Marie-Laurence Mazure

Polynomial spaces revisited via weight functions

Keywords: Bernstein bases, Bernstein operators, Extended Chebyshev spaces, generalised derivatives, weight functions, blossoms, polynomial and rational spaces Primary 41A36; Secondary 41120, 41A50, 65D17

Extended Chebyshev spaces are natural generalisations of polynomial spaces due to the same upper bounds on the number of zeroes. In a natural approach, many results of the polynomial framework have been generalised to the larger Chebyshevian framework, concerning Approximation Theory as well as Geometric Design. In the present work, we go the reverse way: considering polynomial spaces as examples of Extended Chebyshev spaces, we apply to them results specifically developed in the Chebyshevian framework. On a closed bounded interval, each Extended Chebyshev space can be defined by means of sequences of generalised derivatives which play the same rôle as the ordinary derivatives for polynomials. We recently achieved an exhaustive description of the infinitely many such sequences. Surprisingly, this issue is closely related to the question of building positive linear operators of the Bernstein type. As Extended Chebyshev spaces, one can thus search for all generalised derivatives which can be associated with polynomials spaces on closed bounded intervals. Though this may a priori seem somewhat nonsensical due to the simplicity of the ordinary derivatives, this actually leads to new interesting results on polynomial and rational Bernstein operators and related results of convergence.

§1. Introduction

In the introduction I will denote a fixed given non-trivial real interval, and P n the restriction to I of the polynomial space of degree n. Moreover, for the sake of simplicity, all functions we consider will be assumed to be infinitely many times differentiable on I.

On the interval I,t h ec l a s sC of all Extended Chebyshev spaces of any dimension is the natural generalisation of the class P of all polynomial spaces P n , n ≥ 0, with which they share the same upper bounds on the number of zeroes of non-zero elements. In particular, though not as simple to handle, the class C can serve as a useful substitute for the class P to obtain unique solutions to Hermite interpolation problems. Indeed, it presents real advantages compared to P, due both to its great variety and to the shape parameter(s) each Extended Chebyshev space offers. In front of such a generalised situation, the logical approach for a mathematician consists in taking each single known result of the class P, and try to extend it to the larger class C. However, this turns out to be not so easy, for two underlying facts are crucial in many ways in the polynomial framework. Firstly, we have a nested sequence

P 0 ⊂ P 1 ⊂ P 2 ⊂ ••• ⊂ P n . (1.1)
Secondly, the class P is closed under the ordinary derivative D since

DP n := P n-1 ,n ≥ 1. (1.2)
To quote only a most classical example, given any x 0 ,...,x n ∈ I,t h en-th order divided difference based on x 0 ,...,x n of any function F on I is defined as the leading coefficient of the unique polynomial P n ∈ P n which interpolates F at x 0 ,...,x n , in the Hermite sense. The famous recurrence relations for divided differences readily follows from (1.1), along with the so important Newton expression of the interpolating polynomial P n . No similar possibility naturally appears in the class C, unless we more closely imitate the class P by restricting ourselves to nested sequences

E 0 ⊂ E 1 ⊂ E 2 ⊂ ••• ⊂ E n , (1.3)
where each E i is a (i + 1)-dimensional Extended Chebyshev space on I. Indeed, each such sequence (1.3) naturally gives sense to the concept of leading coefficient in each E i by selecting once and for all a function U i ∈ E i \ E i-1 for each i ≥ 0, with E i-1 := {0}. This in turn naturally leads to the concept of associated generalised divided differences, associated recurrence relations and Newton-type formula for the unique element F n ∈ E n which interpolates F at x 0 ,...,x n . Concerning this specific issue, definitions and proofs can easily be adapted from the polynomial framework to all Chebyshevian situations (1.3). This corresponds to the subclass C 0 ⊂C formed by the so-called Extended Complete Chebyshev spaces on I. What about the case where x 0 = ... = x n = ξ, corresponding to Taylor interpolation of order n of F at ξ? As is well known, the corresponding divided difference is equal to F (n) (ξ)/(n!). This obviously fails to be true in general in the class C 0 . What is thus the corresponding result? This is actually strongly connected with the question: what is the analogue of (1.2) within the class C? Well, we precisely face there a major difference between the polynomial and the Chebyshevian frameworks: unlike P,t h ec l a s sC is not closed under D. Applying D to an Extended Chebyshev space decreases the dimension only when it contains constants, but even so, there is absolutely no guarantee that the resulting space will be an Extended Chebyshev space on I.

To overcome this difficulty, one can try to go still closer to the class P by considering generalised monomials associated with any given sequence (w 0 ,...,w n ) of positive functions on I, defined on I by U 0 (t):=w 0 (t)a n d

U i (t):=w 0 (t) t a w 1 (ξ 1 ) ξ1 a w 2 (ξ 2 ) ... ξi-1 a w i (ξ i )dξ i ...dξ 1 ,i =1,...,n, (1.4)
where a is any point in I.E a c hs p a c eE i := span(U 0 ,...,U i ), 0 ≤ i ≤ n, can be proved to be an (i + 1)-dimensional Extended Chebyshev space on I. Accordingly, these spaces all belong to the class C 0 .N o w ,i fL 0 stands for the division by w 0 , the operator DL 0 transforms the corresponding nested sequence (1.3) into the shorter nested sequence DL 0 E i ,1≤ i ≤ n, built exactly the same way, but now from the generalised monomials associated with (w 1 ,...,w n ), instead of (w 0 ,w 1 ,...,w n ). Iterating the process, we obtain a sequence DL i , i =0 ,...,n, of differential operators which is to the nested sequence E i := span(U 0 ,...,U i ), i =0,...,n, the analogue of what the sequence D i , i ≤ n+1, is to the sequence (1.1). Note that the corresponding generalised divided difference based on x 0 ,...,x n when they all are equal to some ξ is equal to L n F (ξ). We thus have the answers to our questions within the subclass C 1 of C 0 composed of all Extended Chebyshev spaces spanned by generalised monomials. The good news is that both classes C 1 of C 0 coincide. Classically, all approximation properties of Extended Chebyshev spaces, e.g., Taylor expansions, generalisedconvexity, ..., were developed within the class C 0 , by means of generalised monomials [START_REF] Karlin | Tchebycheff Systems: with applications in analysis and statistics[END_REF][START_REF] Mühlbach | A recurrence formula for generalized divided differences and some applications[END_REF][START_REF] Schumaker | Spline Functions[END_REF][START_REF] Lyche | A recurrence relation for Chebyshevian B-splines[END_REF]. Now, what about our initial class C? Well, it does coincide too with the class C 0 , but only if the interval I is a closed and bounded interval [a, b]. Let us now work under this requirement, which is not a real limitation for most issues. Within C, we are thus now in the situation (1.3), which seems the exact counterpart of (1.1). However, in spite of the similarities, in practice a huge difference still exists between C and P. Life is quiet in the polynomial class P because, in the space P n ,t h e nested sequence (1.1) is well identified, just as the sequence of ordinary derivatives D i to decrease the dimension down to zero. These useful tools are so simple that nobody would reasonably think of searching for replacements. Life is much less quiet in the larger class C. Indeed, with a given (n + 1)-dimensional space E n of this class one can associate infinitely many nested sequences (1.3). In general none of them seems preferable to the others. The problem is that a result proved using the generalised derivatives associated with one of these nested sequences may not be intrinsic. This motivated the search of all possible such nested sequences, that is, all possible generalised derivatives which can be associated with E n . The result was achieved in [START_REF] Mazure | Finding all systems of weight functions associated with a given Extended Chebyshev space[END_REF]. Surprisingly, this issue is very closely connected to the construction of all linear operators of the Bernstein-type based on E n .T h i s is the reason why each E n of the class C, of dimension at least three, provides us with infinitely many (Chebyshevian) Bernstein operators. In general, none of them seems preferable to the others. This may be seen as an inconvenience compared to the one and only positive linear operator allocated to the space P n after S. Bernstein [START_REF] Bernstein | Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités[END_REF]. We do prefer to interpret this as a real advantage and a great richness of the Chebyshevian world C. R e t u r n i n gt ot h ep o l y n o m i a lc l a s sP, the object of the present article is to give it the benefit of the results achieved is the larger Chebyshevian class C via difficult tools and techniques which are a priori unjustified in P. It is worthwhile mentioning that it is not the first time we develop a similar approach. Indeed, the knowledge of all systems of generalised derivatives in each polynomial space P n already proved to be fruitful for CAGD purposes in [START_REF] Mazure | Polynomial splines as examples of Chebyshevian splines[END_REF]. We explained there how to use them to deduce necessary and sufficient conditions on the connection matrices for the associated space of geometrically continuous polynomial splines to be suitable for design. Here we present some implications concerning either polynomial or rational Bernstein operators in the Chebyshevian sense, with corresponding results of convergence. This is done is Sections 4 and 5, respectively. Beforehand, Section 2 gathers some technical preliminary results related to the convergence of the classical polynomial Bernstein operators which will be very useful to achieve convergence results in Sections 4 and 5. Section 3 briefly surveys the question of generalised derivatives and Bernstein operators based on Extended Chebyshev spaces. Section 6 presents a few final remarks for future related work. It is worthwhile mentioning that an interesting class of rational Bernstein operators was introduced by P. Pit ¸ul and P. Sablonnière in [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF]. They are actually Bernstein operators in the sense of Extended Chebyshev spaces. Some of our results are closely related to them and a detailed comparison is carried out in Section 5.

§2. A brief reminder about polynomials

In this section we fix some notations and we mainly remind readers with known results and related proofs, all connected with the famous Bernstein operators. They will strongly be involved in Sections 4a n d5 .

Throughout the rest of the article, for any nonnegative integer k, P k denotes the degree k polynomial space restricted to [0, 1], and (B k 0 ,...,B k k ) the polynomial Bernstein basis

B k i (t):= k i t i (1 -t) k-i ,t ∈ [0, 1],i =0,...,k. (2.1)
We also denote by B * k : C 0 ([0, 1]) → P k the classical Bernstein operator of degree k [START_REF] Bernstein | Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités[END_REF][START_REF] Devore | The Approximation of Continuous Functions by Positive Linear Operators[END_REF][START_REF] Lorentz | Bernstein Polynomials[END_REF][START_REF] Devore | Constructive Approximation[END_REF]. As is well-known, it is defined by

B * k F := k i=0 F (t * k,i )B k i for all F ∈ C 0 ([0, 1]), with t * k,i := i k ,i =0,...,k. (2.2)
G i v e na ni n t e g e rn ≥ 1, let X n denote the monomial function

X n (t): =t n ,f o rt ∈ [0, 1].
For any k ≥ n, we can consider the Bézier points of X n , in the sense of the coefficients of X n when expanded in terms of the Brenstein polynomial basis of degree k. Let us denote them by x n,k,i ,0≤ i ≤ k,s o that

X 1 = k i=0 x 1,k,i B k i = k i=0 t * k,i B k i for any k ≥ 1,X n = k i=0 x n,k,i B k i , for any k ≥ n ≥ 2.
Any polynomial Bernstein basis being normalised,thatis, k i=0 B k i =1 I, (where 1I denotes the constant function 1I(t) = 1 for all t ∈ [0, 1]), the left equality above is the reason why any Bernstein operator of degree k ≥ 1 reproduces P 1 , in the sense that B k F = F for all F ∈ P 1 .

We will see in Theorem 2.2 that Lemma 2.1 below permits an easy proof of the well-known convergence of the sequence B * k , k ≥ 1, via Korovkin's theorem. However this is not the reason why we need to cite it here. We actually need it for Proposition 2.3 which will be strongly involved in the proofs of the convergence results we will establish in Sections 4 and 5.

le:Xn Lemma 2.1. For any n ≥ 2, we have

lim k→+∞ max 0≤i≤k X n (t * k,i ) -x n,k,i =0. (2.3)
Proof. Let us select any k ≥ n. Considered as a polynomial of degree at most k, the monomial X n possesses a blossom in k variables [START_REF] Ramshaw | Blossoms are polar forms[END_REF]. Let us denote it as x n,k . It is the unique function of k variables which is symmetric, of degree at most one in each variable, and which gives X n on the diagonal. It is therefore given by

x n,k (t 1 ,...,t k )= 1 k n 1≤i1<i2<•••<in≤k t i1 t i2 ...t in .
With the notation x [j] for x repeated j times, we know that

x n,k,i := x n,k 0 [k-i] , 1 [i] , 0 ≤ i ≤ k.
Blossoms thus provide an easy way to obtain the known formula

x n,k,i = ( i n ) ( k n )
,t h a ti s ,

x n,k,i =0 for0≤ i ≤ n -1,x n,k,i = i(i -1) ...(i -n +1) k(k -1) ...(k -n +1) for n ≤ i ≤ k.
We thus have to check that

lim k→+∞ max n≤i≤k i k n - i(i -1) ...(i -n +1) k(k -1) ...(k -n +1) =0.
(2.4)

For n = 1 there is nothing to prove. Besides, for i ≥ n +1, we have

i k n+1 - i(i -1) ...(i -n) k(k -1) ...(k -n) ≤ i k i k n - i -1 k -1 n + i k i -1 k -1 n - (i -1) ...(i -n) (k -1) ...(k -n) .
Given that i k

n - i -1 k -1 n ≤ 2 n-1 (k -i)i k(k -1) ,
the proof of (2.4) readily follows by induction on n.

th:bernstein Theorem 2.2 ([2]). For any

V ∈ C 0 ([0, 1]), lim k→+∞ V -B * k V ∞ =0. Proof.
Since each positive linear operator B n reproduces 1I,X 1 , Korovkin's theorem [START_REF] Korovkin | Linear operators and approximation theory[END_REF][START_REF] Devore | The Approximation of Continuous Functions by Positive Linear Operators[END_REF][START_REF] Devore | Constructive Approximation[END_REF] says that it is sufficient to prove the claimed result for F = X 2 .N o w ,

B k X 2 -X 2 ∞ = k i=0 X 2 (t * k,i ) -x 2,k,i B k i ∞ ≤ max 0≤i≤k X 2 (t * k,i ) -x 2,k,i . Lemma 2.1 guarantees that lim k→+∞ B k X 2 -X 2 ∞ =0.
As mentioned earlier the following consequence of Lemma 2.1 will be essential in the subsequent sections.

B k X n ≥ L k (B k X n ) ≥ X n ≥ L k X n ,k ≥ n. It follows that X n -L k X n ∞ ≤ L k (B k X n ) -L k X n ∞ =m a x 0≤i≤k X n (t * k,i ) -x n,k,i ,k ≥ n. Lemma 2.1 proves that lim k→+∞ X n -L k X n ∞ =0.
§3. The Chebyshevian framework: generalised derivatives vs Bernstein operators

In the present section, we briefly present Extended Chebyshev spaces and the few main tools which this work relies on: Bernstein bases, Bernstein operators, weight functions and associated generalised derivatives, along with how they are connected together. For further acquaintance with this question readers are referred to [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF][START_REF] Mazure | Finding all systems of weight functions associated with a given Extended Chebyshev space[END_REF] and also [START_REF] Aldaz | Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces[END_REF], and more generally for further acquaintance with Extended Chebyshev spaces, to [START_REF] Karlin | Tchebycheff Systems: with applications in analysis and statistics[END_REF][START_REF] Schumaker | Spline Functions[END_REF][START_REF] Lyche | A recurrence relation for Chebyshevian B-splines[END_REF][START_REF] Pottmann | The geometry of Tchebycheffian splines[END_REF][START_REF] Mazure | Blossoms and optimal bases[END_REF][START_REF] Mazure | Chebyshev spaces and Bernstein bases[END_REF][START_REF] Mazure | Ready-to-blossom bases in Chebyshev spaces[END_REF].

Extended Chebyshev spaces

Throughout the present section I stands for any non-trivial real interval, and E ⊂ C n (I)isan( n +1)dimensional linear space. Such a space is said to be an Extended Chebyshev space on I (for short, EC-space on I) if any non-zero F ∈ E vanishes at most n times on I, multiplicities included up to (n + 1), or, equivalently, if E permits Hermite interpolation on I, that is, if any Hermite interpolation problem in (n +1) data in I has a unique solution in E.

Example 3.1. Let E be the null space of any linear differential operator of order (n +1 ) w i t h real constant coefficients. If the characteristic polynomial has only real roots, then E is an (n +1)dimensional EC-space on the whole of R. If some of the roots are not real, then E is also an EC-space, however only on sufficiently small intervals, and at least on any interval of length less that π divided by the maximum of the imaginary parts of all non-real roots. Any power functions with pairwise distinct exponents also span EC-spaces, this time on any interval not containing 0.

re:closure Remark 3.2. The class of all EC-spaces on a fixed interval I is closed under integration (Rolle's theorem), as well as under multiplication by sufficiently differentiable positive functions. As indicated in the introduction, it is important to observe that it is not closed under the ordinary differentiation D, and this is one of the difficulties encountered in the Chebyshevian world. This is made clear by the classical example of the space spanned by 1I, cos, sin: it is an EC-space on EC-spaces can be characterised by the existence of special bases resembling polynomial Bernstein bases, see Theorem 3.3 of [START_REF] Mazure | Chebyshev spaces and Bernstein bases[END_REF] or Theorem 12 of [START_REF] Mazure | Ready-to-blossom bases in Chebyshev spaces[END_REF].

th:BLB Theorem 3.3. The space E is an EC-space on I if and only if for any c, d ∈ I, c<d, E possesses a Bernstein-like basis relative to (c, d),t h a ti s ,ab a s i s(B 0 ,...B n ) such that for each i =0 ,...,n, B i vanishes exactly i times at c and exactly (n -i) times at d,a n di sp o s i t i v eo n]c, d[.

To complete Remark 3.2, consider again the EC-space E spanned by 1I, cos, sin, this time on [0,π]. The functions

B 0 (t):=1+cost, B 1 (t):=sint, B 2 (t):=1-cos t,
form a Bernstein-like basis of E relative to (0,π). All other Bernstein-like bases relative to (0,π)a r e of the form (α 0 B 0 ,α 1 B 1 ,α 2 B 2 )w i t ha n ypo s i t i v eα 0 ,α 1 ,α 2 . None of them is normalised. This is the reason why the space E cannot be used for CAGD purposes on [0,π]. Actually, as recalled below, this is connected to that fact that the space DE fails to be an EC-space on [0,π] (see Theorem 4.2 of [START_REF] Mazure | Chebyshev spaces and Bernstein bases[END_REF]).

th:BB Theorem 3.4. Assume that E is an (n +1)-dimensional EC-space on I. The following properties are equivalent:

(i) for any c, d ∈ I, c<d , E possesses a Bernstein basis relative to (c, d), that is, a Bernstein-like basis relative to (c, d) which is normalised;

(ii) the space E contains constants and the (n-dimensional) space DE is an EC-space on I.

Definition 3.5. An EC-space on I is said to be good for design when it satisfies (ii) of Theorem 3.4.

Behind this definition is hidden the fact that condition (ii) of Theorem 3.4 characterises the existence of blossoms in the space E. Blossoms are functions of n variables generalising polynomial blossoms. However, unlike them they are not defined by algebraic properties, but in a geometrical way by means of intersections of osculating flats [START_REF] Pottmann | The geometry of Tchebycheffian splines[END_REF][START_REF] Mazure | Ready-to-blossom bases in Chebyshev spaces[END_REF]. As soon as they exist, blossoms permit the development of the classical CAGD algorithms like polynomial blossoms, and they guarantee that the Bernstein basis (B 0 ,...,B n ) relative to any given c, d ∈ I, c<d , is totally positive on [c, d](i.e., for any

c ≤ x 0 <x 1 < ••• <x n ≤ d,
all minors of the matrix (B i (x j )) 0≤i,j≤n are non-negative). It is even the optimal normalised totally positive basis of E restricted to [c, d] (that is, the normalised B-basis in the sense of [START_REF] Carnicer | Total positivity and optimal bases[END_REF]), see [START_REF] Mazure | Blossoms and optimal bases[END_REF][START_REF] Mazure | Chebyshev spaces and Bernstein bases[END_REF][START_REF] Mazure | Ready-to-blossom bases in Chebyshev spaces[END_REF]. In order to facilitate the reading we will avoid saying more on blossoms, though they are strongly involved in many proofs.

Example 3.6. Following from Remark 3.2 we can say that the trigonometric space spanned by 1I, cos, s i ni sa nE C -s p a c eg oodf o rd e s i g no n l yo na n yi n t e r v a l[ a, b] such that 0 <b-a<π.

We now have to recall a classical procedure to build EC-spaces on I, see [START_REF] Schumaker | Spline Functions[END_REF]. This will complete and explain the presentation given in the introduction. Start with a system (w 0 ,...,w n ) of weight functions on I, i.e.,f o re a c hi, w i ∈ C n-i (I)a n di sp o s i t i v eo nI.T h e generalised derivatives associated with this system are obtained by alternating division by a weight function and ordinary differentiation as follows:

L 0 F := F w 0 ,L i F := DL i-1 F w i ,i=1,...,n. (3.1)
For each i ≤ n, L i is a linear differential operator of order i. Due to the class of all EC-spaces on I being closed under integration and multiplication by positive functions (see Remark 3.2), the set of all functions F ∈ C n (I)f o rw h i c hL n F is constant on I is an (n + 1)-dimensional EC-space on I,o f which a basis is formed by the generalised monomials (1.4). We denote it as E = EC(w 0 ,...,w n ).

As an instance, if I = R and w i := 1If o ri =0 ,...,n, the generalised derivatives are simply the ordinary derivatives and the space EC(w 0 ,...,w n ) is simply the space of all polynomials of degree at most n. From the recursive definition of generalised derivatives (3.1) one can derive that w 0 ∈ EC(w 0 ,...,w n )a n dt h a tD (EC(1I,w 1 ,...,w n )) = EC(w 1 ,...,w n ). Hence, any space EC-space of the form EC(1I,w 1 ,...,w n )i sg oodf o rd e s i g no n[ a, b]. Within the class of all EC-spaces on I which are defined by means of systems of weight functions, the generalised derivatives are appropriate tools to diminish the dimension. However, if the interval is not closed and bounded this class is strictly contained in the class of all EC-spaces on I. By contrast, both classes coincide if the interval is closed and bounded. This is the case surveyed in the subsection below.

Weight functions and Bernstein operators on closed bounded intervals

In this section we assume that the interval

I is a closed bounded interval [a, b].
It is known that each EC-space on [a, b] can be associated with systems of weight functions via the procedure described in the previous subsection. We are even able to build all such systems, see [START_REF] Mazure | Finding all systems of weight functions associated with a given Extended Chebyshev space[END_REF]. Surprisingly, the search for all such systems is somehow equivalent to the search for all Bernstein operators, as recalled in Theorem 3.10 below.

Unless we explicitly state it differently, we assume that

E ⊂ C n ([a, b]
) is an (n + 1)-dimensional EC-space good for design on [a, b], in which (B 0 ,...,B n ) stands for the Bernstein basis relative to (a, b). When no ambiguity is possible, we will omit "relative to (a, b)".

BF = n i=0 F (ζ i )B i ,F ∈ C 0 ([a, b]), (3.2)
is said to be a Bernstein operator based on E if there exists a two-dimensional EC-space U on [a, b] which is reproduced by B, in the sense that BF = F for all F ∈ U.

re:BO Remark 3.8. A Bernstein operator B based on E reproduces constants. Since it cannot reproduce three linear independent functions (see Proposition 3.7 of [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF]), the two-dimensional EC-space reproduced by B must contain constants. It is thus of the form E =s p a n ( 1 I,U)w h e r eU is a strictly increasing function. Determining all Bernstein operators based on E amounts to determining all non-constant functions they reproduce [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF].

Remark 3.9. A Bernstein operator based on E is a positive operator ( i.e., for any non-negative

F ∈ C 0 ([a, b]), the function BF is non-negative on [a, b]
). Furthermore, due to the Bernstein basis being totally positive on [a, b], any Bernstein operator based on E is also variation diminishing [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF]. See [START_REF] Goodman | Total positivity and the shape of curves, in: Total Positivity and its Applications[END_REF] for further acquaintance with variation diminishing issues.

The following result explains the links between weight functions and Bernstein operators. Its proof, based on Theorem 3.4, strongly involves blossoms and their properties (see Theorem 4.8 of [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF] and Theorems 3.2 and 5.1 of [START_REF] Mazure | Finding all systems of weight functions associated with a given Extended Chebyshev space[END_REF]). (v) U (a) <U(b) and there exists a nested sequence

E 1 := span(1I,U) ⊂ E 2 ⊂ ••• ⊂ E n-1 ⊂ E n := E, (3.3)
where, for each p =1,...,n, E p is a (p +1)-dimensional EC-space on [a, b];

(vi) the function w 1 is positive on [a, b] and there exists a system (w 2 ,...,w n ) of weight functions on [a, b] such that E = EC(1I,w 1 ,w 2 ,...,w n ).

Theorem 3.10 calls for some comments, listed below.

re:zeta Remark 3.11. The equivalence between the first two properties in Theorem 3.10 show that, when (i) is satisfied, the Bernstein operator based on E which reproduces U is given by (3.2) with

ζ i := U -1 (α i ), 0 ≤ i ≤ n. (3.4)
allBO Remark 3.12. Each strictly increasing sequence of Bézier points (α 0 ,...,α n ) generates a unique Bernstein operator based on E. Actually, this establishes a one-to-one correspondence between the set of all Bernstein operators based on E and the set of all equivalence classes of strictly increasing sequences of (n + 1) real numbers under the equivalence relation (α 0 ,...,α n ) ∼ (α 0 ,...,α n )i fa n d only if there exist α, β such that α i = αα i + β for i =0 ,...,n. Accordingly, as soon as n ≥ 2, the space E provides us with infinitely many Bernstein operators. In general none of them has a special meaning for the space E. We conclude this section with the subsequent observation: the three properties (iii) and (iv), and (vi) of Theorem 3.10 can be considered as properties of the EC-space DE. We can thus restate them as follows: (ii) w 0 is positive on [a, b] and the space L 0 E obtained by division by w 0 is good for design;

(iii) there exists a system (w 1 ,...,w n ) of weight functions on [a, b] such that E = EC(w 0 ,w 1 ,...,w n ). re:BLO Remark 3.17. When the (n + 1)-dimensional EC-space E is not assumed to be good for design on [a, b], instead of Bernstein operators one can introduce Bernstein-like operators based on E. Such an operator is associated with a unique two-dimensional EC-space EC(w 0 ,w 1 ) which it reproduces.

However there are specific difficulties due to the fact that Bernstein-like bases are unique only up to multiplication by positive numbers. Therefore, given any w 0 satisfying (i) of Theorem 3.15, it is more convenient to define the Bernstein-like operator B reproducing EC(w 0 ,w 1 )a s

L 0 BF := B (L 0 F ) ,F ∈ C 0 ([a, b]),
where B is the unique Bernstein operator based on L 0 E which reproduces EC(1I,w 1 ). We will not go further into this matter, preferring to refer the reader to [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF]. §4. View P n as an EC-space good for design on [0, 1]

In the present section as well as in the following one, we apply recently obtained results recalled in Section 3 to polynomial spaces on a closed bounded interval, say on [0, 1] without loss of generality. We are thus revisiting the space P n , viewing it as an instance of EC-space on [0, 1]. This revisit will prove to be fruitful since all results we will deduce from it concerning Bernstein operators, either polynomial (present section) or rational (next section) are new.

In the present section, we actually consider P n as an element of the class of all EC-spaces good for design on [0, 1], which corresponds to writing P n under the general form

P n = EC(1I,w 1 ,...,w n ), (4.1)
where (w 1 ,...,w n ) is an appropriate system of weight functions on [0, 1]. Any such equality should be understood as the replacement of the ordinary derivatives (corresponding to the equality P n = EC(1 I, 1I,...,1I

)) by generalised ones.

Polynomial Bernstein operators

As an EC-space good for design on [0, 1], we can apply Theorem 3.

U (t)= 3 i=0 u i B 3 i (t)=B 3 0 (t)+2B 3 1 (t)+4B 3 2 (t)+7B 3 3 (t)=3t 2 +3t +1,t ∈ [0, 1].
The sequence (u 0 ,u 1 ,u 2 ,u 3 )=( 1 , 2, 4, 7) of its Bézier points being strictly increasing, we know that there exists a unique polynomial Bernstein operator B which reproduces U . In order to compute

BF = 3 i=0 F (ζ i )B 3
i we have to solve the equations U (ζ i )=u i for i =0,...,3. This yields

BF := F (0)B 3 0 + F -3+ √ 21 6 B 3 1 + F -1+ √ 5 2 B 3 2 + F (1)B 3 3 ,F ∈ C 0 ([0, 1]).
Note that, via Theorem 3.10, this operator B corresponds to all possible equalities of the form

P 3 = EC(1I,
U (t):=2t 3 -3t 2 +3t -1= 3 i=0 α i B 3 i (t)=-B 3 0 (t)+B 3 3 (t),t ∈ [0, 1].
Since U ′ (t)=3 t 2 +(t -1) 2 , the function w 1 := U ′ is positive on [0, 1]. We can thus introduce the first order differential operator L 1 V := DV w1 . The three-dimensional space L 1 P 3 , obtained by division of P 2 by a positive C ∞ function, is an EC-space on [0, 1] (see Remark 3.2). However, it is not good for design on [0, 1], i.e., the two-dimensional space DL 1 P 3 is not an EC-space on [0, 1]. In other words, L 1 is not a generalised derivative in P 3 , which means that we cannot find any system (w 2 ,w 3 )o f weight functions on [0, 1] permitting to write P 3 = EC(1I,w 1 ,w 2 ,w 3 ). This is due to the sequence (α 0 ,α 1 ,α 2 ,α 3 )=( -1, 0, 0, 1) not being strictly increasing. On the other hand, U being strictly increasing, clearly we cannot expect to find 0 <ζ 1 <ζ 2 < 1 such that U (ζ i )=α i =0f o ri =1 , 2. Hence, the function U cannot be reproduced by a Bernstein operator based on P 3 . Nevertheless, out of dimension elevation we can write U as

U = -B 4 0 - 1 4 B 4 1 + 1 4 B 4 3 + B 4 4 .
The Bézier points of U viewed as an element of P 4 thus form a strictly increasing sequence. Therefore, U is reproduced by a Bernstein operator based on P 4 . Equivalently, we can say that there exists systems (w 2 ,w 3 ,w 4 ) of weight functions on [0, 1] such that P 4 = EC(1I,w 1 ,w 2 ,w 3 ,w 4 ). In other words, the four-dimensional EC-space L 1 P 4 is good for design on [0, 1], that is, DL 1 P 4 is a three-dimensional EC-space on [0, 1]. Accordingly, L 1 is indeed a generalised derivative in P 4 . From Theorem 3.10 we can also say that it is possible to find infinitely many EC-spaces E 2 , E 3 , of dimension 3, 4, respectively, such that span(1I,U)

⊂ E 2 ⊂ E 3 ⊂ P 4 ,
but none of them is included in P 3 .

Convergence of sequences of polynomial Bernstein operators

According to Remark 3.14, a two-dimensional EC-space E 1 which is reproduced by a polynomial Bernstein operator of degree n will automatically be reproduced too by a polynomial Bernstein operator of any degree k ≥ n. This observation gives sense to considering sequences of polynomial Bernstein operators all reproducing the same two-dimensional EC-space. This is what we do in Theorem 4.5 below which proves that convergence in the sense of Theorem 2.2 is not specific to the classical Bernstein operators (2.2).

th:CONV Theorem 4.5. Any infinite sequence of polynomial Bernstein operators of increasing degrees which all reproduce the same two-dimensional EC-space on [0, 1] permits uniform approximation of any continuous function on [0, 1].

Proof. We start with a function U ∈ P n for some n ≥ 1, assumed to have strictly increasing Bézier points u n,0 ,...,u n,n . We know the existence of a nested sequence

E 1 := span(1I,U) ⊂ E 2 ⊂ ••• ⊂ E n := P n ⊂ P n+1 ⊂ ••• ,
As observed in Remark 3.14, for any k ≥ n, U is reproduced by a polynomial Bernstein operator B k of degree k. We want to prove that the sequence B k , k ≤ n,s a t i s fi e s lim k→+∞

F -B k F ∞ =0 foran yF ∈ C 0 ([0, 1]). (4.
2)

The case n = 1 gives the sequence of classical polynomial Bernstein operators B * k , k ≥ 1. We can therefore assume that n ≥ 2. Since each B k , k ≥ n, reproduces the two functions 1I,U, Korovkin's theorem ensures that it is sufficient to prove (4.2) for any V ∈ E 2 . Actually, we will more generally select any V ∈ P n and prove (4.2) for this V .

Let us expand U and V in the Bernstein bases as

U = k i=0 u k,i B k i ,V = k i=0 v k,i B k i ,k ≥ n.
For each k ≥ n, the polynomial Bernstein operator B k is defined on C 0 ([0, 1]) by

B k F = k i=0 F (t k,i )B k i , with U (t k,i ):=u k,i for i =0,...,k.
We thus obtain, for any k ≥ n,

B k V -V ∞ = k i=0 [V (t k,i ) -v k,i ] B k i ∞ ≤ max 0≤i≤k |V (t k,i ) -v k,i | .
In order to make sure that lim k→+∞ V -

B k V ∞ = 0, it is sufficient to prove that lim k→+∞ max 0≤i≤k |V (t k,i ) -v k,i | =0. (4.3)
Now, from Proposition 2.3 we know that

lim k→+∞ max 0≤i≤k U (t * k,i ) -u k,i = 0 = lim k→+∞ max 0≤i≤k V (t * k,i ) -v k,i . (4.4)
The leftmost equality can be replaced by The function U being strictly increasing on [0, 1], we can consider its inverse U -1 . Using the uniform continuity of V • U -1 on the interval [U (0),U(1)], (4.5) implies that

lim k→+∞ max 0≤i≤k U (t * k,i ) -U (t k,i ) =0. ( 4
lim k→+∞ max 0≤i≤k V • U -1 U (t * k,i ) -V • U -1 U (t k,i ) = lim k→+∞ max 0≤i≤k V (t * k,i ) -V (t k,i ) =0.
To obtain (4.3) we just have to additionally use the rightmost part of (4.4) after observing that

|V (t k,i ) -v k,i |≤ V (t k,i ) -V (t * k,i ) + V (t * k,i ) -v k,i .
Remark 4.6. The latter result suggests that it might be interesting to do polynomial approximation of continuous functions with shape parameters. The shape parameters would be provided by the strictly increasing sequence (u n,0 ,...,u n,n ) we start with. Our purpose is not to thoroughly investigate this question, but rather to briefly illustrate this with an example. Consider the function

F (t):=sin πt - π 2 ,t ∈ [0, 1]. (4.6)
For any n ≥ 3, let B n be the Bernstein operator based on P n which reproduces the function U α depending on the parameter α ∈]0, 1[

U α (t):=-B 3 0 (t) -αB 3 1 (t)+αB 3 2 (t)+B 3 3 (t)=(2t -1) 1+(3α -1)t(1 -t) . (4.7)
Observe that, for α = 1 3 , U α (t)=2 t -1. Accordingly, for this specific value of the parameter α,t h e operator B n is the classical Bernstein operator of degree n,t h ati sB * n . Figure 1 presents the graphs of the function B 3 F for various values of the parameter α, by comparison with the graph of the function F itself. Visually speaking, B 3 F is all the closer to F as α is closer to 1. In particular, the function B 3 F obtained for α = 5 6 seems a much better approximant than B * 3 F . §5. Simply view P n as an EC-space on [0, 1]

In the present section, we forget about the fact that the EC-space P n is good for design, that is, about the fact that DP n = P n-1 itself is an EC-space on [0, 1]. In terms of weight functions, this means that we no longer require that the first weight function be equal to 1I. We are now rather interested in all polynomials Ω ∈ P n ensuring equalities of the form P n = EC(Ω,w 1 ,...,w n ).

(5.1)

Rational spaces and weight functions

In order to fix the notations which will be used subsequently it is worthwhile rewriting Theorem 3.15 as follows:

th:Omega Theorem 5.1. Given Ω:= n i=0 ω i B n i ∈ P n , the following three properties are equivalent:

(i) ω i > 0 for i =0,...,n;

(ii) Ω is positive on [a, b] and the space P n obtained from P n by division of all its elements by Ω is an EC-space good for design on [0, 1];

(iii) there exist systems (w 1 ,...,w n ) of weight functions on [0, 1] such that (5.1) holds.

From now on, the function Ω is fixed, and we assume that all ω's are positive. Then, division of both sides of the equality Ω := n i=0 ω i B n i by Ω shows that the sequence

B i := ω i B n i Ω ,i =0,...,n,
is normalised and presents the same property of zeroes as the classical Bernstein basis (B n 0 ,...,B n n ). Accordingly, it is the Bernstein basis relative to (0, 1) in P n .T h es p a c e P n is thus composed of all functions

F (t)= n i=0 α i ω i B n i (t) Ω(t) = n i=0 α i ω i B n i (t) n i=0 ω i B n i (t) ,t ∈ [0, 1]. (5.2)
In other words, (5.2) proves that P n is the rational space of degree n based on the positive weights ω 0 ,...,ω n . To avoid confusion with weight functions, we will rather say that it is based on Ω. For a classical approach of rational spaces, see for instance [START_REF] Farin | NURBS: from Projective Geometry to Practical Use[END_REF][START_REF] Fiorot | Rational curves and surfaces: applications to CAD[END_REF][START_REF] Piegl | The Nurbs Book[END_REF]. On account of (i) ⇒ (ii), Theorem 5.1 ensures that the rational space P n is an EC-space good for design on [0, 1]. The interesting part of this statement is that Theorem 5.2. ( [START_REF] Mazure | Extended Chebyshev spaces in Rationality[END_REF]) The first derivative D P n of a rational space P n is an EC-space on [0, 1]. re:rational Remark 5.3. We thus observe that building all rational spaces of degree n can exactly be viewed as searching for all weight functions Ω permitting an equality of the form (5.1). It also amounts to finding all first order differential operators DL 0 ensuring the same property as the ordinary derivative D when applied to P n , i.e., decrease the dimension by one within the class of all EC-spaces on [0, 1].

Convergence of rational Bernstein operators

In the previous subsection we have seen that the space P n is an EC-space good for design on [0, 1]. As so, it provides us with Bernstein operators based on it, and we now focus on these operators. From Theorem 3.10 we know that they correspond to all weight functions w 1 permitting to write the rational space P n as P n = EC(1I,w 1 ,w 2 ,...,w n ), that is, the polynomial space P n itself as P n = EC(Ω,w 1 ,w 2 ,...,w n ). From Remark 3.17 we also know that they correspond to all Bernsteinlike operators based on P n which reproduce a two-dimensional space of the form EC(Ω,w 1 ). We will more precisely be interested in convergence. This is why we will now write Ω as

Ω= k i=0 ω k,i B k i for all k ≥ n,
so that in particular ω n,i = ω i for i =0,...,n. Applying the degree elevation process to Ω shows that, for each k ≥ n,an deac hi =1,...,k, ω k+1,i is a strictly convex combination of ω k,i-1 , ω k,i .M o r e o v e r ω k+1,0 = ω k,0 and ω k+1,k+1 = ω k,k . By induction we can therefore state that, for each k ≥ n,t h e Bézier points of Ω viewed as an element of P k can be expressed as convex combinations of the initial positive Bézier points ω 0 ,...,ω n . Hence they all are positive. Accordingly, the nested sequence

P n = 1 Ω P n ⊂ P n+1 := 1 Ω P n+1 ⊂ •••⊂ P k := 1 Ω P k ⊂ P k+1 := 1 Ω P k+1 ⊂ ••• (5.3)
is a nested sequence of rational spaces, all based on the same function Ω. According to Remark 3.14, a two-dimensional EC-space which is reproduced by a rational Bernstein based on P n will automatically be reproduced too by a rational Bernstein operator based on P k for all k ≥ n. This situation (5.3) is the most natural one to build infinite sequences of rational Bernstein operators, as will be done in the theorem below.

CONVRat Theorem 5.4. Consider an infinite sequence B k , k ≥ n, of Bernstein operators respectively based on the rational spaces P k , k ≥ n, introduced in (5.3), assumed to all reproduce the same two-dimensional EC-space on [0, 1]. Then, for each

F ∈ C 0 ([0, 1]), the sequence B k F , k ≥ n, converges to F uniformly on [0, 1].
Proof. Let E 1 be the two-dimensional EC-space on [0, 1] which is reproduced by each operator B k , k ≥ n. For any k ≥ n,l e t( B k,0 ,..., B k,k ) denote the Bernstein basis of the rational space P k .I n E 1 we select a strictly increasing function U , expanded as

U = k i=0 u k,i B k,i ,k ≥ n.
For each k ≥ n,t h eB ézier points u k,0 ,..., u k,k of U form a strictly increasing sequence, and the Bernstein operator B k based on the rational space P k which reproduces E 1 is given by

B k F := k i=0 F ( t k,i ) B k,i , with U ( t k,i )= u k,i for i =0,...,k.
All positive operators B k , k ≥ n, reproduce 1Ia n d U . According to Korovkin's theorem, in order to prove the claimed result, it suffices to show that lim k→+∞

B k V -V ∞ C 0 ([0, 1]
) chosen so that the functions 1I, U,a n dV span a three-dimensional EC-space on [0, 1] (see, for instance, [START_REF] Devore | Constructive Approximation[END_REF], Theorem 4.2). Without loss of generality, we can assume that n ≥ 2 and prove the desired property for any P ∈ P n . Select any such P in the rational space P n , expanded as

P = P Ω = k i=0 p k,i B k,i = k i=0 p k,i ω k,i B k,i , with P = k i=0 p k,i B k,i ,k ≥ n. (5.4)
Clearly, it is sufficient to prove that

lim k→+∞ max 0≤i≤k P ( t k,i ) -p k,i =0. ( 5.5)
Now, applying Proposition 2.3 both to P and Ω, we can assert that

lim k→+∞ max 0≤i≤k P (t * k,i ) -p k,i = 0 = lim k→+∞ max 0≤i≤k Ω(t * k,i ) -ω k,i . (5.6)
Using (5.4) we can write

P (t * k,i ) -p k,i = P (t * k,i ) Ω(t * k,i ) - p k,i ω k,i = ω k,i P (t * k,i ) -p k,i + p k,i ω k,i -Ω(t * k,i ) ω k,i Ω(t * k,i ) . (5.7)
As already observed, for each k ≥ n,a n df o re a c hi =0 ,...,k, ω k,i can be calculated as a convex combination of ω n,0 ,...,ω n,n and each p k,i can be calculated via the same convex combination of p n,0 ,...,p n,n . It follows that, for each k ≥ n and each i =0,...,k,w eh a v e

0 <m:= min(ω n,0 ,...,ω n,n ) ≤ ω k,i ≤ M := max(ω n,0 ,...,ω n,n ), |p k,i |≤Q := max(|p n,0 |,...,|p n,n |). (5.8)
On the other hand, for any t 1 ,...,t n ∈ [0, 1], the value ω n (t 1 ,...,t n ) of the blossom ω n of Ω ∈ P n can be calculated as a convex combination of the Bézier points ω n,0 ,...,ω n,n of Ω, via an n-step generalised version of the de Casteljau evaluation algorithm. This holds in particular for each Ω(t)=ω n (t [n] ), t ∈ [0, 1]. In other words we can also state that 0 <m≤ Ω(t) ≤ M for all t ∈ [0, 1].

(5.9)

Taking (5.8) and (5.9) into account, (5.7) leads to

P (t * k,i ) -p k,i ≤ M max 0≤j≤k P (t * k,j ) -p k,j + Q max 0≤i≤k Ω(t * k,i ) -ω k,i m 2 ,
thus proving that lim

k→+∞ max 0≤i≤k P (t * k,i ) -p k,i =0.
(5.10)

In particular, for P = U, this yields .11) As in the proof of Theorem 4.5, uniform continuity arguments eventually lead to the fact that

lim k→+∞ max 0≤i≤k U (t * k,i ) -U ( t k,i ) =0. ( 5
lim k→+∞ max 0≤i≤k P ( t k,i ) -P (t * k,i ) =0.
This proves (5.5) via an appropriate triangular inequality and (5.10).

sion elevation Remark 5.5. Generalising Remark 2.4, observe that the convergence of the so-called dimension elevation algorithm generated by the infinite sequence (5.3) can efficiently be obtained as a consequence of (5.5). Subsequently, we use exactly the same notations as in Theorem 5.4 and in its proof. Given any P ∈ P j for any given j ≥ n, consider the sequence L k P , k ≥ j, defined by

L k P ( t k,i )= p k,i for i =0,...,k, L k P is affine on t k,i , t k,i+1 for i =0,...,k-1,
where, for each k ≥ j, p k,i , i =0 ,...,k,a r et h eB ézier points of P in P k .T h es e to fa l l t k,i , k ≥ n, 0 ≤ i ≤ k, being dense in [0, 1] (see (5.11)), convergence of the dimension elevation algorithm can be viewed as the uniform convergence of the sequence L k P , k ≥ n,t o P on [0, 1], for any such P . Select a nested sequence E 2 ,..., E n-1 of EC-spaces on [0, 1] of increasing dimension, so that

E 1 ⊂ E 2 ⊂ ••• ⊂ E n-1 ⊂ P n .
This is made possible by Theorem 3.10. Setting E k := P k for k ≥ n, for any j ≥ 2, we can then choose a function U j ∈ E j \ E j-1 so that U j will be U -convex, that is, so that the function U j • U -1 will be convex on [ U (0), U(1)]. Out of linearity, it suffices to prove uniform convergence for each U j , j ≥ 2. Taking the U -convexity into account we have (see [START_REF] Mazure | Bernstein-type operators in Chebyshev spaces[END_REF])

B k U j ≥ L k B k U j ≥ U j ≥ L k U j ,k ≥ j.
Similar arguments as those in Remark 2.4 enable us to state that lim k→+∞ U j -L k U j ∞ =0.

An example

Any (n + 1)-dimensional EC-space E a s s u m e dt ob eg o o df o rd e s i g no n[ 0 , 1] permits to obtain any straight line segments in R d , d ≥ 2, as examples of parametric curves: it suffices to choose any Bézier points in a monotonic way on the segment in question. However such segments cannot be described as graphs of a function with values in R d-1 , unless P 1 ⊂ E. This property is generally referred to as linear precision. The question of linear precision for rational spaces was solved in [START_REF] Farin | Linear precision of rational Bézier curves[END_REF]; also see [START_REF] Fiorot | Linear precision of BR-curves[END_REF]. Linear precision can be useful in many ways, and this is why we devote this section to it. For instance, recently rational spaces with linear precision were successfully used to build rational shape-preserving Hermite interpolants in [START_REF] Merrien | Rational splines for Hermite interpolation with shape constraints[END_REF].

As a preliminary result we need to recall the corresponding obvious observation in order to make forthcoming formulae natural. Since P n contains constants, it satisfies linear precision if and only if the identity belongs to P n . The very definition of the space P n makes the following statement obvious.

Lemma 5.6. The rational space P n satisfies linear precision ( i.e., P 1 ⊂ P n )i fa n do n l yi fΩ is of degree at most (n -1).

Let us come back to the infinite nested sequence of rational spaces introduced in (5.3). Clearly, if P n satisfies linear precision, so does any P k , k ≥ n. Nevertheless, it should be observed that, in case P n fails to satisfy linear precision, i.e., in case Ω is of exact degree n, then each further space P k , k ≥ n + 1, does satisfy linear precision all the same. This is why, for the rest of this section, without loss of generality we do assume that Ω is of degree at most (n -1). Equivalently, we assume the existence of real numbers ω * 0 ,...,ω * n-1 (the Bézier points of Ω as an element of P n-1 ) leading to the positive ω 0 ,...,ω n via degree elevation from P n-1 to P n , i.e., ensuring that

ω 0 := ω * 0 ,ω i := i n ω * i-1 + n -i n ω * i for 1 ≤ i ≤ n -1,ω n := ω * n-1 .
(5.12)

Remark 5.7. Note that, in order to ensure the positivity of all ω 0 ,...,ω n , it is sufficient to require that ω * 0 ,...,ω * n-1 be positive. However, this is not necessary. For instance, for n =3 ,t h ec h o i c e

ω * 0 = ω * 2 =2 ,ω * 1 = -1 2 , yields ω 2 = ω 3 = 1 3
. Accordingly, though the space P n-1 := 1 Ω P n-1 is well defined due to Ω being positive on [0, 1], this space is not necessarily a rational space of degree (n -1).

The problem we want to tackle here is the following one: how should we select ω * 0 ,...,ω * n-1 so that the nested sequence (5.3) provides us with a corresponding sequence B k , k ≥ n, of Bernstein operators all reproducing the space P 1 ? From Remark 3.14 we know that it is sufficient to ensure reproduction of P 1 by a Bernstein operator based on P n . From Theorem 3.10 we know that this holds if and only if the Bézier points of the identity X 1 as an element of P n form a strictly increasing sequence. To solve this question, we start with the following simple lemma on polynomial blossoms. where, for i =1, 2, p i denotes the blossom in n i variables of P i ∈ P ni , and where the sum in (5.13) is taken over all sequences of indices

1 ≤ i 1 < ••• <i n1 ≤ n, 1 ≤ j 1 < ••• <j n2 ≤ n with i r = j s for 1 ≤ r ≤ n 1 , 1 ≤ s ≤ n 2 .
Proof. Due to the properties of the blossoms p i , i =1, 2, the function appearing in the right-hand side of (5.13) is clearly symmetric and affine in each variable t 1 ,...,t n , and clearly too, on the diagonal of [0, 1] n we have p(t

[n] )=p 1 (t [n1] )p 2 (t [n2] )=P 1 (t)P 2 (t)=P (t).
Let ω * be the blossom of Ω ∈ P n-1 , that is, its blossom in (n -1) variables

p(a [n-i] ,b [i] )= i n bω * (a [n-i] ,b [i-1] )+ n -i n aω * (a [n-1-i] ,b [i] ), 1 ≤ i ≤ n -1.
Let us apply the latter formula with a =0,b =1. TheB ézier points p 0 ,...,p n of P are thus given by

p 0 =0,p i = p(0 [n-i] , 1 [i] )= i n ω * i-1 for i =1,...,n. (5.14)
Accordingly, as an element of the rational space P n ,t h eB ézier points of the identity X 1 = P Ω are given by (see (5.12))

p 0 ω 0 =0, p i ω i = iω * i-1 iω * i-1 +(n -i)ω * i , 1 ≤ i ≤ n. (5.15)
The positivity of ω * 0 and ω * n-1 enables us to conclude that we always have

0= p 0 ω 0 < p 1 ω 1 , p n-1 ω n-1 < p n ω n .
We thus have to find conditions on ω * 0 ,...,ω * n-1 ensuring that iω

* i-1 iω * i-1 +(n -i)ω * i < (i +1)ω * i (i +1)ω * i +(n -i -1)ω * i+1 , 1 ≤ i ≤ n -2.
(5.16)

Assume that (5.16) is satisfied. Both denominators in (5.16) being positive -see (5.12)-, we can see that as soon as ω * i-1 > 0 for some integer i,1≤ i ≤ n-1, then ω * i is positive in turn. We already know that ω * 0 ,ω * n-1 are positive. Hence, we now know that ω * 0 ,...,ω * n-1 are all positive. Moreover, again due to the positivity of the denominators, condition (5.16) can equivalently be written as follows:

i(n -i -1)ω * i-1 ω * i+1 < (i +1)(n -i)ω * i 2 for i =1,...,n -2.
(5.17)

On the other hand, we know that the positivity of ω * 0 ,...,ω * n-1 guarantees that of ω 0 ,...,ω n obtained via (5.12). Accordingly, we can summarise all previous discussion as follows.

th:reproP1 Theorem 5.9. Given Ω= n i=0 ω i B n i ∈ P n , the following properties are equivalent: (i) the function Ω is positive on [0, 1] and (5.3) is a sequence of rational spaces which provides us with a sequence B k , k ≥ n, of rational Bernstein operators reproducing P 1 ;

(ii) ω 0 ,...,ω n are given by (5.12), where ω * 0 ,...,ω * n-1 are any positive numbers satisfying

i(n -i -1)ω * i-1 ω * i+1 < (i +1)(n -i)ω * i 2 for i =1,...,n-2.
One interesting advantage of reproduction of the identity is that it directly yields explicit expressions of the Bernstein operators. We can indeed complete Theorem 5.9 by the following one, which follows from (5.15) and (5.12).

:reproP1bis Theorem 5.10. Assume that Ω= n i=0 ω i B n i ∈ P n , and that (ii) of Theorem 5.9 holds. Then, for each k ≥ n the Bernstein operator B k based on the rational space P k which reproduces the identity is given by

B k F = k i=0 F iω k-1,i-1 kω k,i B k,i ,F ∈ C 0 ([0, 1]), (5.18)
where, for each k ≥ n -1, ω k,0 ,...,ω k,k are the (positive) Bézier points of Ω as an element of P k .

Remark 5.11. As observed in Remark 5.7, when P 1 ⊂ P n the space P n-1 := 1 Ω P n-1 may fail to be a rational space of degree (n -1). However, as soon as P 1 is reproduced by a Bernstein operator based on P n ,a l lω * 0 ,...,ω * n-1 being positive (Theorem 5.9), P n-1 is indeed a rational space. Remark 5.12. All suitable ω * 0 ,...,ω * n-1 ensuring condition (ii) of Theorem 5.9 can, for instance, be obtained by first choosing any positive ω * 0 , ω * 1 , and then successively selecting ω * 2 ,...,ω * n-1 such that

0 <ω * i < i(n -i +1) (i -1)(n -i) ω * i-1 2 ω * i-2
for i =2,...,n-1.

Comments on rational Bernstein operators

We would like to pay tribute to P. Pit ¸ul and P. Sablonnière for their interesting work [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF] on a class of rational Bernstein operators, to which we will compare the previous subsection. As a special case of Theorem 5.4, we can say that the sequence of rational Bernstein operators defined by (5.18)

satisfies lim k→+∞ B k F -F ∞ =0forallF ∈ C 0 ([0, 1]).
(

We would like to mention that this sequence is one instance of the sequences of rational positive operators studied in [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF] (see also [START_REF] Sablonnière | Rational Bernstein and spline approximation. A new approach[END_REF]). For each integer k, the authors of [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF] considered a positive operator R k

R k F = k i=0 w k,i F (t k,i )B k i Ω k ,F ∈ C 0 ([0, 1]), with Ω k := k-1 i=0 w k,i B k-1 i , (5.20)
the w k,i ,0≤ i ≤ k, being any positive numbers chosen so as to satisfy

w k,i-1 w k,i+1 w k,i 2 < i +1 i k -i k -i -1 , 1 ≤ i ≤ k -2.
(5.21)

Moreover in the numerator, the w k,i , t k,i ,a r eg i v e nb y

w k,i = w k,0 , w k,k = w k,k-1 , w k,i = i k w k,i-1 + 1 - i k w k,i , (5.22)
and t k,0 := 0,t k,k := 1,t k,i = iw k,i-1 iw k,i-1 +(k -i)w k,i , (

That Ω k belongs to P k-1 and has positive Bézier points is required to guarantee that the space 1 Ω k P k is a rational space of degree k with linear precision. All other requirements (5.21), (5.22), (5.23), are intended to ensure reproduction of P 1 by the operator R k , along with strictly increasing t k,i ,0≤ i ≤ k. All these conditions are satisfied by our operators (5.18). The operators R k form an infinite sequence of Bernstein operators in the Chebyshevian sense. However, while in our case Ω k =Ω∈ P n-1 for all k ≥ n, in [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF] we have a sequence of Bernstein operators respectively based on the rational spaces 1 Ω k P k ,e a c hΩ k being a priori chosen independently of the others. Nevertheless, in the more general situation addressed in [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF], the main result of uniform convergence of R k F to F for any continuous function F ([26], Theorem 7.1), is obtained under the additional requirement that there exists a function ϕ ∈ C 0 ([0, 1]) such that Ω k = B * k-1 ϕ for all k.

(5.24)

Clearly, Theorems 5.9 and 5.10 are not within this context, unless our Ω is reproduced by B * k-1 for all k ≥ n, that is unless Ω belong to P 1 . If so, (5.18) can be rewritten with ω k,i =Ω( i k ). Assume that we are not in the latter special case. Using the notations introduced in Remark 2.4, we can always state that

Ω=B * k-1 ϕ k , with ϕ k := L k-1 Ω,k ≥ n -1.
We know that the sequence ϕ k , k ≥ n, is uniformly convergent to Ω on [0, 1]. Accordingly, in the special situation addressed in Theorems 5.9 and 5.10, the convergence result of Theorem 5.4 illustrates Remark 7.2 of [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF]. Indeed, it is mentioned there that condition (5.24) can be weakened and replaced by Ω k = B * k-1 ϕ k where ϕ k is a sequence of continuous functions uniformly convergent to some function ϕ on [0, 1].

To conclude this section, let us mention that, prior to [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF], rational operators of the Bernstein type were generally defined as

Q k F := k i=0 w k,i F i k B k i k i=0 w k,i B k i ,F ∈ C 0 ([0, 1]), (5.25)
where the w k,i , i =0,...,k are any positive numbers [START_REF] Farin | NURBS: from Projective Geometry to Practical Use[END_REF][START_REF] Fiorot | Rational curves and surfaces: applications to CAD[END_REF][START_REF] Piegl | The Nurbs Book[END_REF]. This is indeed an easy way to obtain uniform rational approximation of continuous functions as a consequence of Korovkin's theorem and (2.3). However, in general such operators are not Bernstein operators in the Chebyshevian sense. We would like to draw the reader's attention on the fact that, even if the w k,i 's are chosen so as to have linear precision in the corresponding rational space, the operator Q k defined in (5.25) cannot reproduce P 1 unless it is the classical Bernstein operator B * k ,t h a ti s ,i fa l lw k,i 's are equal. §6. Final remarks

It can be advantageous to regard polynomial spaces on closed bounded intervals as instances of ECspaces, and thereby to replace the simple ordinary derivatives by generalised ones, more difficult to handle. We hope that we have convinced the reader of this fact via the question of Bernstein operators. Further issues concerning approximation by polynomial or rational Bernstein operators could be of interest. Below we cite only a few possible ones:

-It may be worthwhile looking deeper into the question of polynomial (rational) approximation with shape parameters.

-As in [START_REF] Pit | A family of univariate rational Bernstein operators[END_REF] we have considered reproduction of the identity by rational Bernstein operators. Via Lemma 5.8, one could address the more general problem: U ∈ P n being given, how to determine Ω so that the corresponding rational space P n will provide us with a Bernstein operator reproducing U ?

-Is it possible to build polynomial or rational Bernstein operators with prescribed ζ 0 ,...,ζ n ?

-In connection with the previous question, can we choose positive w k,i , i =0 ,...,k, so that the operator Q k defined in (5.25) will be a Bernstein operator? Finally we would like to conclude this article by mentioning that similar results do exist for polynomial and rational Schoenberg operators. In that case, they are obtained when considering polynomial splines as examples of piecewise Chebyshevian splines ( i.e., splines with pieces taken from different EC-spaces) which are good for design (in the sense that they possess refinable B-spline bases, or blossoms as well) [START_REF] Mazure | Piecewise Chebyshev-Schoenberg operators: shape preservation, approximation and space embedding[END_REF].

  [a, b] if and only if b -a<2π, while the space spanned by cos, s i ni sa nE C -s p a c eo n[ a, b] if and only if b -a<π.

  def:BO Definition 3.7. Given a strictly increasing sequence a = ζ 0 <ζ 1 < ••• <ζ n = b, the linear operator B : C 0 ([a, b]) → E defined by

th1

  Theorem 3.10. Given any integer n ≥ 2, we assume that E ⊂ C n ([a, b]) is an (n +1)-dimensional EC-space good for design on [a, b], and we denote by (B 0 ,...,B n ) its Bernstein basis (relative to (a, b)). Consider a function U ∈ E, expanded as U = n i=0 α i B i ∈ E, and its first derivative w 1 := DU.T h e following six properties are then equivalent: (i) α 0 ,...,α n ( i.e.,t h eB ézier points of U relative to (a, b)) form a strictly increasing sequence; (ii) w 1 is positive on [a, b], and there exists a (unique) Bernstein operator based on E reproducing U ; (iii) the function w 1 is positive on [a, b] and, if we define the first order linear differential operator L 1 by L 1 V := (DV )/w 1 , then the n-dimensional space L 1 E is an EC-space good for design on [a, b]; (iv) the coordinates of w 1 in any Bernstein-like basis relative to (a, b) of the space DE are positive;

th: w0

 w0 Theorem 3.15. Let E ⊂ C n ([a, b]) be an (n +1)-dimensional EC-space on [a, b] (not necessarily good for design). Given w 0 ∈ E, the following properties are equivalent: (i) the coordinates of w 0 in any given Bernstein-like basis of E relative to (a, b) are all positive;

re:BBab Remark 3 . 16 .

 316 Theorem 3.4 says that, among all EC-spaces containing constants on a given interval I, those which are good for design can be characterised by the presence of Bernstein bases relative to any pair of distinct points of I. On a closed bounded interval [a, b], one can more simply characterise them by the presence of a Bernstein basis relative to (a, b). This is obtained when applying (i) ⇒ (ii) of Theorem 3.15 with w 0 := 1I.

Figure 1 :

 1 Figure 1: The function F defined in (4.6) (red graph), and its approximant B 3 F ,w h e r eB 3 is the polynomial Bernstein operator of degree 3 reproducing U α defined in (4.7), with α := i 6 .F r o mt o pt o bottom in the left part of the graphs, i =1; 2 ( i.e., B 3 = B * 3 ); 3; 4; 5.

  Iteration of the equivalence (i) ⇔ (iii) provides us with all systems (w 1 ,...,w n )o f weight functions on[a, b] such that E = EC(1I,w 1 ,...,w n ). In general none of the associated sequences of generalised derivatives has a special meaning for the space E. Each such system yields a nested sequence (3.3), defined by E i := EC(1I,w 1 ,...,w i ), i =1 ,...,n. Conversely, via repeated iteration of (i) ⇒ (iii), each sequence (3.3) leads to a system of weight functions (w 1 ,w 2 ,...,w n ) such that E = EC(1I,w 1 ,w 2 ,...,w n ), each w i being unique up to multiplication by a positive constant. In particular, the class of all (n + 1)-dimensional EC-spaces which are good for design on [a, b]c o i n c i d e s with the class of all spaces of the form EC(1I,w 1 ,...,w n ), where (w 1 ,...,w n ) ranges over the set of all systems of weight functions on[a, b]. By differentiation it also follows that the class of all (n +1)dimensional EC-spaces on [a, b] coincides with the class of all spaces of the form EC(w 0 ,w 1 ,...,w n ).

allweightsgood Remark 3.13. re:nestedBO Remark 3.14. Assume that E ⊂ E * ,w h e r eE * is an (n + 2)-dimensional EC-space good for design on [0, 1]. Let E 1 be a two-dimensional EC-space which is reproduced by a Bernstein operator based on E. From the existence of a nested sequence (3.3) we can assert that E 1 is automatically reproduced too by a Bernstein operator based on E * .

  10 to the polynomial space P n . The classical Bernstein operator (2.2) is a Bernstein operator in the sense of Definition 3.7 since it reproduces P 1 . This corresponds to the fact that the identity X 1 has strictly increasing Bézier points i n , i =0,...,n. On account of all results and remarks in the previous section, we can state: Theorem 4.1. For each n ≥ 2, there exist infinitely many polynomial Bernstein operators of degree n (that is, infinitely many Bernstein operators based on the space P n ), characterised by the twodimensional EC-space they reproduce, among which the classical Bernstein operator (2.2) is the one which reproduces P 1 .Remark 4.2. We thus observe that building all polynomial Bernstein operators of degree n can exactly be viewed as searching for all weight functions w 1 permitting an equality of the form (4.1). In particular w 1 =1 I corresponds to the classical Bernstein operator (2.2). It also amounts to finding all first order differential operators L 1 , obtained by composition of the ordinary derivative and division by a positive function, which, applied to P n , ensures the same property as the ordinary derivative D, i.e., decrease the dimension by one within the class of all EC-spaces good for design on [0, 1].The use of generalised derivatives or of polynomial Bernstein operators different from the classical ones may seem surprising. It is therefore certainly useful to give a few detailed examples. Example 4.3. Assume that n = 3 and consider the polynomial function

  :blosprod Lemma 5.8. Given any integers n 1 ,n 2 ≥ 1,a n df o ri =1 , 2, given P i ∈ P ni ,l e tP ∈ P n be defined by P := P 1 P 2 ,w i t hn := n 1 + n 2 . Then, the blossom p in n variables of P is given byp(t 1 ,...,t n )= 1 (t i1 ,...,t in 1 )p 2 (t j1 ,...,t jn 2 ),t 1 ,...,t n ∈ [0, 1], (5.13)

	n	p 1
	n1

le

  . Let P ∈ P n be defined by P (t):=tΩ(t), t ∈ [0, 1]. The blossom p of P (in n variables) is thus given by

	p(t 1 ,...,t n )=	1 n	n i=1	t i ω

* (t 1 ,...,t i-1 ,t i+1 ,...,t n ),t 1 ,...,t n ∈ [0, 1]. As a special case, given distinct a, b ∈ [0, 1], we obtain


T Oberlin

S Meignen

S Mclaughlin

A NOVEL TIME-FREQUENCY TECHNIQUE FOR MULTICOMPONENT SIGNAL DENOISING

Keywords: Time-frequency, ridge, synchrosqueezing, denoising, multicomponent signals

Multicomponent signals, i.e. superpositions of modulated waves, arise in many physical or biological systems. Exploiting the particular structure of these signals, denoising methods based on time-frequency distributions often outperform standard techniques such as those based on diagonal estimation or sparsity approaches. Recently, a simple denoising technique based on local integration in scale of the wavelet transform was proposed. In spite of its behaviour being better compared to classical techniques for medium noise levels, it does not perform so well in other cases. We propose here a method to improve denoising behaviour based on a more accurate mode reconstruction technique. The method is detailed for time-frequency representation given by shorttime Fourier and continuous wavelet transforms, with the emphasis placed on their differences.

INTRODUCTION

Many signals from the physical world can be modeled as a sum of amplitude-and frequency-modulated (AM-FM) waves, called multicomponent signals, which have been the focus of much interest in the past few decades. As a result of their simplicity and efficiency, linear time-frequency (TF) transforms such as short-time Fourier transform (STFT) and continuous wavelet transform (CWT) have received particular attention. The STFT and CWT of multicomponent signals draw so-called ridges in the TF plane which, once detected, allow for the reconstruction of the different components by considering the transform on the ridges [START_REF] Delprat | Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies[END_REF]. More recently, it was shown in [START_REF] Meignen | A new algorithm for multicomponent signals analysis based on synchrosqueezing: With an application to signal sampling and denoising[END_REF][START_REF] Meignen | Multicomponent signal denoising with synchrosqueezing[END_REF] that local frequency integration improved the robustness to noise of the reconstruction.

Yet, these methods do not perform very well when the frequency modulation is strong, i.e. when the modes locally behave like linear chirps. We propose here to adapt the size of the integration domain to frequency modulation and noise level.

Section 2 introduces the notation, and recalls the usual first-order approximation of the CWT and STFT of multicomponent signals. Then, section 3 extends these approximations to strong frequency modulations, while section 4 shows how to reconstruct the components in a noisy context. Finally numerical results are given in section 5, demonstrating the efficiency of the method on the one hand, and putting the emphasis on the differences between STFT and CWT on the other.

DEFINITIONS

Short-Time Fourier Transform

In the following, we denote by L 1 (R) and L 2 (R) the space of real integrable and square-integrable functions. Given a signal s ∈ L 1 (R), its Fourier transform is defined by: ŝ(η) := R s(t) e -2iπηt dt.

(

) 1
Taking a window g ∈ S(R), the (modified) Short-Time Fourier Transform (STFT) of a signal s is defined by

V s (η, t) := R s(τ )g(τ -t)e -2iπη(τ -t) dτ. (2)
The STFT admits the following synthesis formula:

s(t) = 1 g(0) R V s (η, t) dη, (3)
provided that η → V s (η, t) is integrable, which will always be the case in this paper.

Continuous Wavelet Transform

Taking an admissible wavelet

ψ ∈ L 2 (R) (i.e. C Ψ := ∞ 0 | ψ(ξ)| 2 ξ dξ < ∞) and letting ψ a,t (τ ) := 1 a ψ τ -t a
, we define the CWT of the signal s by:

W s (a, t) = s, ψ a,t = 1 a R s(τ ) ψ τ -t a dτ, (4)
where z denotes the complex conjugate of z. We suppose that ψ is analytic, i.e. Supp( ψ) ⊂ [0, ∞[, so that the WT W s of a real signal s is the half of the WT of its analytic signal s an = s + iH(s), where H stands for the Hilbert transform (see [START_REF] Mallat | A wavelet tour of signal processing[END_REF] for details). We recall the Morlet formula (obtained by taking a Dirac for synthesis, see [START_REF] Mallat | A wavelet tour of signal processing[END_REF] for instance):

s an (t) = 1 C ′ ψ ∞ 0 W s (a, t) da a , (5)
where

C ′ ψ = ∞ 0 ψ(ξ) dξ ξ .
The real signal is easily obtained by s = 1 2 Re(s an ).

Multicomponent signals and ridges

A general modulated wave writes h(t) = a(t)e 2iπφ(t) , with a(t) > 0 and φ ′ (t) > 0. When a and φ ′ are slow-varying functions, both the CWT and STFT of such signals have been studied for decades, and can be well approximated in the vicinity of time t by considering the pure wave

h1 (t + τ ) = a(t)e 2iπ[φ(t)+φ ′ (t)τ ] . (6)
We then get

V h (η, t) ≈ h(t) ĝ(η -φ ′ (t)) W h (a, t) ≈ h(t) ψ(aφ ′ (t)), (7)
The frequency center of ψ being assumed to be one without any loss of generality. Dealing with multicomponent signals, i.e. superpositions of modes, is no more complicated as soon as the different components are separated in frequency, which means their TF distribution do not overlap. Let s(t) = K k=1 a k (t)e 2iπφ k (t) be a multicomponent signal, we recall the separation condition for both STFT and CWT:

Proposition 2.1. The signal s is separated for STFT if |φ ′ k (t) -φ ′ l (t)| > Supp ĝ, k = l, (8)
and for CWT if

|φ ′ k (t) -φ ′ l (t)| |φ ′ k (t) + φ ′ l (t)| > 1 2 Supp ψ, k = l, (9)
where Supp ĝ (resp. ψ) denotes the size of the frequency bandwidth of g (resp. ψ). The difference arises here because of the logarithmic frequency description used by CWT. Also, studying separated multicomponent signals amounts to studying single components, which we will do in the sequel.

Gaussian window and wavelet

Due to its optimal TF resolution, the Gaussian function is often used as window g. Let us define the Gaussian window and its corresponding complex Gaussian (also called Morlet) wavelet:

g(t) = σ -1/2 e -π t 2 σ 2 , (10)
ψ(t) = σ -1/2 e -π t 2 σ 2 e 2iπt . (11)
The parameter σ enables the choice of the size of the window/wavelet in accordance with ( 8) and ( 9). The next section shows how to use these particular functions to extend approximations [START_REF] Bleistein | Asymptotic expansions of integrals[END_REF] to the case of strong frequency modulations.

RECONSTRUCTING STRONG FREQUENCY-MODULATED SIGNALS

Let us now assume that φ ′′ (t) is no longer negligible, then first order approximations [START_REF] Bleistein | Asymptotic expansions of integrals[END_REF] no longer hold, and more involved techniques are needed. For that purpose, an asymptotic method in time domain, the stationary phase approximation, has been used successfully in [START_REF] Delprat | Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies[END_REF]; a different technique introduced in [START_REF] Oberlin | Analysis of strongly modulated multicomponent signals with the short-time Fourier transform[END_REF] is to expand the phase up to the second order. The next section derives the corresponding approximation of the STFT and CWT magnitude, on which our denoising technique is based.

Exact formulae for a Gaussian window/wavelet

We now aim to extend formulae [START_REF] Bleistein | Asymptotic expansions of integrals[END_REF] when the frequency modulation is not negligible. The simplest way to proceed is to approximate the mode h in the vicinity of t by its secondorder Taylor expansion:

h2 (t + τ ) = a(t)e 2iπ[φ(t)+φ ′ (t)τ + 1 2 φ ′′ (t)τ 2 ] . (12)
To study the STFT of such chirps, we need to compute the Fourier transform of a Gaussian modulated linear chirp.

Proposition 3.1. Consider the function u(t) = e -πzt 2 , where z = re iθ with cos θ > 0, so that the function is integrable. Then its Fourier transform is

û(ξ) = r -1 2 e -i θ 2 e -π re iθ ξ 2 . ( 13
)
Proof. One can proceed as in the case when z is real: it suffices to differentiate u and consider the Fourier transform of the obtained differential equation (see Appendix A of [START_REF] Kammler | A first course in Fourier analysis[END_REF] for instance).

Theorem 3.1. For a Gaussian window or its associated Morlet wavelet, the magnitude of the STFT and CWT transform of h2 admits the following closed-form expressions:

|V h2 (η, t)| = |h(t)|σ 1 2 (1 + σ 4 φ ′′ (t) 2 ) -1 4 e - πσ 2 (η-φ ′ (t)) 2 1+σ 4 φ ′′ (t) 2 |W h2 (a, t)| = |h(t)|σ 1 2 (1 + σ 4 a 4 φ ′′ (t) 2 ) -1 4 e - πσ 2 (1-aφ ′ (t)) 2 1+σ 4 a 4 φ ′′ (t) 2 .
Proof. For a fixed time t, equation ( 12) and Proposition 3.1 give 14)

V h2 (η, t) = h(t)σ -1 2 r -1 2 e -i θ 2 e -π r e -iθ (η-φ ′ (t)) 2 ,(
with r = ( 1 σ 4 + φ ′′ (t) 2 )
1 2 and θ = arctan(-φ ′′ (t)σ 2 ). Using the identity cos arctan x = 1 √ 1+x 2 , one finally obtains V h2 , and W h2 in the same manner. This shows that the magnitude of the STFT of a linear chirp is also a Gaussian function centered in η = φ ′ (t). The difference with equations ( 7) lies in the magnitude and the width of this Gaussian. The formula for the CWT is very similar, except that σ is replaced by aσ.

Stationary phase approximation

We have explicit formulae for the STFT and the CWT of a linear chirp, when both g and ψ are Gaussian. However, in a more general context such formulae are no longer available, but the transforms can still be well approximated by the stationary phase approximation [START_REF] Bleistein | Asymptotic expansions of integrals[END_REF]. Let us suppose that φ ′ (t) is strictly monotonic, then for any (η, t) there exists at most one time t c such that φ ′ k (t c ) = η. If t c exists, then the stationary phase approximation gives

|V h (η, t)| ≈ |h(t c )| g(t c -t) |φ ′′ (t c )| . ( 15
)
We can conduct the same reasoning with the CWT, where η is replaced by 1/a, which gives

|W h (a, t)| ≈ |h(t c )| ψ tc-t a a |φ ′′ (t c )| . ( 16
)
The method presented here can be adapted to any general window, using these formulae.

DENOISING MULTICOMPONENT SIGNALS

The previous section gave details about how the information is localised around the ridges. We aim here at using this information for denoising ; so we consider a multicomponent signal corrupted by a white Gaussian noise. To denoise the signal we propose the following method, also used in [START_REF] Meignen | Multicomponent signal denoising with synchrosqueezing[END_REF]:

• Estimate the ridge at time t, φ ′ (t).

• Compute the integration domains I(t), so that for η ∈ I(t), |V h (η, t)| is higher than the noise level.

• Integrate the STFT or CWT of the noisy signal on the domain, to get an estimate of h(t).

• Iterate the process for all times t and all components.

This process is illustrated in Figure 1, where the STFT and CWT of an AM-FM mode are displayed along with a slice of both transforms for a fixed time, showing the integration domain. The following sections will explain how to choose the frequency integration domains I(t), considering either the first-or second-order approximation.

Noise distribution in the transformed domain

Let us start by quantifying the noise distribution on STFT and CWT. We consider a noisy realisation s(t) = h(t) + n(t), where n is a Gaussian process with 0 mean and variance σ 2 n . For a fixed frequency η or scale a, a simple calculation leads to

Var(|V n (η, t)|) = σ 2 n Var(|W n (a, t)|) = 1 a σ 2 n .
(17)

Integration domain, first order

According to equation ( 7), η → |V h (η, t)| almost reaches a maximum at φ ′ (t), and is symmetric with respect to φ ′ (t). Thus, we aim at computing

∆ 1,V such that |V h (φ ′ (t) ± ∆ 1,V , t)| = σ n . Using equation (17) and remarking that √ σ|h(t)| = |V h (φ ′ (t), t)| = V max (t), we get ∆ 1,V = 1 σ √ π -log σn Vmax(t) . ( 18
)
To compute the analogue for the CWT leads to integrate on [

1-∆ 1,W φ ′ (t) , 1-∆ 1,W φ ′ (t) ],
with

∆ 1,W = 1 σ √ π -log σn Wmax(t) √ a , (19)
where a =

1±∆ 1,W φ ′ (t)
is the corresponding scale. An easy way to remove a from the right-hand side is to approximate a by φ ′ (t).

Second order model

A similar computation for the second-order model gives

∆ 2,V = √ 1+σ 4 φ ′′ (t) 2 σ √ π -log σn Vmax(t) , ∆ 2,W = √ 1+a 4 σ 4 φ ′′ (t) 2 σ √ π -log σn Wmax(t) √ a , (20)
where we can still use the approximation a ≈ φ ′ (t).

NUMERICAL RESULTS

This section shows the efficiency of the denoising method on synthetic signals. Methods will be denoted by STFT1, STFT2, CWT1 or CWT2, depending on the transform and the approximation order. We will compare them with a general TF denoising technique: Block-Thresholding [START_REF] Yu | Audio denoising by time-frequency block thresholding[END_REF], denoted by BT hereafter. Ridges are estimated using a simple heuristic search like in [START_REF] Oberlin | Analysis of strongly modulated multicomponent signals with the short-time Fourier transform[END_REF], whereas φ ′′ k (t) is estimated by ridge differentiation after a regularizing spline-fitting step. The code implementing the method can be downloaded from http://www-ljk.imag.fr/membres/ Thomas.Oberlin/Eusipco13.tar.gz, together with Matlab scripts that plot all the figures of this paper.

A first example

Let us first assess the efficiency of first-order methods.

STFT vs CWT, and window's size

Let us here stress the differences between STFT and CWT. First, note that the denoising performance depends on the ability of the transform to sparsely represent the signal. If the signal is made of monochromatic waves, taking a large window will ensure a quasi-perfect TF representation (i.e., made of three thin lines). But if the components are frequency modulated, a very large window will cause strong time diffusion, which hampers the quality of the representation. Thus, one has to choose σ so as to achieve a trade-off between time and frequency localization.

The main difference between STFT and CWT lies in how they handle frequency modulation: STFT performs well provided φ ′′ is low, whereas CWT needs φ ′′ /φ ′ to be low (see the discussion in [START_REF] Mallat | A wavelet tour of signal processing[END_REF] section 4.5 for more details). The other difference concerns mode separation condition (equations ( 8) and ( 9)), which must be linear for STFT and logarithmic for CWT. This is illustrated here through the study of two different 3-component signals: one is composed of polynomial chirps and is well adapted to STFT, whereas the other contains exponential chirps that can be well processed by CWT. The STFT and CWT of both signals are displayed in Figure 3, together with their respective denoising performances using the second-order approximation. Figure 3 (c) shows that methods based on STFT representation are well adapted to polynomial chirps, whereas a poor separation in the time-scale plane hampers CWT2. However, according to Figures 3 (d) and (f), STFT-based methods do not manage to denoise the signal properly because of heavy time diffusion effects. Note parameters σ are chosen carefully for each method to get the best possible results, i. e. they must ensure that (8) and ( 9) are satisfied. .

Contribution of the second-order model

This section illustrates how to take into account second-order terms enables better signal denoising in case of strong frequency modulation. For this purpose, the two signals of Figure 3 are denoised using either the first or second order methods, and the results are displayed in Figure 4. This shows that the second-order approximation offers little improvement when the noise level is high, but is of great interest for low noise. Future works should include a broader comparison with other denoising methods, based for instance on the Wigner-Ville transform [START_REF] Stankovic | The Wigner distribution of noisy signals with adaptive time-frequency varying window[END_REF], wavelet packet dictionaries [START_REF] Krim | On denoising and best signal representation[END_REF] or the Empirical Mode Decomposition [START_REF] Kopsinis | Development of EMD-based denoising methods inspired by wavelet thresholding[END_REF].

Fig. 1 .

 1 Fig. 1. Illustration of STFT-or CWT-based denoising. (a) STFT of a noisy AM-FM wave. (b) a slice of the STFT at time t = 1 2 , along with the mean noise level and the integrated part. (c) and (d): idem for CWT, scales being displayed on a logarithmic scale.

Figure 2 (Fig. 2 .

 22 Fig. 2. Denoising performances on a synthetic 3-components signal. (a) STFT with window σ = 0.05. (b) CWT with σ = 7. (c) comparison between 3 methods.

  This paper analysed the magnitude of the STFT and CWT of strongly modulated multicomponent signals for signal denoising. A component reconstruction method by local integration was proposed which took into account local frequency modulation and noise level. Numerical experiments demonstrated the effectiveness of the approach for the denoising of multicomponent signals with polynomial or exponential phase, respectively from their STFT or CWT.

Fig. 3 .

 3 Fig. 3. Analysis of signals with strong frequency modulations. (a): STFT of a superposition of polynomial chirps; (b): CWT of the same signal; (c): corresponding denoising results for STFT2 and CWT2. (d), (e) and (f): STFT, CWT and corresponding denoising performance for a superposition of exponential chirps. The parameters are σ = 0.08 for STFT and σ = 5 for CWT.

Fig. 4 .

 4 Fig. 4. Importance of second-order terms for strong frequency modulation. (a): denoising performance for STFT1 and STFT2 applied to the signal of Figure 3 (a) and (b), with σ = 0.08. (b): denoising performance for CWT1 and CWT2 applied to the signal of Figure 3 (d) and (e), with σ = 5.


Dr Michel Grangeat
email: michel.grangeat@ujf-grenoble.fr

Ass. Pr David Cross

Dr Nadia Nakhili

A Professional Community of Science Teachers and Researchers for Enacting Change in Classroom Assessment

This paper aims to specify conditions that seem fruitful for enacting change in schools. It addresses a project involving teachers from lower secondary and primary schools. The project aims to develop formative assessment within science inquiry-based teaching methods. The elements underpinning this project are described. The effects on teacher conceptions and practices are shown.

Data results from an analysis of the project material and of focus groups with the teachers involved. The methodology is based on the activity theory framework.

The results are aligned with other research that showed how fruitful it is to involve teachers within cooperative settings where they can reflect about specific professional questions and tackle complex problems with other colleagues, researchers and teacher educators from university.

From Professional Community to Inquiry Community

Current research findings concur with the claim that collective settings spur teacher development. Nevertheless, fruitful collective settings are to respect some conditions [START_REF] Grangeat | Teaching as a collective activity: analysis, current research and implications for teacher education[END_REF]. This paper aims to specify and evaluate conditions that seem productive for enacting change in schools, particularly in science education. [START_REF] Hargreaves | Professional Capital: Transforming Teaching in Every School[END_REF] noticed that 'schools [as well as teachers] improve when they collaborate with and learn from other schools [and teachers] -but not always ' (p. 136). They showed that change failed to be valuable for learners if teachers are pushed to transform their practices by implementing new methods, even if such a change is underpinned by teachers' enrolment in collective based programs. They claimed that teachers need to be pulled in improving their practices when they participate in 'important work with committed and excited colleagues and leaders engaged in activities that require creativity to solve complex problem ' (p. 151). They demonstrated how teachers and schools become collectively concerned by students' learning matters when they are embedded in local networks. They concluded that professional learning communities need to be appropriately designed if they aim to be productive.

Three conditions seem crucial:

-concerning professionals from a same educational sector who share common interest and motivation in resolving a specific educational problem, -stimulating cooperation among several schools and among different kinds of professional (teachers, head of schools, inspectors, researchers and teacher educators from the university), -underpinning the long process of teacher learning and teaching transformation by extending the time span of the program. We propose to call this kind of collaborative settings 'Inquiry Based Continuing Professional Development program' since all the professionals involved in this IB-CPD program endeavor to resolve a common problem but none of them know the solution of this problem.

This paper addresses a specific IB-CPD program that concerns teachers from two low secondary and four primary schools of a socially disadvantaged sector in France. A first question is to describe the elements that underpin this program. A second question is to grasp the effects on teacher conception and practices.

Evaluating the Efficiency of a Specific IB-CPD Program

The sample consists of sixteen teachers involved in the EvaCoDICE project that aim to develop formative assessment method within science inquiry-based teaching method. This EvaCoDICE project is included in a network of collaborative settings in France --ruled by the French Institute of Education--and is part of a European project --ASSIST-ME-about assessment within science education. The EvaCoDICE project is planned to last three years from 2012 to 2015. Thus this paper concerns the first steps of the study.

A first kind of data results from an analysis of the material that was shared by the actors through a specific website and three seminars per year. This will lead to describe the structure of the studied collaborative setting.

A second kind of data is drawn from focus groups with the teachers involved in the project. This will elicit the evolution of teachers' approaches.

Results

The methodology is based on the activity theory framework after [START_REF] References Engeström | Expansive learning at work: toward an activity theoretical reconceptualization[END_REF]. According to this theory, one actor's activity is understood as a system that links several key elements. This system is organized upon a main axis that links the actors, the action's object and the action's effects. The actors create meaning about the action by reflecting on their goals and their results, in other words on the difference between the initial object and the final effects. Nevertheless, as it is well known, the action and its meaning are always mediated by conceptual and physical artefacts (e.g. constructs, guidelines, tools, data base). Overall, this action is influenced by the labor organization (e.g. leadership style), by the culture of the community (e.g. the will of teacher team to spur students' achievement), and by the repertoire of actions that is available for the actors (e.g. innovative practices). These elements and this system represent a construct for understanding teacher professional knowledge [START_REF] Grangeat | Elaborating a Model for a better Understanding of Science Teachers' Approaches and Practices towards Inquiry Based Science Teaching and Learning[END_REF].

Elements that underpin IB-CPD program

In this section, we will describe the different elements of the system in which teachers' activities were embedded.

Actors

The EvaCoDICE project concerns three types of professional team:

-A practitioner team comprised of 11 science teachers from 2 lower secondary schools and 4 primary teachers from 4 schools; all these schools were part of one educational sector that concerns socially disadvantaged people. -A research team comprised of 5 teacher educators and 6 researchers from the university in charge of teacher education; this research team was complemented by a PhD student. -An institutional team comprised of 2 principals of lower secondary schools, 2 persons (inspector and counselor) who rule the primary schools, and 3 persons from the regional authorities for secondary schools (inspectors and counselor).

Artefacts, tools, and resources

The EvaCoDICE project is based on three types of resources:

-A dedicated website which allows actors to share the project documents (courses preparations, assessment tools, meeting minutes, scientific articles, etc.). -Three quarterly one-day seminars which concerns all the actors. These sessions were designed by the research team, with the institutional team's agreement, and their agendas were communicated to the practitioner team through the website. A report of each seminar is available on the website. -Many meetings were set up with teacher educators in order to improve teaching material and assessment tools.

Repertoire of actions

Many different actions were designed in order to pull the actors within the project.

-Two rules underpin the EvaCoDICE project: the cooperation was horizontal thus no one can refer to his or her position in order to convince others; teachers' knowledge was always valuable -and improvable-since teachers experience teaching in classrooms everyday along the project. -Seminars take place in a room that is very flexible. Chairs and tables are moved several times by seminar. Walls are used to pin posters. Works in-progress are shared and discussed easily through a video projector. -Regional and national meetings had required actors from the three teams to present the EvaCoDICE project to other teachers, schools, and stakeholders.

Community and division of labor

The EvaCoDICE project was implemented in a sector were teachers and principals were used to cooperate. For many years, primary and lower secondary teachers met together in order to align their goals regarding pupils' careers. Inspectors describe themselves as counselors and critical friends who don't want to interfere with the project progress.

Effects on teacher conceptions and practices

The study will analyse the elements of the activity system which shape the action of the teachers involved in the project [START_REF] References Engeström | Expansive learning at work: toward an activity theoretical reconceptualization[END_REF]. Firstly it will grasp the objects of teacher actions and the effects they noticed. Secondly, it will describe the tools and resources that seem useful for the teachers. Thirdly, and finally, it will analyze the rules that shape teachers' approaches and practices.

Teachers' objects and purposes

The first element of the activity system consists of the teachers' objects and purposes. During the focus groups, teachers were asked to elicit how their approaches of assessment had changed during the project first year, in terms of objects and purposes. Four main modifications are noticed.

-Assessing for learning is becoming more important than assessing for marking or ranking.

-Assessment is becoming more focused and aligned on what is essential during the course.

-Assessment is conceived as an on-going process undertaking the inherent long time needed for learning. -Sharing a common vocabulary through the different teachers and contents is seen as essential in facilitating students' understanding.

Teachers have suggested to dedicate the project's second year to design, share, test and improve other formative assessment tools and methods.

Artefacts, tools, constructs and resources

The second element of the activity system consists of tools and resources that mediate teacher actions. During the focus groups, teachers were asked to elicit what tools and resources were useful in their approaches transformation. Six main sources of modification are noticed.

-The volition for making explicit and for sharing teaching material (course preparations and assessment tools) have been steered by the seminars themselves. Teachers explained that they had always wanted to act like that but without a deadline that contrives them, they had always postponed this kind of interactions with their colleagues. -Designing the seminars and the entire project as an inquiry has made the teachers more comfortable and confident in undertaking IBST methods with their own students. -The positive interactions among the actors and teams and the entire respect of teacher knowledge have supported the teachers in improving their practices, and have supported the necessary efforts required by the program. -The pedagogical content knowledge provided by the research team was ranked as very helpful since, for instance, it had allowed teacher to know and to foresee the students' usual mistakes or misconceptions with respect to a specific content. -The goals and methods of the research team that had been explained and stressed as limited and focused have supported the teachers in the limitation and focus of their expectation in their own classrooms. Teachers have suggested improving the students' responsibility in the assessment process. They like to design tools and methods that allow their students to formulate the criteria for assessing knowledge, skills and attitude toward a scientific problem.

Repertoire of actions

The third element of the activity system consists of repertoire of actions that shape teacher actions. During the focus groups, teachers were asked to explain what practical changes have occurred during the past year. Changes were grasped into three main types:

-Lesson's anticipations are deepened since teachers focused on the knowledge that is requested by the problem resolution, on the nature of the hypothesis that might be tested by students and on the warrants that might be used to support their claims. -Lessons are more aligned with student progression than with official guidelines. Thus, teachers anticipated assessment methods allowing students to monitor and check their learning activities. -Awareness of each student activity and reflection is shared by all teachers. They explained that all their students enter within the activity and are able to justify their strategies while most of them were passive at the beginning of the year and all the other past years. Teachers have suggested making more explicit for students what will be expected during the lesson (As a teacher, I'd like you to be able to…) and after the lesson (Strategies for homework).

Community and division of labour

As expected, teachers didn't explain any deep changes towards these elements. The only transformation concerned the way teachers conceived the research team: educational research methods and actors had appeared as reachable and understandable. Some teachers seemed interested in educational studies: they read articles and watch video on internet.

Conclusions

These results are aligned with other research that show how it is fruitful to involve teachers within cooperative settings where they can reflect about specific professional questions with other colleagues, researchers and teacher educators from university [START_REF] Jaworski | Theory and Practice in Mathematics Teaching Development: Critical Inquiry as a Mode of Learning in Teaching[END_REF][START_REF] Van Der Valk | Scaffolding Science Teachers in Open-inquiry Teaching[END_REF]. These cooperative settings tend to create inquiry communities between teachers and educationalists in order to promote and develop specific methods within the classroom. The key feature of these inquiry communities is that participants acknowledge and address issues and tensions within their approaches and practices. This critical alignment is both a goal and an outcome of the collective work. These studies report the way these inquiry communities supports teachers towards their goals, such as fostering students' self-esteem or involving students in their own assessment, despite their old habits and the social context which could impede these changes.

The first main characteristic of this kind of IB-CPD program is the 'push-pull architecture' of the project: teachers and schools were pulled into by 'voluntary participation, common bonding, professional inspiration, peer assistance, practical strategies, and technical supports' (Hargreaves & Fullan, 2012, p.138). The second is the project focus on a shared professional question that underpins the collective inquiry.

These results encourage in seeking new ways to design and implement educational research in order to support educational changes that might be researched based.

Acknowledgment

The study was part of the ASSIST-ME project, which has received funding from the European Community's Seventh Framework Program [FP7, project number 321428].