ) -iλ sin(ξ), and

F BW e i ξ = 1 - 3 2 λ + 1 2 λ 2 + 2λ -λ 2 e -i ξ + -λ + λ 2 2 e -i 2 ξ .
And after some computations, we get

F LW e i ξ 2 = 1 -4λ 2 (1 -λ 2 ) sin 4 ξ 2 , F BW e i ξ 2 = 1 -4λ(2 -λ)(1 -λ) 2 sin 4 ξ 2 ,
from which we deduce by convexity that F O3 (κ) belongs to D for all κ ∈ S 1 . In fact, further computations lead to

F O3 e i ξ 2 = 1 - 4 9 λ(2 -λ)(1 -λ 2 ) sin 4 ξ 2 3 + 4λ(1 -λ) sin 2 ξ 2 ,
such that F O3 (κ) belongs to S 1 only when κ = 1, that is K = 1 in the notation of Assumption 3 with κ 1 = 1. We find that the relation ( 8) is satisfied with:

α 1 = λ , β 1 = λ(2 -λ)(1 -λ 2 ) 24 , µ 1 = 2,
and that β 1 > 0 for λ ∈ (0, 1). Furthermore, note that Assumption 5 is trivially satisfied since I 1 = {z 1 } with z 1 = 1. The spectral curve F O3 (S 1 ) is illustrated in Figure 9 in the case λ = 1/2. Next, we have that

A -2 (z) = λ(1 -λ 2 ) 6 , and
A 1 (z) = λ(2 -λ)(1 -λ) 6 ,
Figure 10: Illustration of the scaling factor in the generalized Gaussian bounds provided by Theorem 1 in the case of the O3 scheme (blue circles) and the ∞ scheme (orange squares). We plot log 10 sup j∈Z G n j as a function of log 10 (n) together with a best linear fit for each scheme for n ranging from 1 to 10 3 . For the O3 scheme we find a slope of -0.2496 while for the ∞ scheme we find a slope of -0.4983 which compare well with the theoretical -1/4 and -1/2 scaling factors.

such that Assumption 3 also holds true for each λ ∈ (0, 1). We can therefore apply Theorem 1 which, in the case of (43), yields the uniform generalized Gaussian bound:

G n j ≤ C n 1 4 exp -c |j -λ n| 4 3 n 1 3 , n ≥ 1 , j ∈ Z .
This behavior is illustrated in Figure 9 in the case λ = 1/2 where we compare the Green's function G n j to the generalized Gaussian profile H n j defined as

H n j := C n 1 4 exp -c |j -λ n| 4 3 n 1 3 , n ≥ 1 , j ∈ Z , (44)
with two fixed constants C > 0 and c > 0 independent of j and n which we have set to C = 0.8 and c = 1.1765 respectively. Note that the scaling factor n -1 4 in the generalized Gaussian bound is further demonstrated in Figure 10 where we represent sup j∈Z G n j in logarithmic scale. Using a best linear fit, we numerically obtain a slope of -0.2496 which is in good agreement with the theory.

Example 4: the ∞ scheme

We complete our series of examples with a last explicit scheme, which we shall call the ∞ scheme in reference to the spectral curve associated with it (see Figure 11 for an illustration). The ∞ scheme corresponds to the operators:

Q ∞ 1 := I, Q ∞ 0 := 1 16 S -
such that we have r = p = 3 in that case. The definition (6) reduces here to:

F ∞ e i ξ = 7 8 cos(ξ) + 1 8 cos(3 ξ) -i 2 sin(2 ξ) .

Computing:

F e i ξ 2 = 1 -sin 2 (ξ) 1 -sin 2 (2ξ) 16 , we find that F (κ) belongs to D for all κ ∈ S 1 , and F (κ) belongs to S 1 for such κ if and only if κ = ±1. We thus have (7) with κ 1 = 1 and κ 2 = -1, and the reader can check that (8) is satisfied with:

α 1 = 1 , α 2 = -1 , β 1 = β 2 := 1 2 , µ 1 = µ 2 = 1 ,
which means that Assumption 2 is satisfied. As a consequence, we also have z 1 = z 2 = 1, and we immediately see that Assumption 5 is also satisfied: both sets I 1 and I 2 equal {1, 2} and α 1 α 2 = -1 < 0. This is entirely similar to the behavior of the implicit scheme (42) except that (45) corresponds to a finitely supported convolution operator. The spectral curve F (S 1 ) is illustrated in Figure 11. Since the modulus of F (κ) attains its maximum at two points of S 1 and that α 1 = α 2 , we cannot apply the uniform Gaussian bound from [START_REF] Diaconis | Convolution powers of complex functions on Z[END_REF] or [START_REF] Randles | Convolution powers of complex functions on Z d[END_REF]. The improvement of Theorem 1 is thus meaningful here. As far as Assumption 3 is concerned, we compute:

A -3 (z) = - 1 16 , A 3 (z) = - 1 16 ,
so Assumption 3 is satisfied again. We also note that Assumptions 1 and 4 are satisfied since the ∞ scheme is explicit. We can therefore apply Theorem 1 which, in the case of (45), yields the uniform Gaussian bound:

G n j ≤ C √ n exp -c (j + n) 2 n + exp -c (j -n) 2 n n ≥ 1 j ∈ Z .
This behavior is illustrated in Figure 11. Note that the scaling factor n -1 2 in the Gaussian bound is further demonstrated in Figure 10 where we represent sup j∈Z G n j in logarithmic scale. Using a best linear fit, we numerically obtain a slope of -0.4983 which is in good agreement with the theory.

4.5 Extending Theorem 1 to the case where some α k can be zero

In the explicit case, it is possible to extend Theorem 1 to the case where some α k can be zero. The idea is to shift the operator L = Q 0 such that for some integer J ∈ Z the shifted operator L := S J Q 0 verifies the assumptions of Theorem 1 for which we obtain a generalized Gaussian estimate which is necessarily of the form

∀ n ∈ N * , ∀ j ∈ Z , L n δ j ≤ C K k=1 1 n 1/(2 µ k ) exp   -c |j -(α k + J) n| n 1/(2 µ k ) 2 µ k 2 µ k -1   .

But we observe that

∀ n ∈ N * , ∀ j ∈ Z , L n δ j = (L n δ) j-nJ , which gives a generalized Gaussian estimate for L n δ. More precisely, we introduce a relaxed version of Assumption 2 where some α k could vanish.

Figure 11: Left: Spectrum (blue curve) σ(L ) = F (S 1 ) for the ∞ scheme (45). Right: The absolute value of the Green's function (marked points) at different time iterations for the ∞ scheme (45) compared with two fixed Gaussian profiles centered at j = n and j = -n (solid lines). We started with an initial condition given by the Dirac mass δ.

Assumption 6. The function F defined in (6) satisfies Assumption 2 but possibly with some α k in (8) that can be zero.

We then have the following corollary.

Corollary 2. Assume that Q 1 is the identity, and that Q 0 satisfies Assumption 6. If there exists some J ∈ Z such that S J Q 0 satisfies Assumptions 2, 3 and 5, then there exist two constants C > 0 and c > 0 such that

∀ n ∈ N * , ∀ j ∈ Z , (L n δ) j ≤ C K k=1 1 n 1/(2 µ k ) exp   -c |j -α k n| n 1/(2 µ k ) 2 µ k 2 µ k -1   .
Typically, when Q 1 is the identity and K = 1 with α 1 = 0 in Assumption 2, we can always apply Corollary 2 by choosing J sufficiently large (in that case, we have r = 0 and p > 0 for the shifted operator S J Q 0 , A 0 (z) = z and A p (z) is a nonzero constant).

Further extensions

For the sake of clarity, we have focused here on the case of scalar iterations, but the techniques developed in this article should apply to some multistep iterations of the form

Q s+1 u n+s+1 = Q s u n+s + • • • + Q 0 u n , n ∈ N , u 0 , . . . , u s ∈ 2 (Z) ,
where s ∈ N is a given fixed integer and there are now s + 2 convolution operators with finite support involved. Of course, the statement of the assumptions should be suitably modified (for instance, Assumption 1 now bears on Q s+1 and not on Q 1 ).

We have focused here on numerical schemes for which the modulus of the amplification factor F is not constant on S 1 and such that the local behavior of F near a point where its modulus attains its maximum is dictated as in [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF]. We recall that for explicit operators (Q 1 = I) of the form (2), the main result in [START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF] shows that Assumption 2 is necessary and sufficient for Q 0 to be power bounded from 1 (Z; C) to 1 (Z; C) (or equivalently from ∞ (Z; C) to ∞ (Z; C)). A major advantage of the approach developed in this article is that it gets rid (more or less) of Fourier analysis. In particular, we have used the above strategy in [START_REF] Coulombel | Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions[END_REF] for proving a sharp stability result on a discretized transport equation with numerical boundary conditions under a degenerate version of the so-called Kreiss-Lopatinskii condition (see [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF][START_REF] Kreiss | On the stability definition of difference approximations for the initialboundary value problem[END_REF][START_REF] Gustafsson | Time dependent problems and difference methods[END_REF] for some background on numerical boundary conditions for hyperbolic equations). The problem considered in [START_REF] Coulombel | Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions[END_REF] is set in 2 (N; C) rather than 2 (Z; C), which makes many Fourier based techniques useless. Eventually, the above strategy is used in [START_REF] Coeuret | Local limit theorem for complex valued sequences[END_REF] to sharpen the local limit theorem of [START_REF] Randles | On the convolution powers of complex functions on Z[END_REF] and prove generalized Gaussian estimates for the remainder in the local limit theorem of [START_REF] Randles | On the convolution powers of complex functions on Z[END_REF].

Proof of Lemma 16. We first consider a point κ ∈ S 1 such that f (κ) has modulus 1. Writing:

f κ e ζ = f (κ) exp ( g(ζ) ) ,
for some holomorphic function g on a neighborhood of 0, we can conclude that there exists a power series a n x n with real coefficients and a positive radius of convergence, such that for any sufficiently small real number ξ, there holds:

f κ e i ξ = exp + ∞ n=0 a n ξ n .
Since f (κ) has modulus 1, we have a 0 = 0. Using now the condition (46), we can conclude that either all the coefficients a n , n ∈ N, are zero or there exists a smallest nonzero even integer 2 p such that:

a 2 p < 0 and a 0 = a 1 = • • • = a 2 p -1 = 0 .
In particular, for any ξ in some interval (-α, α) with α > 0, there holds:

f (κ e i ξ ) ≤ 1 - |a 2 p | 2 ξ 2 p .
If all the coefficients a n are zero, we can conclude that there exists an interval (-α, α) with α > 0, such that for any ξ ∈ (-α, α), the modulus of f (κ exp(i ξ)) equals 1. We have thus classified the two possible behaviors of the modulus of f near any point κ ∈ S 1 at which the modulus of f attains its maximum. where it attains its maximum shows that the set O is closed. Since S 1 is connected, O is either empty or equal to S 1 . Let us now prove the claim of Lemma 16. The case O = S 1 corresponds to the first possibility where f (κ) has modulus 1 for any κ ∈ S 1 . We thus assume that the modulus of f is non-constant on S 1 and therefore O is empty. Then any point κ admits an open neighborhood V in S 1 such that:

∀ κ ∈ V \ {κ} , | f (κ) | < 1 .
The conclusion follows from the compactness of S 1 .

A.2 The Bernstein type inequality

This subsection is devoted to the proof of Lemma 6. A proof of Lemma 6 is provided in [START_REF] Coulombel | Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions[END_REF] in the particular case where the coefficients a ,0 are real and K = 1, κ 1 = 1. We explain below why the result holds in the broader context of complex valued sequences and an arbitrary number K of tangency points. The result of Lemma 6 is a variation on the so-called Courant-Friedrichs-Lewy condition for numerical approximations of hyperbolic equations [START_REF] Courant | Über die partiellen Differenzengleichungen der mathematischen Physik[END_REF]. This condition, which bears on continuous and numerical domains of dependence, is known to be related to the von Neumann stability condition and the Bernstein inequality for trigonometric polynomials, see, for instance, [START_REF] Queffélec | On Bernstein's inequality for polynomials[END_REF] for a proof and historical comments on the Bernstein inequality. We refer to [START_REF] Strang | Trigonometric polynomials and difference methods of maximum accuracy[END_REF]page 152] for the link between the CFL condition and the Bernstein inequality.

,

  with b ∈ 1 (Z; C). One of the simplest such operators is the so-called shift operator S defined by: S : u j j∈Z -→ u j+1 j∈Z .The two definitions of operators L a and S b are closely related. Namely, given b ∈ 1 (Z; C), we have S b = L a where the sequence a ∈ 1 (Z; C) is defined by a := b -for all ∈ Z. Our convention, which is different from [DSC14, RSC15, RSC17], is the reason for the minus sign in (8) in the termi α k ξ (compare, for instance, with [DSC14, equation (3.3)]).

Figure 1 :

 1 Figure1: Geometrical illustrations of the set S given in Corollary 1 in the case K = 3. The curve of essential spectrum σ(L ) (dark red curve) is tangent to the unit circle precisely at z k for k = 1, 2, 3 (black crosses) and otherwise strictly contained in the unit disk D. For ∈ (0, ), each ball B (z k ) intersects σ(L ) at two points (black stars in the right panel) and the dashed dark blue line represents the circle passing through the point with largest modulus. We then fix δ > 0 such that the circle {z | |z| = e -δ } (magenta curve) is contained in between the unit circle and the circle passing through the point with largest modulus.

  , together with ∂Φ(Im(τ ), τ p ) ∂Im(τ ) ≤ C , for each |Im(τ )| ≤ and |τ p | ≤ .

Figure 3 :

 3 Figure 3: Illustration of the contour used in the case -η 2 ≤ ρ ζ γ ≤ 0 when n δ ≤ j ≤ n r. The contour is composed of Γ out -∪ Γ in -∪ Γ p ∪ Γ in + ∪ Γ out + .The contours Γ in ± and Γ out ± are the portions of the segment Re(τ ) = -η which lie either inside B (0) or outside B (0) with |Im(τ )| ≤ π while Γ p is defined in (35) and intersects the real axis at τ p .

Figure 4 :

 4 Figure 4: Illustration for the explicit (left) and implicit (right) cases of the different domains in the (j, n) plane where generalized Gaussian estimates are obtained; here K = 3 with α 1 < 0 and α 2 = α 3 > 0.Generalized Gaussian estimates are derived near each j ≈ n α k , k = {1, 2, 3} in the interior of the sectors delimited by the magenta lines. In the explicit case, below the lines j = -n p and j = n r (blue), the Green's function G n j vanishes.

Figure 5 :

 5 Figure 5: Left: typical contour used when (j, n) ∈ D 3 in the case depicted in Figure 4 where all α k are distinct α 1 < α 2 < α 3 (case A). Middle: typical contour used when (j, n) ∈ D 2 = D 3 in the case α 1 < α 2 = α 3 (case B). Right: typical contour used when (j, n) ∈ D 1 = D 2 = D 3 in the case α 1 = α 2 = α 3 (case B). Here K = 3.

Figure 6 :

 6 Figure 6: Illustration of the contour Γ k k used in the proof of Lemma 14 and Lemma 15.

  With this preliminary fact at our disposal, let us now consider the set:O := κ ∈ S 1 | ∃ α > 0 , ∀ ξ ∈ (-α, α) , f κ e i ξ =1 . The set O is clearly open and the previous argument on the local behavior of |f | near any point of S 1

It is actually a necessary and sufficient condition, see[START_REF] Thomée | Stability of difference schemes in the maximum-norm[END_REF].

When discretizing partial differential equations with real coefficients, these numbers are real.

For scalar problems, this is even equivalent to the so-called von Neumann stability condition[START_REF] Richtmyer | Difference methods for initial value problems[END_REF][START_REF] Gustafsson | Time dependent problems and difference methods[END_REF].

Note that I k always contains {k}.

We assume, of course, that Q0 and Q1 are irreducible.

Following the analysis of spectral projectors in[START_REF] Kato | Perturbation theory for linear operators[END_REF], we shall say that a vector space E(z) ⊂ C N that depends on a complex variable z for z in an open set O ⊂ C and that has constant dimension n, depends holomorphically on z if, for any z ∈ O, there exists a neighborhood V of z in O and a basis e1(z), . . . , en(z) of E(z) that depends holomorphically on z in V . This amounts to saying that the vector bundle defined by E over O is holomorphic. A typical example is the case E(z) = P (z) C N where P (z) is a projector on C N that depends holomorphically on z in an open set O.

Here we also use Assumption 3 to get a uniform local bound for Ap(z) -1 , including in the case z ∈ S 1 for which z can come inside the unit disk.

Here we use again that Ap(z) does not vanish in the ball B (z k ) up to restricting the radius .

The constant c here refers to the same one as in (25).

This is possible because the curves Γp are symmetric with respect to the real axis.

µ 2 µ-1 < 0 .

µ 2 µ-1

In the theory of finite difference schemes, it is referred to as the Courant-Friedrichs-Lewy parameter[START_REF] Courant | Über die partiellen Differenzengleichungen der mathematischen Physik[END_REF].

We do not use the absolute value here since all coefficients G n j are nonnegative.

Acknowledgements

We are indebted to the referee for an extremely careful reading of an earlier version of the manuscript, which has greatly helped us to improve the presentation. We also wish to thank Lucas Coeuret for his numerous comments and for suggesting the argument of Corollary 2 (which is also used in [Coeu21]).

Research of J.-F. C. was supported by ANR project Nabuco, ANR-17-CE40-0025. G.F. acknowledges support from the ANR via the project Indyana under grant agreement ANR-21-CE40-0008-01, from an ANITI (Artificial and Natural Intelligence Toulouse Institute) Research Chair and from Labex CIMI under grant agreement ANR-11-LABX-0040.

A Appendix. Proof of intermediate and related results

A.1 Behavior of the amplification factor on the unit circle

The aim of this subsection is to prove the following result, which generalizes the classification obtained by Thomée [Tho65, page 280] for trigonometric polynomials. Lemma 16 below shows that this classification only depends on the holomorphy of the considered function on a neighborhood of the unit circle.

Lemma 16. Let δ > 0 and let f be a holomorphic function on the annulus:

Then one of the following is satisfied:

• there exists a finite set of points {κ 1 , . . . , κ K }, K ≥ 1, in S 1 such that f (κ k ) has modulus 1 for any k ∈ {1, . . . , K} and:

Assumption 2 in our work thus excludes the case where the rational function F in (6) has modulus 1 on the whole unit circle S 1 (an example of such functions are the so-called Blaschke products [START_REF] Rudin | Real and complex analysis[END_REF]).

Proof of Lemma 6. Since Q 1 is the identity, we have F = Q 0 . Given an integer k ∈ {1, . . . , K}, we introduce a polynomial function P k defined by:

Assumption 2 implies that P k is a non-constant holomorphic function on C and, furthermore, the modulus of P k is not larger than 1 on S 1 . By the maximum principle for holomorphic functions [Rud87, chapter 12], P k maps D onto D. In particular, there holds:

and we easily obtain P k (1) = 1. We now use (8) and compute:

from which we get P k (1) = rα k and the asymptotic expansion:

as (the real number) tends to zero. Arguing by contradiction, this gives rα k ≥ 0 since P k maps D onto D. We now show that α k can not equal r and argue again by contradiction. Assuming α k = r, (47) reduces to:

and we thus obtain 13 , for real positive values of tending to zero:

This leads to a contradiction because exp(i exp(i π/(2 µ k ))) belongs to D for any > 0 and β k has positive real part.

In order to prove the inequality α k > -p, we introduce the complex reciprocal polynomial Q k of P k (see, again, [START_REF] Queffélec | On Bernstein's inequality for polynomials[END_REF]):

By the same argument as the one used for P k , we have Q k (1) = 1 and Q k maps D onto D. We also compute:

and we therefore have p + α k ≥ 0 because Q k maps D onto D. We now show that α k can not equalp and argue again by contradiction. Assuming α k = -p, we use (47) and compute:

= e i (p + r) exp(i π/(2 µ k )) exp i (rα k ) exp(i π/(2 µ k )) +

We are again led to a contradiction. The proof of Lemma 6 is thus complete. 13 The crucial fact here is that (47) holds for either real or complex values of ξ.