Laurent Baratchart
email: laurent.baratchart@inria.fr

Wei-Xiong Mai

Tao Qian

GREEDY ALGORITHMS AND RATIONAL APPROXIMATION IN ONE AND SEVERAL VARIABLES

Keywords: 2000 Mathematics Subject Classification. 45P05, 30H Rational Orthogonal System, Takenake-Malmquist System, Szegö Kernel, Hardy Spaces

We will review the recent development of rational approximation in one and several real and complex variables. The concept rational approximation is closely related to greedy algorithms, based on a dictionary consisting of Szegő kernels in the present context.

Introduction

Traditionally, rational approximation is mostly restricted to approximation to functions of one complex variable in various domains. Recently a number of adaptive or sparse representation methods were developed, including in principle the greedy algorithms and those based on learning theory, including the SVM method ( [START_REF] Mo | Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem[END_REF][START_REF] Mo | Sparse Representation in Szegő Kernels through Reproducing Kernel Hilbert Space Theory with Applications[END_REF]), that all fit into the concept of rational approximation. This article mainly concerns the greedy algorithm method in relation to a dictionary in the context at hand. Under the greedy algorithm method, rational approximation is generalized to include approximation in several complex and real variables. This idea, in particular, is applicable to function spaces with a Cauchy structure, as well as to reproducing kernel Hilbert spaces. Below we discuss this concept in a number of individual contexts.

Preliminaries on Greedy algorithm in Hilbert Spaces

Let H be a Hilbert space with a dictionary, where, by a dictionary, we mean a set D ⊂ H satisfying (i) e ∈ D implies e = 1; and (ii) span{D} = H.

The most basic greedy algorithm would be the following. Let ρ ∈ (0, 1] be fixed. Let f ∈ H and g 1 = f. Choose e 1 ∈ H such that

| g 1 , e 1 | ≥ ρ sup{| g 1 , e | | e ∈ D}.
When ρ ∈ (0, 1), a desired e 1 always exists. While for ρ = 1, a desired e 1 satisfying the above requirement may not exist. In the context of the present paper, where a Cauchy structure prevails, the desired e 1 for ρ = 1 always exists. Our discussion, however, is for the general case ρ ∈ (0, 1]. Subsequently, we have the decomposition

f = g 1 , e 1 e 1 + g 2 ,
where g 2 is the standard remainder:

g 2 = f -g 1 , e 1 e 1 .
Obviously, g 2 is orthogonal with g 1 , e 1 e 1 , and hence,

f 2 = g 1 , e 1 e 1 2 + g 2 2 = | g 1 , e 1 | 2 + g 2 2 .
Therefore, to minimize g 2 2 is to maximize | g 1 , e 1 | 2 . If we apply the same reduction to g 2 , we obtain the standard remainder g 3 , where f = g 1 , e 1 e 1 + g 2 , e 2 e 2 + g 3 , and

f 2 = | g 1 , e 1 | 2 + | g 2 , e 2 | 2 + g 3 2 ,
where e 2 is chosen so that

| g 2 , e 2 | ≥ ρ sup{| g 2 , e | | e ∈ D}.
Repeating the same procedure on g 3 we get g 4 , and so on. Inductively we obtain f = n k=1 g k , e k e k + g k+1 and

f 2 = n k=1 | g k , e k | 2 + g k+1 2 ,
where e k is chosen to make

| g k , e k | ≥ ρ sup{| g k , e | | e ∈ D},
while e 1 , ..., e k-1 were previously chosen. The theory of greedy algorithms ( [START_REF] Devore | Some remarks on Greedy Algorithms[END_REF], [START_REF] Temlyakov | Greedy algorithm and m-Term trigonometric approx-imation[END_REF]) asserts that

f = ∞ k=1 g k , e k e k and f 2 = ∞ k=1 | g k , e k | 2 .
The above described algorithm is the so called General Greedy Algorithm. A refinement of the general greedy algorithm is the so called Orthogonal Greedy Algorithm. The difference is only that the g k are replaced by the orthogonal remainders gk , defined through the relation Among the well known greedy algorithms, the most effective one would be the just described Orthogonal Greedy Algorithm. We note that neither the General Greedy Algorithm, nor the Orthogonal Greedy Algorithm can repeatedly select dictionary elements: Repeated selections or even selecting one in the linear span of the already selected will give nil contribution to the energy approximation.

f = n k=1 gk , B k B k + gk+1 , (2.1) where {B 1 , ...B k-1 , B k } is the G-S orthogonalization of {B 1 , ..., B k-1 , e k },
When the space is a reproducing kernel Hilbert space, consisting of functions defined, say, in a set D, and if the dictionary consists of the normalized reproducing kernels e a , where a ranges over a set D ⊂ C N for some N, then for any f ∈ H, f, e a = r(a) f (a), where r(a) > 0 is the normalizing constant that makes e a = 1.

In such a case, at each recursive step of the General Greedy and Orthogonal Greedy Algorithms, one seeks a suitable e a such that

r(a)| f (a)| ≥ ρ sup{r(b)| f (b)| | b ∈ D}.
Numerically this is easy to achieve through computation based on the information on the known function f.

We will show that under certain assumption there exists a variation of the Orthogonal Greedy Algorithm that allows repeated selection of the variable a. Repeating the selection of the variable a corresponds to selecting directional derivatives, of order one and higher, of the dictionary elements e a .

This new greedy algorithm proposed in [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF], called Pre-Orthogonal Greedy Algorithm (P-OGA), is formulated as follows.

Let {e a 1 , ..., e a n-1 } be the (n -1)-tuple of the previously selected dictionary elements, and

{B 1 , ..., B n-1 } its G-S orthogonalization. Sometimes B k = B {a 1 ,...,a k } is more precisely written as B a k {a 1 ,...,a k-1 } . The selection criterion for a n is | f n , B a n {a 1 ,...,a n-1 } | ≥ ρ sup{| f n , B a {a 1 ,...,a n-1 } | | a ∈ D}, (2.2)
where f n is the standard remainder with respect to the orthonormal system {B 1 , ..., B n-1 }. We note that under such a selection criterion f k is different from gk defined through (2.1). Now we add an assumption under which the machinery that we design will work. Namely, we assume for any f ∈ H and a 1 , ...,

a n-1 ∈ D that lim a→∂D | f, B a {a 1 ,...,a n-1 } | = 0, (2.3)
where ∂D indicates the boundary of D in C N ∪ ∞. We show that under this assumption, if the dictionary is suitably extended, then the threshold ρ = 1 can be reached, and the following equality holds:

| f n , B a n {a 1 ,...,a n-1 } | = max{| f n , B a {a 1 ,...,a n-1 } | | a ∈ D}. (2.4)
The extension of the dictionary consists in adjoining the directional derivatives of the kernels e a with respect to a, in all directions and for all a ∈ D. This of course only makes sense if D is open and e a has some smoothness with respect to a. Hereafter we assume that a → e a is smooth as a function D → H, in particular the derivatives of e a again lie in H as limits of divided differences in H. In all examples that we shall deal with, e a is even analytic with respect to a, and this warrants the discussion below.

In fact, the Cauchy-Schwarz inequality gives

| f n , B a {a 1 ,...,a n-1 } | ≤ f n .
Thus, there exists a sequence of points, a (l) , converging to an interior or a boundary point of D, lim l→∞ a (l) = a n , such that

lim l→∞ | f n , B a (l) {a 1 ,...,a n-1 } | = sup{| f n , B a {a 1 ,...,a n-1 } | | a ∈ D}. (2.5)
With the assumption (2.3) the limiting point a n of a (l) must be an interior point of D unless f n ≡ 0. In the latter case, our contention trivially holds. Otherwise, the relation (2.5) becomes with one of a 1 , ..., a n-1 , we are in this case. We note that none of the e a (l) is in span{B 1 , ..., B n-1 }, for, otherwise, B a (l) {a 1 ,...,a n-1 } = 0, and | f n , B a (l) {a 1 ,...,a n-1 } | will have no contribution towards the maximum. We consequently have Extracting a subsequence if necessary, we may suppose that (e a (l)e a n )/ e a (l)e a n converges to some unit vector ν ∈ H. If ν is not in span{B 1 , ..., B n-1 }, then we can take the limit in the above expression as a (l) → a n , to obtain that B a (l) {a 1 ,...,a n-1 } converges to B ∂ v a n {a 1 ,...,a n-1 } , where the notation ∂ v indicates that we computed the directional derivative of e a in the direction v. When ν ∈ span{B 1 , ..., B n-1 }, then the above expression has indeterminate limiting form 0/0, and higher order derivatives must be computed that we will not discuss further. Observe there may be several directions ν, accounting for the fact that a n needs not be unique. In particular, the directions along the real or purely imaginary axes induce partial derivatives of the reproducing kernels ( [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF]).

lim l→∞ | f n , B a (l) {a 1 ,...,a n-1 } | = sup{| f n , B a {a 1 ,...,a n-1 } | | a ∈ D.} (2.
B a (l) {a 1 ,...,a n-1 } = e a (l) -P {a 1 ,...,a n-1 } e a (l) e a (l) -P a 1 ,...,a n-1 e a (l) = e a (l) -P {a 1 ,...,a n-1 } e a (l) -e a n -P {a 1 ,...,a n-1 } e a n e a (l) -P a 1 ,...,a n-1 e a (l) -e a n -P {a 1 ,...,a n-1 } e a n = e a (l) -e a n -P {a 1 ,...,a n-1 } e a (l
The just described theory mainly grows out from the study of the Hardy space H 2 on the open unit disc. It is then applied, at least in part or adaptively, to other contexts in one or several complex variables, as well as in the Clifford algebra setting. That helps to form a concept of rational approximation in various contexts. In the following sections we will briefly review the particulars of each individual context.

The Hardy H 2 (D) case

The so called AFD and Pre-Orthogonal Greedy Algorithm were originated from this context. In this section D represents the open unit disc. Among the equivalent definitions of the Hardy H 2 (D) space we will cite only

H 2 (D) = { f (z) = ∞ k=0 c k z k | c k ∈ C, f H 2 := ∞ k=0 |c k | 2 < ∞, |z| < 1}.
f ∈ H 2 (D) implies that f is holomorphic in D, and, as an important property, the partial sum of the infinite series on the unit circle has a L 2 (∂D)-limit that equals to the non-tangential boundary limit lim r→1-f (re it ) almost everywhere. Those boundary limits form a closed subspace of the L 2 space on the boundary circle, denoted by H 2 (∂D). The mapping from H 2 (D) to H 2 (∂D) is an isometry. We hence identify the space H 2 (∂D) with H 2 (D). If we start from a function f in L 2 (∂D) with Fourier coefficients c k , k = 0, ±1, ±2, ..., then we have the so called Hardy spaces decomposition f = f + + f -, where

f + (e it ) = ∞ k=0 c k e ikt , f -(e it ) = -∞ k=-1 c k e ikt .
If f is real-valued on the circle, then we have

f = 2Re f + -c 0 .
The last relation shows that approximation of functions in L 2 (∂D) can be reduced to that of functions in the Hardy class. In other contexts we have analogous relations, so the case of Hardy spaces on which we concentrate below will be a prototypical example.

In H 2 (D), the normalized reproducing kernels (also known as Szegő kernels) are the rational functions

e a (z) = 1 -|a| 2 1 -az , a ∈ D. (3.1)
Then {e a } a∈D is a dictionary of H 2 (D). Let A = (a 1 , ..., a n ) ∈ D n be a n-tuple. If the a k are all distinct, we associate to A the n-tuple (e a 1 , ..., e a n ). of normalized reproducing kernels.

More generally, if

A consists of m < n distinct points b 1 , • • • , b m where b k is repeated l k times with l 1 + • • • + l m = n, then we associate to A the n-tuple (3.2) (E {b 1 ,1} , ..., E {b 1 ,l 1 } , E {b 2 ,1} , ..., E {b 2 ,l 2 } , • • • , E {b m ,1} , , • • • , E {b m ,l m } )
defined as follows. We set E {0, j} (z) = z j , and if a k 0 then E {a k , j} (z) = c(a k , j) (1-a k z) j , where c(a k , j) is the constant making E {a k , j} = 1. The orthogonalization of (3.2) is the so called Takenaka-Malmquist (TM-) system, or orthogonal rational system, (B 1 , ..., B n ), where

B k (z) = B {a 1 ,...,a k } (z) = 1 -|a k | 2 1 -a k z k-1 l=1 z -a l 1 -a l z [36, Lecture V].
We call each B k a modified Blaschke product. One recognizes that the rational function

√ 1-|a n | 2
1-a n z in front is the Szegő kernel and the product thereafter is the Blaschke product with the zeros a 1 , ..., a k-1 . The claim is: when studying rational approximation in H 2 (D), TM systems are unavoidable. This is no wonder, because every rational function is a linear combination of E b k , j for some b k . Existing studies on TM systems include Laguerre and Kautz systems. The traditional works on TM systems deal with the condition

∞ k=1 (1 -|a k |) = ∞. (3.3)
This is called the hyperbolic non-separability condition. For any p ∈ [1, ∞), it is necessary and sufficient for the corresponding TM system to be complete in H p (D) ( [START_REF] Bultheel | Orthogonal Rational Functions[END_REF]). Also, for 1 < p < ∞ and any sequence (a 1 , ..., a n , ...), the corresponding TM system is a Schauder basis of the closure of span{B n } ∞ n=1 in H p (D) ( [START_REF] Qian | Rational Orthogonal Systems are Schauder Bases[END_REF]). The difference with the current study is that the parameters a 1 , ..., a n , ... used to approximate a given function are not fixed before hand, nor are they required to satisfy the hyperbolic non-separability condition (3.3), and, correspondingly, the induced TMsystem {B n } is not necessarily a basis. Instead, we adaptively select the parameters a n , as in greedy algorithms, and thus formulate expansions of given signals with fast convergence. Below we will introduce our main algorithm in the unit disc case called Adaptive Fourier Decomposition, abbreviated as AFD ( [START_REF] Qian | Adaptive fourier series-a variation of greedy algorithm[END_REF]).

Let f be any function in H 2 (D). Letting f 1 = f, recursively and for any n complex numbers a 1 , ..., a n in D, we have

f (z) = f 1 , e a 1 e a 1 (z) + f 2 (z)
ze a 1 1a 1 z ,

and f 2 (z) = f 2 , e a 2 e a 2 (z) + f 3 (z) z -e a 2 1 -a 2 z ,
etc. so that we arrive at

f (z) = n k=1 f k , e a k B {a 1 ,...,a k } (z) + f n+1 (z) n k=1 z -a k 1 -a k z . (3.4)
This identity gives rise to an interpolating rational function to f for the interpolating points a 1 , ..., a n where repeated points correspond to interpolation with derivatives of the function.

The identity, furthermore, gives rise to fast approximation in energy if one selects a k , once a 1 , ..., a k-1 have been fixed, according to the formula:

a k = arg max{| f k , e a | 2 | a ∈ D}. (3.5)
The following energy relation is to be noted:

f 2 = n k=1 | f k , e a k | 2 + f n+1 2 .
Under the selection criterion (3.5) we can show that in the energy sense ( [START_REF] Qian | Adaptive fourier series-a variation of greedy algorithm[END_REF]), namely in the sense of strong convergence in H 2 (D),

f = ∞ k=1 f k , e a k B k .
The above described AFD, or Core AFD algorithm was published in 2012 and lately, in 2015, found to be equivalent with the Pre-Orthogonal Greedy Algorithm ( [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF]). The motivation of AFD is characterizing positive-instantaneous-frequency decomposition, or mono-component decomposition of signals. There followed two elaborations on AFD of which one is the called unwinding AFD ( [START_REF] Nahon | Phase Evaluation and Segmentation[END_REF][START_REF] Qian | Intrinsic mono-component decomposition of functions: An advance of Fourier theory[END_REF][START_REF] Coifman | Nonlinear phase unwinding of functions[END_REF]) and the other is geablack towards n-best rational approximation ( [START_REF] Qian | Optimal Approximation by Blaschke Forms, Complex Variables and Elliptic Equations[END_REF][START_REF] Qian | Cyclic AFD Algorithm for Best Approximation by Rational Functions of Given Order[END_REF]).

Unwinding AFD dwells on Core AFD and the principle of energy front-loading for Nevanlinna outer functions in one complex variable. The latter principle addresses the following fact: if F(z) = ∞ k=0 c k z k is a H 2 (D)-function with the Nevanlinna factorization (see [START_REF] Garnett | Bounded Analytic Functions[END_REF], [START_REF] Weiss | A derivation of the main results of the theory of Hp-spaces[END_REF]): F(z) = I(z)O(z), where I(z) and O(z) are, respectively, its inner and outer factors, and if we write O(z) = ∞ k=0 d k z k , then there holds for any positive integer N that

N k=0 |c k | 2 ≤ N k=0 |d k | 2 . Since ∞ k=0 |c k | 2 = F 2 = O 2 = ∞ k=0 |d k | 2 ,
the above inequality amounts to saying that the outer part of a Hardy function has best polynomial approximation of degree N in H 2 (D) that converges faster than that of the original function. This suggests that when decomposing a function in H 2 (D), it may be a good strategy to perform the Nevanlinna factorization and then decompose the outer part instead of the original function, to finally multiply back by the inner factor which is a finite Blaschke product, at least when the function is continuous on ∂D. Of course, performing the Nevanlinna factorization is not such an easy business as it essentially involves estimating conjugate functions, and fair judgement should be used in each case.

The related theory is developed in [START_REF] Nahon | Phase Evaluation and Segmentation[END_REF][START_REF] Qian | Intrinsic mono-component decomposition of functions: An advance of Fourier theory[END_REF][START_REF] Coifman | Nonlinear phase unwinding of functions[END_REF]. Experiments show that Unwinding AFD is indeed among the best positive frequency decomposition methods ( [START_REF] Qian | Comparison of Adaptive Mono-component Decompositions[END_REF]). The authors became aware late 2015 that the Ph. D. thesis of M. Nahon ( [START_REF] Nahon | Phase Evaluation and Segmentation[END_REF]) at Yale University, 2000, under the guidance of R. Coifman, develops an analogous unwinding algorithm based on the Nevanlinna factorization ( [START_REF] Garnett | Bounded Analytic Functions[END_REF][START_REF] Weiss | A derivation of the main results of the theory of Hp-spaces[END_REF]). In a recent paper by Coifman and Steinerberger, theoretical aspects of the algorithm are further developed ( [START_REF] Coifman | Nonlinear phase unwinding of functions[END_REF]).

Cyclic AFD was designed to approach the problem of n-best rational approximation in H 2 (D). The problem is formulated as follows. Given f ∈ H 2 (D), find a rational function of the form p/q, with deg{p} and deg{q} not exceeding n, and q zerofree inside the unit disc, such that fp/q is minimum among all possible rational functions of the same kind. The latter are just the rational functions of degree no larger than n in H 2 (D). Existence of such minimizing rational functions was proved a long time ago, but a theoretical algorithm for finding p/q to give rise to the minimum is still an open issue. A detailed account of the problem may be found in [START_REF] Baratchart | Existence and Generic Properties of L 2 Approximants for Linear Systems[END_REF][START_REF] Baratchart | A criterion for uniqueness of a critical point in H 2 rational approximation[END_REF][START_REF] Baratchart | A remark on uniqueness of best rational approximation of degree 1 in L2 of the circle[END_REF][START_REF] Baratchart | Asymptotic uniqueness of best rational aproximants of given degree to Markov functions in L 2 of the circle[END_REF][START_REF] Baratchart | Asymptotic uniqueness of best rational approximations to complex Cauchy transforms in L2 of the circle, Re-cent Treds in Orthogonal Polynomialsand Approximation Theory[END_REF]. Both the RARL2 algorithm (which extends to the matrix-valued case, see www-sop.inria.fr/apics/RARL2/rarl2.html for a description and tutorial as well as [START_REF] Baratchart | Identification and rational L2 approximation, a gradient algorithm[END_REF][START_REF] Baratchart | Critical points and error rank in matrix H2 ra-tional approximation and the strong diRerential consistency of output error identification from white noise inputs[END_REF][START_REF] Fulcheri | Matrix rational H 2 approximation: a gradient algorithm based on Schur analysis[END_REF] for further references) and the one through Cyclic AFD ( [START_REF] Qian | Cyclic AFD Algorithm for Best Approximation by Rational Functions of Given Order[END_REF]) provide practical algorithms. RARL2 is a descent algorithm using Schur parameters to describe Blaschke products of given degree along with a compactification thereof to ensure convergence to a local optimum. It is used in identification and design of microwave devices, see [START_REF] Olivi | Identification of microwave filters by analytic and rational H 2 approximation[END_REF][START_REF] Seyfert | Deembedding of filters in multiplexers via rational approximation and interpolation[END_REF]. The algorithm using Cyclic AFD is parameterized by the zeros of the denominator polynomial, and uses the fact that the expansion as a sum of modified Blaschke products

n k=1 f k , e a k B {a 1 ,...,a k } (z)
is a rational function of degree no larger than n by construction. The theory and algorithmic scheme of Cyclic AFD are definitely simpler (though the 1-D search over a ∈ D iterated at each step to reach a fixed point is nontrivial), but convergence to a local minimum is still an issue. Cyclic AFD corresponds to simultaneous optimal selection of n parameters, while Core AFD corresponds to sequential selection of n parameters ( [START_REF] Qian | Cyclic AFD Algorithm for Best Approximation by Rational Functions of Given Order[END_REF]). For some related studies we refer to [START_REF] Baratchart | A remark on uniqueness of best rational approximation of degree 1 in L2 of the circle[END_REF][START_REF] Baratchart | Minimax principle and lower bounds in H2rational approximation[END_REF][START_REF] Baratchart | Asymptotic uniqueness of best rational approximations to complex Cauchy transforms in L2 of the circle, Re-cent Treds in Orthogonal Polynomialsand Approximation Theory[END_REF][START_REF] Baratchart | Weighted extremal domains and rational approximation[END_REF]. Other algorithms based on fixed point heuristics or balanced truncation of Hankel operators to approach rational H 2 (D)approximation can be found, e.g. in [START_REF] Van Dooren | H2 optimal model reduction of MIMO systems[END_REF][START_REF] Gugercin | H2 model reduction for large-scale linear dynamical systems[END_REF].

The above considerations and algorithms extend to the context of the half-plane rather than the disk, by means of conformal mapping. The reason is that a conformal map from the disk to the half-plane is a rational function of degree 1 (i.e. a Möbius transform) and therefore it preserves rationality and the degree, see [START_REF] Garnett | Bounded Analytic Functions[END_REF]Ch. I]. There is also a parallel approach by using TM systems and the corresponding maximal principle in the half-plane. More generally, the relevant extension of what precedes to more general domains is that of best meromorphic approximation with n poles, see [START_REF] Baratchart | 2-D inverse problems for the Laplacian: A meromorphic approximation approach[END_REF].

For functions of multivalent variables, finding a basis is equivalent to finding a uniqueness set. That is a fundamental task and therefore of great interest in mathematical analysis. It is, however, in many cases difficult to achieve. On the other hand, Szegő kernels are usually simple rational functions, and fast representation of signals as linear combinations of Szegő kernels has great significance in relation to applications. The principles outlined in the last two sections are valid and the results available in a number of contexts for several real and complex variables, and with functions valued in vectors and matrices. We give a brief introduction to this circle of ideas in the following sections.

Quaternionic and Clifford Contexts for Functions of Several Real Variables

Denote by Q the quaternion algebra, and by A m the Clifford algebra of linear dimension 2 m . The analogues of AFD have been formulated for Hardy spaces in the setting of Q and of A m in [START_REF] Qian | Adaptive fourier decomposition of functions in quaternionic hardy spaces[END_REF] [START_REF] Wang | Approximation of monogenic functions by higher order szegö kernels on the unit ball and the upper half space[END_REF] respectively. The first setting is very much in the spirit of AFD.

The second, however, is more in the spirit of General Greedy Algorithm with dictionary consisting of the higher order Szegő kernels, due to the fact that the inner product of Clifford algebra-valued functions is not necessarily scalar-valued. That prevents the G-S orthogonalization process from being applied.

Define R m+1

+ = {x = (x 0 , x) ∈ R m+1 ; x 0 > 0, x ∈ R m }.
We briefly introduce the AFD (General Greedy Algorithm) in the Hardy space H 2 (R 4 + ) of the upper-half space (resp. H 2 (R m+1 + )), which consists of Q-valued (resp. A m -valued) left monogenic functions [START_REF] Gilbert | Clifford algebras and Dirac operators in harmonic analysis[END_REF]. For the parallel theory in the Hardy spaces on the unit ball, see [START_REF] Qian | Adaptive fourier decomposition of functions in quaternionic hardy spaces[END_REF][START_REF] Wang | Approximation of monogenic functions by higher order szegö kernels on the unit ball and the upper half space[END_REF]. Denote by φ a (x) the non-normalized Szegő kernel of

H 2 (R 4 + ) (resp. H 2 (R m+1 + )),i.e. φ a (x) = x+a |x+a| 4 (resp. φ a (x) = x+a |x+a| m+1 ). Regarding H 2 (R 4 + ), although Q is a non-commutative algebra, it is not difficult to apply the P-OGA to {φ a (x), a, x ∈ R 4 + } to obtain an orthonormal system {B n } ∞ n=1 parameterized by the selected sequence {a (n) } ∞ n=1 in R 4 + . Similar to AFD, one has lim n→∞ || f -n k=1 B k f, B k || = 0.
We note that the study of AFD in the setting of Q predates that of P-OGA: the latter is a generalization of the former.

As to H 2 (R m+1 + ), we introduce the completion of the Szegő kernel dictionary given by

D = ψ β,a (x) ||ψ β,a || , β = (β 1 , ..., β m+1 ), a, x ∈ R m+1 + , where ψ β,a (x) = ∂ |β| ∂a β 1 0 •••∂a β m+1 m+1 φ a (x). For each f ∈ H 2 (R m+1 + ), applying the General Greedy Algorithm with D one has lim n→∞ || f -n k=1 ψ β (k) ,a (k) ||ψ β (k) ,a (k) || g k , ψ β (k) ,a (k)
||ψ β (k) ,a (k) || || = 0, where g k is the standard remainder defined in §2.

The sphere cases are also considered in the quaternionic and Clifford algebra settings ( [START_REF] Qian | Adaptive fourier decomposition of functions in quaternionic hardy spaces[END_REF][START_REF] Wang | Approximation of monogenic functions by higher order szegö kernels on the unit ball and the upper half space[END_REF]).

In both settings it is shown that a global maximal selection of the parameter is attainable at each step of the recursive process (i.e. ρ = 1). In particular, one can obtain rational approximations of functions in L 2 (R 4 ) by applying the corresponding AFD and the well-known Sokhotskyi-Plemelj formula (e.g. [START_REF] Gilbert | Clifford algebras and Dirac operators in harmonic analysis[END_REF][START_REF] Qian | Adaptive fourier decomposition of functions in quaternionic hardy spaces[END_REF]).

Several Complex Variables

Similar approximation schemes in settings involving several complex variables have also been studied. In fact, one can consider functions defined on various classical domains, with values in C N , or C N×M matrices, etc. 5.1. Functions Defined on n-Torus. Denote by T n the n-torus, where T = ∂D. There are two generalizations of AFD in the Hardy space of the n-torus H 2 (T n ) ( [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF]). One merely consists in processing P-OGA in this context (i.e. it is shown that P-OGA is applicable to H 2 (T n )). The other is based on the product-TM system. As to the former, we omit the details as it should be clear from the previous section already how to perform. Below, we give a brief introduction to the latter. For simplicity, we consider only the case n = 2.

Denote by B a k (z) the modified Blaschke product (a member of the TM system) associated with the sequence a = {a k } ∞ k=1 in D. We introduce the tensor product type modified Blaschke product {B a k (z) ⊗ B b l (w)}, where a, b ⊂ D and (z, w) ∈ D 2 = D × D. For f ∈ H 2 (T 2 ), we look for a rational approximation of f of separable type given by f ) is proved in [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF] through a technical discussion. In a way similar to previously described AFD, S m ( f ) converges to f in the H 2 -norm if each (a k , b k ) is selected according to criterion (5.1). As an application, one can obtain rational approximations of functions in L 2 (T 2 ). 5.2. Functions Defined on R n in the Setting of Hardy Spaces on Tubes. As mentioned in the previous sections, one can obtain rational approximations of functions in L 2 of the boundary of a domain by applying AFD in the domain. Since R n can be written as the union of 2 n octants, seeking rational approximations of functions in L 2 (R n ) motivates the study of AFD in Hardy spaces on tubes over octants ( [START_REF] Mai | Rational approximation of functions in the hardy spaces on tubes[END_REF]). For the purpose of illustration, it suffices here to investigate AFD in the Hardy space of the tube over the first octant H 2 (T Γ 1 ), where

= lim m→∞ 1≤k,l≤m f, B a k ⊗ B b l B a k ⊗ B b l = lim m→∞ S m ( f ) in the H 2 -norm. Denote by D m ( f ) = S m ( f ) -S m-1 ( f ) the m-partial sum difference.
T Γ 1 = C + × • • • × C + .
Denote by S z (w) the Cauchy-Szegő kernel of H 2 (T Γ 1 ), i.e. S z (w) = n j=1 -1

2πi(w k -z k ) . By using the methodology P-OGA, one can obtain an orthonormal system {B k } ∞ k=1 parameterized by the sequence {z

(k) } ∞ k=1 ⊂ T Γ 1 . Indeed, {B k } ∞ k=1
is the G-S orthogonalization of the selected Cauchy-Szegő kernels and, if necessary, their higher order directional derivatives. It is concluded in [START_REF] Mai | Rational approximation of functions in the hardy spaces on tubes[END_REF] that the attainability of a global maximal selection at each step follows from a certain kind of boundary behavior of functions in H 2 (T Γ 1 ), called "boundary vanishing condition (BVC)" (also see [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF]). After verifying the BVC in H 2 (T Γ 1 ), the convergence follows from the general theory given in [START_REF] Qian | Two-dimensional adaptive fourier decomposition[END_REF]. In [START_REF] Mai | Rational approximation of functions in the hardy spaces on tubes[END_REF] the P-OGA generalization of AFD in the Hardy spaces on tubes over regular cones is also given.

5.3.

Matrix-valued Signals Defined in the Unit Disc. We denote by H p×q 2 the space of p × q matrices with entries in H 2 (D). In a recent paper of D. Alpay, F. Colombo, T. Qian and I. Sabadini they show that it is possible, as in the scalar case, to decompose those functions as linear combinations of suitably modified matrix-valued Blaschke product, in an adaptive way. The procedure is based on a generalization to the matrix-valued case of the maximum selection principle of 1-D AFD, which involves not only selections of suitable points in the unit disc but also suitable orthogonal projections. It can be shown that the maximum selection principle again gives rise to a convergent algorithm ( [START_REF] Alpay | Adaptive orthonormal systems for matrix-valued functions[END_REF]). The analogous parametrization in terms of Schur analysis and tangential interpolation directions was given earlier in [START_REF] Baratchart | On the differential structure of matrix-valued inner functions[END_REF], and has been used to design a matrix-valued version of the RARL2 algorithm, see [START_REF] Fulcheri | Matrix rational H 2 approximation: a gradient algorithm based on Schur analysis[END_REF]. 5.4. Adaptive Decomposition: the Case of the Drury-Arveson Space. Blaschke factors and products have counterparts in the unit ball of C N , and this fact allows us to extend the maximum selection principle to the case of functions in the Drury-Arveson space of functions analytic in the unit ball of C n . This gives rise to an algorithm which is a variation of the higher dimensional AFD. In the corresponding paper of D. Alpay, F. Colombo, T. Qian and I. Sabadini they also introduce infinite Blaschke products in this setting and study their convergence ([2]). 5.5. Matrix-valued Signals Defined on the Polydisc. The polydisc case has been given special attention, due to its connection with image processing. It develops in the context of multi-trigonometric series, like 2-D AFD treated in an earlier subsection. An alternative setting is given in the third paper of D. Alpay, F. Colombo, T. Qian and I. Sabadini where they develop interpolation theory as well as an operator-valued Blaschke product method that offers an adaptive expansion of holomorphic functions in the Hardy space over the polydisc corresponding to signals on the n-torus ([3]).

AFD and Aveiro Method in Reproducing kernel Hilbert Spaces

We first note that the Hardy H 2 space is a reproducing kernel Hilbert space, where the reproducing kernel is given by the Szegő kernel. Subsequently, P-OGA was proposed as an expansion algorithm in reproducing kernel Hilbert spaces (see §2) although in the previous sections we restrict ourselves to AFD in various Hardy spaces. The key of AFD (or P-OGA) is the construction of an orthonormal system by applying the G-S orthogonalization process to the selected reproducing kernels and their higher order derivatives. Such a construction ensures that the approximating function and its derivatives meet interpolation conditions to the approximated function at the selected points.

The study of interpolating functions is closely related to interpolation and sampling problems in reproducing kernel Hilbert spaces, some prototypical aspects of which may be found in [START_REF] Seip | Interpolation and sampling in spaces of analytic functions[END_REF]. We use the notation H K to indicate that H is a reproducing kernel Hilbert space admitting a reproducing kernel K(q, p). Suppose that H K consists of holomorphic functions defined in an open set E ⊂ C. Let further {p k } ∞ k=1 ⊂ E be a sequence of distinct points. The so-called Aveiro Method, proposed by S. Saitoh et al in [START_REF] Castro | A new discretization method by means of reproducing kernels[END_REF], aims at constructing an approximating function to f ∈ H K involving a finite number of sampling points {p 1 , ..., p n }.

Based on this work, the authors of [START_REF] Mai | Aveiro method in reproducing kernel hilbert spaces under complete dictionary[END_REF] propose the so-called "Aveiro Method under complete dictionary (AMUCD)" by combining the ideas of P-OGA with Aveiro Method. Roughly speaking, AMUCD enhances the power of Aveiro Method in that the approximating function given by AMUCD does not require all elements of {p k } ∞ k=1 to be distinct. As in AFD, the representation and its derivatives enjoy interpolation properties at {p k } ∞ k=1 . It is shown in [START_REF] Mai | Aveiro method in reproducing kernel hilbert spaces under complete dictionary[END_REF] that AMUCD is applicable to the classical Hardy spaces and Paley-Wiener spaces. It turns out that AMUCD is, in fact, an alternative representation of AFD. Nevertheless, AMUCD has the advantage not to require working out the related orthonormal system, whereas in many instances of P-OGA one does not know explicit formulas for the related orthonormal system of functions.

  and e k is chosen so that | gk , e k | ≥ ρ sup{| gk , e | | e ∈ D}.

  )e a n e a (l)e a n -P {a 1 ,...,a n-1 } e a (l)e a n = e a (l) -e an e a (l) -e an -P {a 1 ,...,a n-1 } e a (l) -e an e a (l) -e an e a (l) -e an e a (l) -e an -P {a 1 ,...,a n-1 } e a (l) -e an e a (l) -e an .

  The main step is to select (a * m+1 , b * m+1 ) ∈ D 2 according to the maximal problem (a * m+1 , b * m+1 ) := arg sup (a m+1 ,b m+1 )∈D 2 ||D m+1 ( f )|| 2 , (5.1) where {a 1 , ..., a m } and {b 1 , ..., b m } are previously fixed. The existence of (a * m+1 , b * m+1

  Denote by P {a 1 ,...,a n-1 } the projection operator from H to span{B 1 , ..., B n-1 }. Now, there are two cases.1. The limiting point e a n is not in span {B 1 , ..., B n-1 }. In such a case, a n , in particular, does not coincide with any of a 1 , ..., a n-1 , and B a n {a 1 ,...,a n-1 } is just given by , ..., B n-1 , B a n {a 1 ,...,a n-1 } } is the G-S orthogonalization of {B 1 , ..., B n-1 , e a n }. 2. The limiting point e a n lies in span {B 1 , ..., B n-1 }. In particular, if a n coincides

	6)
	Next, we compute
		lim l→∞	B a (l) {a 1 ,...,a n-1 } .
	(2.7)	B a n {a 1 ,...,a n-1 } =	e a n -P {a 1 ,...,a n-1 } e a n e a n -P a 1 ,...,a n-1 e a n	,
	and {B 1


B Talin
email: btalin@piima1.univ-mrs.fr

E Dufour

A Calisti

M A Gigosos

M A González

Del R Ío Gaztelurrutia

J W Dufty+

Molecular dynamics simulation for modeling plasma spectroscopy

Keywords: numbers: 52.65.Yy, 32.70.Jz, 52.27.Aj

The ion-electron coupling properties for a ion impurity in an electron gas and for a two component plasma are carried out on the basis of a regularized electron-ion potential removing the short-range Coulomb divergence. This work is largely motivated by the study of radiator dipole relaxation in plasmas which makes a real link between models and experiments. Current radiative property models for plasmas include single electron collisions neglecting charge-charge correlations within the classical quasi-particle approach commonly used in this field. The dipole relaxation simulation based on electron-ion molecular dynamics proposed here will provide means to benchmark and improve model developments.

Benefiting from a detailed study of a single ion imbedded in an electron plasma, the challenging two-component ion-electron molecular dynamics simulations are proven accurate. They open new possibilities to obtain reference lineshape data.

Introduction

In this paper classical ion-electron plasma models are used to carry out studies preliminary to line spectra simulations for ion emitters. A classical plasma model involves point charges, sometimes in a uniform charge background, interacting through given additive binary forces and parameters characterizing densities and temperatures. It is assumed that quantum mechanics are either negligible or can be incorporated through appropriate modification of the classical Coulomb potential. An example of such a classical plasma is the one component plasma (OCP) developed to study the statistical properties of non degenerate ion fluid. The present study deals with ions of charge Ze and electrons. The inclusion of electrons requires essential quantum mechanical effects to eliminate the divergence of the attractive ion-electron Coulomb potential in the classical plasma. This is accomplished by using a regularized Coulomb potential which remains finite at short distances, representing quantum diffraction effects [1], and allowing the derivation of all the desired properties using the laws of classical mechanics. A detailed study of the time independent statistical properties of an electron gas with an ion impurity has been carried out recently on the basis of such a regularized electron-ion potential [2]. Two standard classical methods have been used for this study: molecular dynamics (MD) and hypernetted chain approximation (HNC) allowing to perform both cross comparisons and interpretations of the results. A first interpretation of the dynamical properties of the electron field autocorrelation function at the impurity is reported in this issue [START_REF] Dufty | Calisti A and[END_REF]. Beside structure and correlation functions MD can provide realistic representations of these stochastic electric forces at the ion. Time dependent electric fields are the necessary ingredients to describe Stark broadening (dipole relaxation) of spectral lines emitted by ions in plasmas. It is of great interest to describe these fields by simulation in order to obtain lineshapes accounting for all the correlations between charged particles. Such simulated lineshapes would provide essential reference data to benchmark more efficient phenomenological lineshape models developed for plasma diagnostics via spectroscopy. Several recent models for line broadening have been developed for investigations at new plasma conditions. They suggest that part of the discrepancies found with experiments is an inadequate description of the ion and electron perturber coupling mechanisms. The main objective of our program is to perform MD simulations with ions of charge Z ≥ 1 and electrons in order to derive all the relevant data required for lineshape simulation accounting for all the interactions between charges. The parameter range explored in this work is chosen to be compatible with hot and dense plasmas diagnostics based on spectroscopy. This is complementary to other MD simulation studies of dense hydrogen [START_REF] Pschiwul | [END_REF]5] with Z = 1 and a much larger electron-electron coupling strength. Two examples are provided to show the potential for application of MD simulation to spectral line broadening. In the next two sections the effects of electron-electron correlations on electron broadening of spectral lines is studied for a massive radiator in an electron gas. All ion broadening is neglected in order to better isolate the electron coupling effects. Next, in section 4 the more realistic case of a radiator in a two component plasma (TCP) of electrons and ions is considered. It is shown that separating the effects of electron and ion broadening, as done in most current theories, requires some care and interpretation.

Ion impurity in an electron gas

In this and the following section we consider a single ion impurity of infinite mass imbedded in an electron gas with an additional uniform positive background for charge neutrality. As mentioned above a more detailed study has been reported elsewhere so only a few remarks will be recalled here. The main statistical properties at the ion are derived using both MD simulation and HNC approximation. The ion-electron Coulomb regularized potential postulated is:

V ie (r) = -Ze 2 (1 -e -r/δ )e -r/λ /r (1)
where δ = (2πh 2 /m e k B T ) 1/2 is the De Broglie wavelength and λ is a decay length long enough to represent Coulomb correlations but short enough to allow periodic boundary conditions (see below). Other parameters are the average electron-electron distance r 0 = (3/4πN e ) 1/3 defined in terms of the electron density N e , the electronelectron coupling constant Γ, the electron plasma frequency ω p , and the ion-electron potential at the origin which, in units of r 0 , becomes: σ = ZΓ/δ. For simplicity, the OCP electron-electron potential is taken to be V ee (r) = e 2 e -r/λ /r. Molecular dynamics simulations are carried out using a number N of electrons in the elementary cubic cell of size c large enough to ensure the following constraints: Z << N , λ > λ D where λ D is the Debye wavelength and λ ≤ c/2. These conditions are required in order that the Coulomb screening mechanisms actually reduce the ionion interaction length. The hypernetted chain set of equations [6] is written for the single ion case, i.e. for a vanishing ion density. As a result, the ion-electron pair distribution function g ie (r) is obtained by iteration as a function of the OCP direct electron-electron correlation function c ee (r) calculated separately. Both MD simulation and HNC approximation are restricted to a limited range of values for the ion charge, density, and temperature. Outside of this range electron trapping and non-convergent iteration could occur invalidating the results. For the parameter space considered MD and HNC results are in good agreement as illustrated in Fig 1 . The electron accretion around the ion impurity shown by the shape of the ion-electron pair distribution function is the expression of screening mechanisms taking place due to the attractive forces. It will be indirectly shown later that for these plasma parameters the screening length is the Debye length.

Implication for dipole relaxation mechanisms

Most frequently models for Stark effect in plasmas are semiclassical, i.e. they rely on the hypothesis that line broadening can be given by the evolution of a quantum system -the emitter -perturbed by the classical plasma electric micro-field. In addition it is supposed that the emitter size is small in respect to the micro-field space fluctuation so that this coupling occurs via dipole interaction with the total electric micro-field of the plasma. Due to the small electron-ion mass ratio, the Stark effect resulting from slow ion and fast electron micro-fields are generally considered separately. In this way the ion Stark effect is approximated by a static field model, while electron broadening is described as a dynamical process resulting from the high frequency stochastic fluctuation of the electron micro-field. In order to illustrate the implication of the present study for line broadening attention is restricted in this section to electron broadening alone, using the simple case of a massive emitter in an electron gas. The usual approach for electron broadening for ion emitters in plasmas relies on a binary collision model involving independent electrons moving on straight trajectories at constant velocity. Electron correlations are accounted for indirectly by screening the electron-emitter potential at the Debye length. The discussion of this approximation is not the purpose here, but rather it is mentioned to motivate the following comparison. Two MD simulations are performed to calculate the electron electric field at the emitter, followed by calculation of the spectral line shape: 1) the electrons are considered as free particles that do not interact with each other but perturb the ion emitter with a field obtained from the potential written in (1) with λ = λ D ; 2) the electrons have Coulomb interactions with each other and couple to the emitter with a field obtained using in (1) with λ = c/2. The latter is, of course, the correct treatment of correlations. Given the fields from these MD calculations a lineshape simulation follows. Although generally not coded in that way this simulation process is equivalent, first, to solve numerically a stochastic equation for the quantum system evolution operator:

dU fi (t) dt = -iL fi (t)U fi (t) U(t) = {U fi (t)} av = 1 N N i=1 U fi (t) (2)
where {f 1 , f 2 ...f N } is a sample set of independent perturbing field histories and L fi (t) is an operator involving the coupling of the emitter dipole with the external field f i . Then, the lineshape results from Fourier transform of the emitter dipole autocorrelation function. It is written in Liouville space as a scalar product in terms

I(ω) = 1 π ∞ 0 d|U(t)|ρd dt (3)
These simulations require a fast integration process based on an earlier work [7] carried out for hydrogenic lines. The example reported is the Balmer alpha line emitted by an hydrogen-like helium emitter in an electron gas with the density N e = 10 18 cm -3 and the temperature T e = 60000K. Figure 2 shows comparisons of independent electrons (case 1) and coupled electrons (case 2). From the field autocorrelation functions plotted on Fig 2-a E(t)•E(0) /E 2 0 , where brackets indicate a classical Gibbs ensemble average and E 0 is the mean electronic field, it can be inferred that the field de-correlation rate of case 1 is lower than for case 2. An alternative interpretation is that the field fluctuation rate is larger when charge-charge correlations are accounted for than if they are not. As the broadening mechanism is mainly a dynamical process, a larger field fluctuation rate results in a narrower line, as shown on Fig. 2b. However, this behavior found for an ionic charge Z = 1 cannot be generalized straightforwardly since the broadening effect depends not only on the field fluctuation rate but also on the field covariance which is an increasing function of Z.

Two component plasma MD simulations

Intuitively, the electron microfield properties investigated above for a single ion in an electron gas will be different if the positive background assumed for charge neutrality is replaced by mobile ions. In this section a charge-neutral two-component plasma (TCP) of electrons and ions of charge Ze will be considered. Classical MD simulation is used to account for correlations among all charges and cross comparisons performed for the single ion case ensure the accuracy of these TCP simulations. The challenging aspect of this study is to move slow and fast particles at the same time. This requires simulations stable over a long period of time, long enough to account for ion motion and based on a time step compatible with electron motion. These simulations are expensive as the particle number in the elementary cubic cell, N i ions and ZN i electron, is large. However, a few simulations have been carried out successfully. That for the same plasma conditions than in Fig 1 is reported here. Before proceeding it is worth contrasting such a MD simulation with other methods of describing the TCP. Simulations involving ions and electrons of a two component plasma have been carried out with the so called quantum molecular dynamics [8,[START_REF]Dharma-Wardana C in Strongly Coupled Plasmas[END_REF]. At each time step the electrons are equilibrated about the ion configuration (Born-Openheimer approximation) using an appropriate density functional model. The ion-ion forces are then calculated from the resulting electron density and these forces are used in classical MD to move the ions. The process is repeated at each time step. Alternatively, the electrons can be equilibrated by the Carr-Parinello method. In both cases the electron dynamics is fully equilibrated at each time step. The advantage of MD simulation, although limited by the semi-classical potential, is that both electrons and ions move dynamically according to the given equations of motion. It appears that the two classes of methods are complementary, with quantum molecular dynamics being more appropriate for lower temperature and MD more appropriate at higher temperatures.

Screening effects

The first point of interest is to verify that the expected ion-electron screening mechanisms take place properly. As noted before, in the MD elementary cubic cell the charge-charge interaction length is half the cell size while the expected effective screening length (Debye) is only a fraction of it. A comparison of the ion-ion pair distribution function for the TCP and for the corresponding ion OCP is made. In the OCP system the ion-ion Coulomb potential is screened at the Debye length, while such screening must be generated by the electrons in the TCP simulations. Figure 3 shows a good agreement between the two simulations giving an indirect evidence of the effectiveness of the ion-ion screening by the electrons. Other calculations not reported here show that the agreement is preserved as well when the temperature varies.

Field dynamical properties

Molecular dynamics simulation allows determination of statistical data for each kind of particle involved. In the electron-ion TCP two kinds of time dependencies coexist, the high frequency and the low frequency dynamics related to electron motion and ion motion, respectively. Both of them are of interest for probing or discussing the simulation accuracy. For the high frequency data the electron field autocorrelation function (FAF) at the ions has been calculated. Figure 4-a shows a comparison of the field autocorrelation function for a single ion in electrons and for the TCP. It can be observed that for the TCP the FAF does not go to 0 as for the impurity case but, at the high frequency time scale used for Fig 4-a, keeps a constant value that denotes the occurrence of a low-frequency component in the electron field. At a low-frequency dynamics time scale it will be seen below that, as expected, this constant value goes to 0. The low-frequency component can be obtained averaging the electron field at the ions on a period of time long enough for the field to loose correlation of the high frequency fluctuation. Contrary to the single ion case, a given ion is no longer a center of symmetry for the smooth electron density that could result from the same high frequency dynamics time average. Due to the screening mechanisms, the structure of this smooth electron density is a blurred image of the ion configuration which remains quasi-static at this time scale. The primary consequence of the occurrence of a low-frequency component in the TCP electron field is that it can no longer be used in lineshape simulations. This will be a motivation to carry out lineshape simulations accounting for the total ion plus electron field. In order to investigate how the low-frequency component is driven by the ion configuration several FAF have been plotted on Fig 4-b: 1) the field autocorrelation function of the low-frequency component, labelled "electron TCP"; 2) the ion FAF for the ion OCP; 3) the ion FAF for the TCP. It can be noted that the low-frequency component FAF goes to 0 with nearly the same rate than the ion FAF curves. For the TCP the ion-ion interaction length is the cell size while for the OCP it is the Debye length. This induces an inadequacy of the actual ion-ion coupling and the pure ion-ion field. Therefore, the ion TCP field appears useless when considered without the screening electron field i.e. without the low-frequency component. This explains the quite large discrepancy for the initial value of the two ion FAF curves. Some differences occur also between the dynamical behavior of the various partial fields. Further results and analysis will be given elsewhere.

Discussion

Classical molecular dynamics simulations of a two component electron-ion plasma based on a regularized potential have been carried out successfully. The step by step motion of slow ions and fast electrons considered at the same time is inherently time consuming. However, MD is a unique way for providing static and time dependent statistical data accounting for all the charge-charge couplings. Among them, the screening mechanisms of forces take place accurately despite the limited size of the MD simulation cell. The genuine MD microfields are directly utilizable in semiclassical linebroadening calculations providing the means to build a clear link between theoretical models and plasma spectroscopy experiments. An example of such a calculation has been reported here for the simple case of electron broadening of a single ion in an electron gas. It shows the magnitude, virtually observable, of the discrepancy induced by the independent quasi-particle approximation. Line profile simulations based on the two component plasma will be a unique way to perform reference data for further developments of fast lineshape models.

Figure 1 .

 1 Figure 1. CVI impurity ion-electron pair correlation function: Z = 5, Ne = 10 21 cm -3 and Te = 232000 0 K from HNC: solid line, MD: circles

Figure 2 .

 2 Figure 2. HeII impurity for independent electrons (dashed line) and coupled electron (solid line): Z = 1, Ne = 10 18 cm -3 and Te = 60000K. a) field autocorrelation function, b) Balmer alpha line

Figure 3 .

 3 Figure 3. CVI ion-ion pair correlation function, same conditions as in Fig 1; TCP: solid line, ion OCP: circles

Figure 4 .

 4 Figure 4. (same as in Fig 1 and 3) a) electron FAF: single ion case and TCP, b) time averaged electron ×5, ion OCP and ion TCP field auto correlation functions

Acknowledgments

Support for this research has been provided by the U. S. Department of Energy grant No. DE-FG03-98DP0218. J. Dufty is grateful for the support and hospitality of the University of Provence.