Salim Bouzebda
email: salim.bouzebda@upmc.fr

Omar El-Dakkak
email: omar.eldakkak@gmail.com

Approximations for general bootstrap of empirical processes with an application to kernel-type density estimation

Keywords: General bootstrap, Brownian bridge, Best approximation, kernel density estimator AMS Classifications: 62G30, 60F17

. As a consequence, we establish an approximation of the bootstrapped kernel-type density estimator.

Introduction and Main Results

Let X 1 , X 2 , . . . be a sequence of independent, identically distributed [i.i.d.] random variables with common distribution function F (t) = P (X 1 ≤ t). The empirical distribution function of X 1 , . . . , X n is

F n (t) = 1 n n i=1 1{X i ≤ t}, -∞ < t < ∞, (1)
where 1{A} stands for the indicator function of the event A. Given the sample X 1 , . . . , X n , let X * 1 , . . . , X * m , be conditionally independent random variables with common distribution function

F n . Let F m,n (t) = 1 m m i=1 1{X * i ≤ t}, -∞ < t < ∞, (2)
denote the classical Efron (or multinomial) bootstrap (see, e.g. [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF] and [START_REF] Efron | An introduction to the bootstrap[END_REF] for more details). Define the bootstrapped empirical process, α m,n , by

α m,n (t) := √ n(F m,n (t) -F n (t)), -∞ < t < ∞.
(3) Among many other things, [START_REF] Bickel | Some asymptotic theory for the bootstrap[END_REF] established weak convergence of the process in (3), which enabled them to deduce the asymptotic validity of the bootstrap method in forming confidence bounds for F (•). [START_REF] Shorack | Bootstrapping robust regression[END_REF] gave a simple proof of weak convergence of the process in (3) [see also [START_REF] Shorack | Empirical processes with applications to statistics[END_REF], Section 23.1]. The Bickel and Freedman result for α m,n has been subsequently generalized for empirical processes based on observations in R d , d > 1 as well as in very general sample spaces and for various set and function-indexed random objects [see, for example [START_REF] Beran | Bootstrap methods in statistics[END_REF], [START_REF] Beran | Confidence sets for a multivariate distribution[END_REF], [START_REF] Beran | Convergence of stochastic empirical measures[END_REF], [START_REF] Gaenssler | Confidence bands for probability distributions on Vapnik-Chervonenkis classes of sets in arbitrary sample spaces using the bootstrap[END_REF], [START_REF] Lohse | Consistency of the bootstrap[END_REF]]. This line of research found its "final results" in the work of Giné andZinn (1989, 1990) and [START_REF] Csörgő | Bootstrapping empirical functions[END_REF]. By now, the bootstrap is a widely used tool and, therefore, the properties of α m,n (t) are of great interest in applied as well as in theoretical statistics. In fact, several procedures can actually be described in terms of the empirical process α n (t), the limit distributions being functionals of B(F (t)), where B is a Brownian bridge.

The fact that the limits may depend on the unknown distribution F (t) makes it important that good approximations of these limiting distributions be found and that is where the bootstrap proved to be a very effective tool.

There is a huge literature on the application of the bootstrap methodology to nonparametric kernel density and regression estimation, among other statistical procedures, and it is not the purpose of this paper to survey this extensive literature. This being said, it is worthwhile mentioning that the bootstrap as per Efron's original formulation (see [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF]) presents some drawbacks. Namely, some observations may be used more than once while others are not sampled at all. To overcome this difficulty, a more general formulation of the bootstrap has been devised: the weighted (or smooth) bootstrap, which has also been shown to be computationally more efficient in several applications. For a survey of further results on weighted bootstrap the reader is referred to [START_REF] Barbe | The weighted bootstrap[END_REF]. Exactly as for Efron's bootstrap, the question of rates of convergence is an important one (both in probability and in statistics) and has occupied a great number of authors (see [START_REF] Csörgő | Strong approximations in probability and statistics[END_REF], [START_REF] Komlós | An approximation of partial sums of independent RV's and the sample DF[END_REF][START_REF] Horváth | Approximations for weighted bootstrap processes with an application[END_REF] and the references therein).

In this note, we will consider a version of the Mason-Newton bootstrap (see [START_REF] Mason | A rank statistics approach to the consistency of a general bootstrap[END_REF], and the references therein). As will be clear, this approach to bootstrap is very general and allows for a great deal of flexibility in applications. Let (X n ) n≥1 be a sequence of i.i.d. random variables defined on a probability space (Ω, A, P). We extend (Ω, A, P) to obtain a probability space (Ω (π) , A (π) , P ). The latter will carry the independent sequences (X n ) n≥1 and (Z n ) n≥1 (defined below) and will be considered rich enough as to allow the definition of another sequence (B * n ) of Brownian bridges, independent of all the preceding sequences. The possibility of such an extension is discussed in detail in literature; the reader is referred, e.g., to [START_REF] Csörgő | Strong approximations in probability and statistics[END_REF], [START_REF] Komlós | An approximation of partial sums of independent RV's and the sample DF[END_REF] and [START_REF] Berkes | An almost sure invariance principle for the empirical distribution function of mixing random variables[END_REF]. In the sequel, whenever an almost sure property is stated, it will be tacitly assumed that it holds with respect the the p.m. P defined on the extended space.

Define a sequence (Z n ) n≥1 of i.i.d. replicae of a strictly positive random variable Z with distribution function G(•), independent of the X n 's. In the sequel, the following assumptions on the Z n 's will prevail:

(A1) E(Z) = 1; E(Z 2 ) = 2 (or, equivalently, Var(Z) = 1).
(A2) There exists an ε > 0, such that E(e tZ ) < ∞ for all |t| ≤ ε.

For all n ≥ 1, let T n = Z 1 + • • • + Z n and define the random weights,

W i;n := Z i T n , i = 1, . . . , n. (4)
The quantity

F * n (t) = n i=1 W i;n 1{X i ≤ t}, for -∞ < t < ∞.
(5) will be called generalized (or weighted) bootstrapped empirical distribution function. Analogously, recalling the empirical process based on X 1 , . . . , X n ,

α n (t) = n 1/2 (F n (t) -F (t)), -∞ < t < ∞, (6)
define the corresponding generalized (or weighted) bootstrapped empirical process by

α * n (t) = n 1/2 (F * n (t) -F n (t)), -∞ < t < ∞. (7)
The system of weights defined in (4) appears in [START_REF] Mason | A rank statistics approach to the consistency of a general bootstrap[END_REF], p.1617 where it is shown that it satisfies assumptions (W I ), (W II ) and (W III ) on p.1612 of the same reference, so that all the results therein hold for the objects to be treated in this note. In particular, weak convergence for the process α * n to a Brownian bridge is proved. For more results concerning this version of the the weighted boostrapped empirical process, we refer the reader to [START_REF] Deheuvels | Asymptotic certainty bands for kernel density estimators based upon a bootstrap resampling scheme[END_REF]. Note that, as a special case of the system of weights we are considering, one can obtain the one used for Bayesian bootstrap (see [START_REF] Rubin | The Bayesian bootstrap[END_REF]).

In what follows, we obtain a KMT rate of convergence for this process in sup norm. More precisely, we consider deviations between the generalized bootstrapped empirical process {α * n (t) : t ∈ R} and a sequence of approximating Brownian bridges {B * n (F (t)) : t ∈ R} on R. Our main result goes as follows.

Theorem 1 Let assumptions (A1) and (A2) hold. Then, it is possible to define a sequence of Brownian bridges {B * n (y) : 0 ≤ y ≤ 1} such that, for all ε, η > 0, there exists N = N (ε, η), such that, for all n ≥ N and all x > 0,

P sup -∞<t<∞ |α * n (t) -B * n (F (t))| > 3n -1/2 (K 1 log n + x) ≤ K 2 exp - K 3 x (1 + ε) 2 + η, (8)
where K 1 , K 2 and K 3 are positive universal constants.

The proof of Theorem 1 is given in Section 3.

Remark 1 Theorem 1 implies the following approximation of the weighted bootstrap:

sup -∞<t<∞ |α * n (t) -B * n (F (t))| = O P log n n 1/2 . ( 9
)
Remark 2 Theorem 1 turns out be useful in obtaining confidence bands for the distribution function of the sample data. We formalize this idea as follows: for 0 < α < 1, one has

lim n→∞ P sup -∞<t<∞ √ n|F n (t) -F (t)| ≤ c(α) = P sup -∞<t<∞ |B(F (t))| ≤ c(α), . (10)
Note that for each fixed t, B(F (t)) is a zero-mean Gaussian random variable with covariance structure

E(B(F (t))B(F (s))) = F (t ∧ s) -F (t)F (s)
where t ∧ s := min(t, s). In practice, c(α) can, of course, not be computed since the covariance structure of B(F (t)) depends on the unknown cdf F . Instead, suppose (Z

(1) 1 , . . . , Z (1)
n ), . . . , (Z

(N ) 1 , . . . , Z (N )
n ) are N independent vectors of i.i.d. copies of Z, sampled independently of the X i 's. Define the random variables

ψ j := sup -∞<t<∞ α * n,j (t) , j = 1, . . . , N, (11)
where α * n,j denotes the generalized bootstrapped empirical process constructed with the sample (Z

(j) 1 , . . . , Z (j)
n ), j = 1, . . . , N . Theorem 1 accounts for the use of the smallest z > 0 such that

1 N N i=1 1 ψ j ≤ z ≥ 1 -α.
as an estimator of c(α).

A direct consequence of Theorem 1 and Theorem 1.5 in [START_REF] Horváth | Approximations for weighted bootstrap processes with an application[END_REF] is the following approximation for α * n (•) based on a Kiefer process Theorem 2 There is a Kiefer process {K(t; x);

0 ≤ t ≤ 1; 0 ≤ x ≤ ∞} such that max 1≤k≤n sup -∞<t<∞ k i=1 (W i;n -1/n)1{X i ≤ t} -K(F (t), k) = O P (n 1/4 (log n) 1/2 ). ( 12
)

An application to kernel density estimation

Let X 1 , . . . , X n be independent random replicae of a random variable X ∈ R with distribution function F (•).

We assume that the distribution function F (•) has a density f (•) (with respect to the Lebesgue measure in R).

First of all, we introduce a kernel density estimator of f (•). To this end, let K(•) be a measurable function fulfilling the following conditions (K1) K(•) is of bounded variation and compactly supported on R;

(K2) K ≥ 0 and K(u)du = 1. Now, define the Akaike-Parzen-Rosenblatt kernel density estimator of f (•) (see [START_REF] Akaike | An approximation to the density function[END_REF], [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] and [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF]) as follows: for all x ∈ R, estimate f (x) by

f n,hn (x) = 1 nh n n i=1 K x -X i h n , (13)
where {h n : n ≥ 1} is a sequence of positive constants satisfying the conditions

h n ↓ 0 and nh n ↑ ∞, as n → ∞.
Secondly, we define the bootstrapped version of f n,hn (•), by setting for all h n > 0 and x ∈ R,

f * n,hn (x) = 1 h n n i=1 W i;n K x -X i h n , (14)
where W i;n is defined in (4). We will provide an approximation rate for the following process

γ * n (x) = nh 2 n f * n,hn (x) -f n,hn (x) , -∞ < x < ∞. ( 15
)
The following theorem, proved in the next Section, shows that a single bootstrap suffices to obtain the desired approximation for non-parametric kernel-type density estimators.

Theorem 3 Let conditions (A1), ( A2), ( K1) and (K2) prevail. Then we can define Brownian bridges {B * n (y) : 0 ≤ y ≤ 1} such that almost surely along X 1 , X 2 , . . . , as n tends to infinity, we have

sup -∞<x<∞ γ * n (x) -K x -s h n dB * n (F (s)) = O P log n √ n . ( 16
)
If, moreover, we suppose boundedness of the unknown density, f, i.e. if we suppose the existence of M > 0 such that sup -∞<x<∞ f (x) ≤ M, then, almost surely along X 1 , X 2 , . . . , as n tends to infinity,

sup -∞<x<∞ γ * n (x) -B * n (F (x)) K(t)dt = O P log n √ n + h n log h -1 n . ( 17
)
Remark 3. Under appropriate conditions, and using the same arguments rehearsed in the proof of Theorem 3, it is possible to obtain an approximation of a smoothed version of F * n .

Proofs

Proof of Theorem 1. In the sequel, we will write

• to indicate sup -∞<t<+∞ | • |. We have that α * n (t) -B * n (F (t)) = √ n(F * n (t) -F n (t)) -B * n (F (t)) . Now, it is easily seen that √ n(F * n (t) -F n (t)) = n T n 1 √ n n i=1 Z i 1{X i ≤ t} -F (t)T n + (F (t) -F n (t))T n , ( 18
) so that α * n (t) -B * n (F (t)) ≤ S 1 (n) + S 2 (n) + S 3 (n), where S 1 (n) := n T n 1 √ n n i=1 Z i 1{X i ≤ t} -T n F (t) -B * n (F (t)) , (19)
where

S 2 (n) := n T n T n √ n (F (t) -F n (t)) , (20)
and where

S 3 (n) := n T n -1 B * n (F (t)) . (21)
We start by dealing with the term S 3 (n). We will treat the cases x > Cn and x ≤ Cn (C being a strictly positive constant) separately. Fix x > Cn arbitrarily. Union bound gives for all n,

P S 3 (n) ≥ n -1/2 (x + c log n) ≤ P S 4 (n) ≥ x 2 √ n + P B * n (F (t)) ≥ x 2 √ n ,
where

S 4 (n) := n T n B * n (F (t)) . (22)
Now, it is known that, for all n ≥ 1 and all x > n ≥ 1, there exists a positive constant c 1 , such that

P B * n (F (t)) ≥ x 2 √ n ≤ c 1 exp - x 2 4n ≤ exp - x 4 . (23)
On the other hand, since strong law of large numbers gives

n T n -1 a.s. → 0,
for all ε, η 0, there exists

N 1 = N 1 (ε, η), such that, for all n ≥ N 1 , P n T n -1 ∈ (0, ε) ≥ 1 -η. (24)
Consequently, denoting the law of n Tn by L n Tn , independence of the Z n 's from the B n 's gives

P S 4 (n) ≥ x 2 √ n = P S 4 (n) ≥ x 2 √ n , n T n -1 ∈ (0, ε) + P S 4 (n) ≥ x 2 √ n , n T n -1 ∈ (0, ε) ≤ P n T n B * n (F (t)) ≥ x 2 √ n | n T n -1 ∈ (0, ε) + P n T n -1 ∈ (0, ε) ≤ 1+ε 1-ε P B * n (F (t)) > x 2 ny 2 | n T n = y L n Tn (dy) + η ≤ P B * n (F (t)) > x 2 n(1 + ε) 2 + η ≤ c 1 exp - x 4(1 + ε) 2 + η, (25)
where, in the last inequality, we have used (23). Combining ( 23) and ( 25), we have that, for all ε, η > 0, there exists N 1 = N 1 (ε, η), such that, for all n ≥ N 1 ,

P S 3 (n) ≥ n -1/2 (x + c log n) ≤ (1 + c 1 ) exp - x 4(1 + ε) 2 + η. ( 26
)
Now we turn to the case 0 < x ≤ Cn. Again, by the union bound,

P S 3 (n) ≥ n -1/2 (x + c log n) ≤ P n T n -1 > x n + P B * n (F (t)) > √ x . (27)
Again by ( 23), we have that for all n,

P B * n (F (t)) > √ x ≤ c 1 exp(-x/2). (28)
On the other hand, by ( 24), for all ε, η > 0, there exists

N 1 = N 1 (ε, η) such that for all n ≥ N 1 , P n T n -1 > x n = P n T n -1 > x n , n T n -1 ∈ (0, ε) + P n T n -1 > x n , n T n -1 ∈ (0, ε) ≤ P n T n T n n -1 > x n , n T n -1 ∈ (0, ε) + η ≤ P T n n -1 > x n(1 + ε) 2 + η (29)
Use Theorem 2.6 in Petrov (1995) to find constants c 2 and c 3 such that

P T n n -1 > x n(1 + ε) 2 ≤ c 2 exp -c 3 x (1 + ε) 2 . ( 30
)
Combining ( 28), ( 29) and ( 30), and plugging in ( 27), we deduce the existence of positive universal constants c 4 and c 5 such that

P S 3 (n) ≥ n -1/2 (x + c log n) ≤ c 4 exp -c 5 x (1 + ε) 2 + η, (31)
so that one concludes, from ( 26) and ( 31), that for all ε, η > 0, there exists N = N (ε, η), such that, for all n ≥ N, and all x > 0

P S 3 (n) ≥ n -1/2 (x + c log n) ≤ c 6 exp -c 7 x (1 + ε) 2 + η, (32)
for some universal constants c 6 and c 7 .

The proof is concluded once we show the existence of universal positive constants c 8 , c 9 , c 10 and c 11 such that, for all ε, η > 0, there exists N 2 = N 2 (ε, η), and N 3 = N 3 (ε, η) such that, for all n ≥ N 2 , and all x > 0

P S 1 (n) ≥ n -1/2 (x + c log n) ≤ c 8 exp -c 9 x (1 + ε) 2 + η, (33)
and for all n ≥ N 3 and all x > 0,

P S 2 (n) ≥ n -1/2 (x + c log n) ≤ c 10 exp -c 11 x (1 + ε) 2 + η. (34)
Since [START_REF] Horváth | Approximations for weighted bootstrap processes with an application[END_REF] combined with arguments similar to those used for the term S 3 (n) imply (33). As for (34), formula (3.5) in [START_REF] Horváth | Approximations for weighted bootstrap processes with an application[END_REF] together with the by now usual ε, η argument conclude the proof.

S 1 (n) = n T n 1 √ n n i=1 Z i 1{X i ≤ t} -T n F (t) -B * n (F (t)) , formula (3.7) in
Proof of Theorem 3. We start by proving ( 16). We have for

x ∈ R nh 2 n f * n,hn (x) -f n,hn (x) = K ((x -s)/h n ) d{n 1/2 (F * n (s) -F n (s))} = K ((x -s)/h n ) dα * n (s).
Integration by parts implies that

K x -s h n dα * n (s) = -α * n (x -th n )dK(t), (35)
and

K x -s h n dB * n (F (s)) = -B * n (F (x -th n ))dK(t). (36)
Now, Theorem 1 together with condition (K1) give

sup -∞<x<∞ α * n (x -th n )dK (t) -B * n (F (x -th n ))dK (t) ≤ sup -∞<u<∞ |α * n (u) -B * n (F (u))| d|K (t) | = O P log n √ n , (37)
thus proving (16). Once ( 16) is at hand, to prove (17), it suffices to bound

B * n (F (x -th n ))dK (t) -B * n (F (x)) ≤ |B * n (F (x -th n )) -B * n (F (x))| dK (t) , (38)

  in probability. By condition (K1), and provided the unknown density f is bounded (by a strictly positive constant, say M ), for n large enough,|B * n (F (xth n )) -B * n (F (x))| ≤ sup |u-v|≤δn |B * n (u) -B * n (v)|(39)where δ n = M h n . Now, it is always possible to define a Brownian Bridge, {B * (y) : 0 ≤ y ≤ 1}, on the same probability space carrying the sequence of Brownian Bridges {B * n (y) : 0 ≤ y ≤ 1} n≥1 , such that for all n, and all ε > 0P {2δ n log δ -1 n } -1/2 sup |u-v|<h sup h∈[0,δn] |B * n (u) -B * n (v)| > 1 + ε = P {2δ n log δ -1 n } -1/2 sup |u-v|<h sup h∈[0,δn] |B * (u) -B * (v)| > 1 + ε .Since δ n → 0, by Theorem 1.4.1 in[START_REF] Csörgő | Strong approximations in probability and statistics[END_REF], we have with probability onen (x) -B * n (F (x)) dK(t) = O P log n √ n + h n log h -1 n ,thus completing the proof of Theorem.

Acknowledgement

The authors are grateful to Professors Paul Deheuvels and Giovanni Peccati for a number of very useful discussions.


Anne Le Nel

Jana Krenkova

Karel Kleparnik

Claire Smadja

Myriam Taverna

Dr Jean-Louis Viovy
email: jean-louis.viovy@curie.fr

Frantisek Foret

On-chip tryptic digest with direct coupling to ESI-MS using magnetic particles

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On-chip tryptic digest with direct coupling to ESI-MS using magnetic particles

As a step toward a fully automated front-end microfluidic chip for MS proteomics, we propose a system capable of performing online tryptic digest and ESI-MS, using a replaceable on-chip digestion microcolumn based on self-assembled magnetic particles.

Keywords: Magnetic beads / MS / Self-organization Proteomics, which represents a major tool in the postgenomics area, is directed toward the identification of proteins (and any structural modifications after translation from the gene or post-translational modification) [START_REF] Liebler | Introduction to Proteomics[END_REF]. This task typically includes the separation of the proteins in a sample with the highest resolution and sensitivity possible and the identification of each analytical ''band'' or ''peak''. The conventional tools for proteomics, 2-D electrophoretic separation followed by tryptic digest of individual spots, peptide mass measurements and database interrogation, are very labor-intensive. Thus, numerous efforts are devoted to the development of more automated strategies involving, e.g. LC or CE coupled to MS. However, in spite of the high resolution of these techniques, they are not sufficient for the unambiguous identification of proteins in a complex mixture, and an additional step, involving enzymatic cleavage at specific sites to yield unique ''fingerprints'', is usually required.

Many different enzymes are routinely applied in the protein characterization protocols, but trypsin is by far the most widely used peptidase [START_REF] Urban | [END_REF]. Tryptic digestion is typically performed in free solution. The substrate/enzyme ratio must be kept high (generally greater than 50), in order to prevent excessive sample contamination by trypsin and its autolysis products. This leads to relatively slow digestion. Typically, processing is performed overnight at 371C. Much faster digestion can be achieved by immobilizing the enzyme on a solid-phase support [3]. A variety of solid-phase supports has been shown to be suitable for enzyme immobilization, including packed beads [4], porous monoliths [5] and magnetic particles [6,7].

For automated, high-throughput analysis, an interesting strategy is to couple online separation, proteolysis and detection. This also minimizes the risk of losses during manual handling of the samples. For example, online coupling of in-capillary digestion on monolith phases to ESI-MS has been proposed [5]. Microfluidic systems or ''labon-chips'' are particularly well adapted to the development of fully integrated, high-throughput analytical systems and pave the route to entirely new molecular analysis systems [8]. Indeed, chips combining LC separation and ESI-MS are now developed [9]. Capillary immobilized microfluidic enzymatic reactors have been directly coupled online with ESI-MS [10,11] or MALDI-MS [12] and extensive studies on ESI-MS interfacing for coupling microfluidics to MS have been reported including various techniques [13]. A recent paper reviewing microfluidic systems coupled with electrospray MS can be found in [14].

Here, we describe the coupling of microfabricated devices based on self-organized magnetic beads, enabling protein digestion, to an ESI mass spectrometer. The use of self-organized magnetic beads offers several advantages over earlier supports: first, they make the packing step particularly straightforward and reproducible; second, they provide an easy tuning of the porosity of the microcolumns for optimal digestion time; third, they allow for an automated replacement of contaminated microreactors, for long-term fully automated operation and cost reduction.

The trypsin was covalently immobilized onto the magnetic beads under the conditions optimized previously [6]. The particles used were non-porous, COOH functionalized polystyrene-coated particles with a magnetic susceptibility of 40 emu/g and a mean diameter of 500 nm.

Microchips were fabricated in polydimethylsiloxane by rapid prototyping technology [15]. The channels were 1 mm wide by 250 mm high and 2 cm long.

Five hundred micrograms of trypsin beads (in solution in phosphate buffer 0.1 M) was introduced into the microchannel with a syringe pump at a flow rate of 33 mL/min. The permanent magnets used for the trapping of the magnetic beads with the immobilized trypsin were positioned around the microchannel in order to yield inside the microchannel a field mostly collinear to the channel's axis, and a gradient also collinear to this axis, as described previously [6,7].

The Mariner TOF mass spectrometer (Applied Biosystems, MA, USA) was used in all experiments. The ESI-TOF measurements were performed in positive ion mode with a scan range of 400-2500 m/z. The list of detected ions (m/z) was used for protein identification by the MS-Fit tool of Protein Prospector database (http://prospector.ucsf.edu).

Two modes of operation were tested. In the off-line mode, cytochrome c was dissolved to a concentration of 0.5 mg/mL in 10 mM ammonium acetate solution, pH 8.5. The protein solution was pumped through the microchip at a flow rate of 170 nL/min using a syringe pump. The digestion was performed at 251C. The peptide fragment solutions were collected in microvials and analyzed by infusion MS analysis conducted using a nanospray (10 mm id fused-silica capillary with a polished tip).

In the online mode, the input and output channels of the PDMS microdevice were connected to the syringe pump and to the electrospray interface, respectively, via a 360 mm od fused-silica tubing (10 cm  75 mm id). This electrospray source was adapted from the commercial microspray interface supplied with the mass spectrometer. Details of the system are apparent from the photographs in Fig. 1. The protein solutions were prepared in 10 mM ammonium acetate, pH 8.5, in the concentration range of 0.1-0.5 mg/ mL. The stream flowing through the microchip was directly delivered to the mass spectrometer microelectrospray interface using a T-union. The eluent from the microreactor was continuously mixed with 2 mL/min spray solution consisting of 50% aqueous ACN and 1% formic acid using a T-union. This arrangement provided practically constant flow rate and pH for optimum ESI, regardless of the sample pH and flow rate. The electrospray signal was additionally stabilized by a flow of nitrogen curtain gas set (1 L/min) and nitrogen nebulizer gas set (0.25 L/min).

Here, we use an original way of organizing magnetic beads inside a microchannel. The PDMS device integrates strong magnets to create a magnetic field parallel to the flow, with a strong gradient pointing through the center of the channel. When exposed to a uniform external field, the magnetic beads self-organize into a supraparticle structure consisting of a columnar clustering in the direction of the field [16]. In the presence of the magnetic field gradient, however, the columns accumulate in a zone of high gradient of about 3-4 mm, and a new phase is obtained: individual magnetic columns gather into tortuous walls parallel to the field direction, and a ''labyrinth-like'' structure is achieved [17]. Compared with previous immobilization methods involving fields perpendicular to the channel's axis, this strategy yields a lower pressure drop, a better resistance to flow and a more uniform permeation. The resulting proteolytic matrix has a small and relatively homogeneous pore size (around 1 mm), well adapted to the protein sizes, thus allowing for fast and reproducible protein reactions. This concept had been applied to enzymatic digestion with trypsin or proteinase K immobilized on beads [6,7]. The results showed a 100-fold increase in digestion rate on-chip compared with digestion in solution, for both trypsin and proteinase K. This system allows easy replacement of the bead matrix by the application of a strong flow. The reproducibility of beads packing and digestion efficiency is very good, with an error of less than 2%. We also demonstrated that, in contrast to the soluble form, enzymes immobilized on magnetic beads can achieve efficient digestion at room temperature without alteration of digestion specificity; consequently, there is no need to perform the digestion at 371C.

In the first series of experiments, to control the activity of the trypsin beads, on-chip tryptic digestion was performed and fragments were collected and then analyzed with off-chip MS. The MS spectrum is shown in Fig. 2. Fragments from protein digestion are detectable revealing the effective activity of the trypsin magnetic beads. The experimental sequence coverage was 79% (82/104 amino acids).

In the second series of experiments, the microchip was connected to the MS device through the T-junction. The magnetic beads with surface-immobilized trypsin were introduced into the microchannel by the syringe pump, trapped by the magnets inside the microchannel and cytochrome c solution was passed through the plug at different flow rates from 670 to 170 nL/min (see Fig. 3) corresponding to three different residence times: 58, 93 and 233 s. Fragments from trypsin digestion were continuously analyzed by ESI-TOF. These experiments confirmed that the lower the flow rate, the more efficient the digestion. While in the flow range of 400-670 nL/min the intact protein envelope was clearly visible, below 300 nL/min only the tryptic peptides were detected, suggesting complete protein digestion. These results were reproducibly repeated in the protein concentration range of 0.1-0.25 mg/mL while refreshing the bead bed, removing and adding new trypsin beads. The matched peptides cover 92% (96/104 amino acids) of the protein sequence. One can note that the noise in Fig. 3C is significantly higher than that achieved off-line (Fig. 2). This difference mainly arises from the use of two different MS interfaces (nanospray off-line and microspray online). For technical reasons, with the flow rates and pressures achievable in our microdigestion system, we were not yet able to adapt it to existing nanospray interface and had to use a microspray interface. This involves significant dilution by a spray buffer between digestion and injection, which is responsible for the decreased sensitivity. However, even with this dilution, the quality of the data achieved with microspray is largely sufficient for accurate peptide mapping down to protein concentrations below mg/mL.

In summary, we demonstrated online peptide mapping using a microfluidic microreactor based on self-assembled magnetic particles and direct ESI-MS coupling. The microdevice with embedded permanent magnets provides a simple means for the formation of a reusable enzymatic reactor. This scheme simplifies considerably the formation of the microreactor compared with earlier systems such as packed beds or monoliths. Also, the beads can be renewed by a strong hydrodynamic flush in a fully automated manner. This may represent a significant advantage in complex microfluidic systems. No leakage of magnetic particles in the ESI was observed for the whole range of flow rates practically interesting for digestion. Performance sequence coverages as good as or better than that of conventional batch process were obtained, at typical flow rates for LC or CE separations. These results pave the route for the development of a fully integrated micro total analysis system dedicated to proteomic analysis. In particular, our comparison between nanospray and microspray (Figs. 2 and 3C) shows that strong gains in sensitivity would be made possible, by integrating magnetic microreactor with nanospray injection.

Figure 1 .

 1 Figure 1. Experimental setup: pictures of the connection to the MS interface (A) and of the microdevice (B). Details of the microdevice with the magnets and the T-union for coupling with the ESI interface (C).

Figure 2 .

 2 Figure 2. ESI-MS spectrum of the digested fragments of cytochrome c in an off-line mode at a flow rate of 170 nL/min.

Figure 3 .

 3 Figure 3. MS spectra of the digested fragments of cytochrome c in the online mode at different flow rates: 667 nL/min (A), 417 nL/min (B), 167 nL/min (C), only the major peptide fragments are indicated.

This work was supported in part by the French Ministry of Research/CNRS Programme ''NMAC: Microproteolab'' and by the European Project ''NEUROTAS''. Additional support was received from the IACH research plan Z40310501, Grant Agency of the Czech Republic (203/06/1685) and Czech Academy of Sciences (KAN400310651). This work was performed as a part of the UMR168 microfabrication clean room. The authors have declared no conflict of interest.


Christophe Hazard
email: christophe.hazard@ensta.fr

A distribution framework for the generalized Fourier transform associated with a Sturm-Liouville operator

Keywords: distributions, generalized Fourier transform distributions, transformation de Fourier généralisée

The generalized Fourier transform associated with a selfadjoint Sturm-Liouville operator is a unitary transformation which converts the action of this operator into a simple product by a spectral variable. For a particular operator defined on the half-line and which involves a step function, we show how to extend such a transformation to generalized functions, or distributions, with a suitable definition of such distributions. This extension is based essentially on the fact that, as the usual Fourier transform, this transformation has the property to exchange regularity and decay between the physical and spectral variables.

Introduction

One of the fundamental properties of the usual Fourier transform of square integrable functions lies in the possibility to be extended to generalized functions, or distributions, which makes it a power tool for the solution of partial differential equations with constant coefficients. The aim of the present paper is to show how such a property holds for the generalized Fourier transform associated with a Sturm-Liouville operator. For the sake of simplicity, we consider a very simple operator defined on the half-line, for which most calculations are explicit. We denote by A the unbounded selfadjoint operator in L 2 (R + ) given by

Aϕ := -ϕ ′′ -k 2 ϕ ∀ϕ ∈ D(A) := {ψ ∈ H 2 (R + ); ψ(0) = 0}, (1)
where k = k(x) is the function defined on R + by

k(x) := k if 0 ≤ x < h, k if x ≥ h, with 0 < k < k. ( 2
)
As we will see, the latter assumption makes possible the presence of eigenvalues in the spectrum of A, denoted by Λ.

A generalized Fourier transform F associated with A is a unitary transformation from the 'physical space' L 2 (R + ) to a spectral space H (which contains functions of the spectral variable λ ∈ Λ) which diagonalizes A in the sense that it converts the action of A into a simple product by λ in H, that is,

A = F * λ F.
The usual Fourier transform plays this role in the case of constant coefficients (here: k = k). For Sturm-Liouville operators with variable coefficients, such a transformation has been introduced in the first half of the last century in the original works of H. Weyl and E.C. Titchmarsh [START_REF] Titchmarsh | Eigenfunction expansions associated with second order differential equations[END_REF]. More generally, for every selfadjoint operator in a Hilbert space, one can construct a generalized Fourier transform, which can be interpreted as a decomposition on a family of generalized eigenfunctions [START_REF] Berezansky | Functional Analysis II[END_REF]. It has numerous applications, for instance in scattering theory [START_REF] Hazard | Generalized eigenfunction expansions for conservative scattering problems with an application to water waves[END_REF] or quantum mechanics.

It is well-known that the decay properties for large λ of the usual Fourier transform ϕ(λ) of a function ϕ(x) is related to the regularity of ϕ(x), and conversely, the regularity of ϕ(λ) is related to the behaviour of ϕ(x) for large x. The main purpose of this paper is to study how this exchange of regularity and decay between the physical and spectral spaces holds for the generalized Fourier transform F. This will allow us to extend F to a space larger than L 2 (R + ) similar to the usual Schwartz space of distributions [START_REF] Schwartz | Théorie des Distributions[END_REF].

A possible application of this extension concerns the propagation of timeharmonic waves in a slab waveguide [START_REF] Bonnet-Ben Dhia | Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition[END_REF], more precisely the solution of Helmholtz equation in a half-plane:

- ∂ 2 u ∂x 2 - ∂ 2 u ∂y 2 -k 2 u = f in R + × R, u(0, y) = 0 ∀y ∈ R,
where k is given by (2): it only depends on x. To solve such a problem, one can use the usual Fourier transform in the variable y, which diagonalizes the y-part of the laplacian. This leads to a differential equation in x with a variable coefficient.

Here we are interested in using instead the generalized Fourier transform in the x-direction. At least formally, the above system can be rewritten as

- ∂ 2 u ∂y 2 + A u = f,
and applying F then yields a differential equation in y with constant coefficients:

- ∂ 2 u ∂y 2 + λ u = f in R,
where u := Fu and f := Ff. But for a fixed y ∈ R, function u(•, y) does not belong to L 2 (R + ) in general. In order to justify the use of F, we have to make clear in what sense we apply this transformation. We show here that it can be interpreted in the sense of distributions by constructing a space of distributions adapted to operator A.

The paper is organized as follows. In §2, we introduce the functional material which is needed in the rest of the paper. In §3, we recall the construction of the generalized Fourier transform F associated with A. Section 4 then investigates the exchange of decay and regularity under the action of F. This provides us the basic tool for the extension of F to a space of distributions, which is the object of §5.

The following notations concerning some usual functional spaces are used hereafter. If X is a subset of R and µ is a measure on X, we denote by L 2 (X; dµ) the space of square integrable functions on X for the measure µ, equipped with the following inner product and associated norm: (ϕ, ψ) X;dµ := X ϕ(x) ψ(x) dµ(x) and ϕ X := (ϕ, ϕ)

1/2 X;dµ .
If µ is the Lebesgue measure on X, we shall omit dµ in these notations. For r ∈ R, we denote by H r (X) the usual Sobolev spaces. If X is unbounded, L 2 comp (X) is the subspace of L 2 (X) composed of functions with compact support, L 2 loc (X) is the set of functions ϕ such that ϕ| K ∈ L 2 (K) for all bounded sets K ⊂ X, and ϕ, ψ

X := X ϕ(x) ψ(x) dx ∀ϕ ∈ L 2 comp (X), ∀ψ ∈ L 2 loc (X).
In the whole paper, √ • denotes the principal branch of the complex square root defined for all z ∈ C \ R -by √ z := |z| 1/2 e i(arg z)/2 with | arg z| < π, which has one-sided limits near every negative real number:

√ z ± i0 := lim εց0 √ z ± iε = ±i |z| 1/2 ∀z ∈ R -.

Preliminary information

We introduce some particular solutions to the following differential equation:

-ϕ ′′ -(k 2 + ζ)ϕ = 0 on R + , (3)

INRIA

where k = k(x) is defined by [START_REF] Bonnet-Ben Dhia | Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition[END_REF] and

ζ ∈ C. We denote r ζ (x) := -k 2 (x) -ζ, that is, r ζ (x) =    ṙζ := -k2 -ζ if 0 ≤ x < h and ζ ∈ C \ [-k2 , +∞), rζ := -k2 -ζ if x ≥ h and ζ ∈ C \ [-k2 , +∞).
Both functions ṙζ and rζ have one-sided limits on their respective branch cuts: ṙλ±i0 = ∓i βλ where βλ := k2 + λ

1/2
for λ ∈ (-k2 , +∞), rλ±i0 = ∓i βλ where βλ := k2 + λ

1/2
for λ ∈ (-k2 , +∞).

We consider a canonical basis of solutions to (3), denoted by c ζ (x) and s ζ (x), associated with point x = h in the sense that they satisfy

c ζ (h) = 1 et c ′ ζ (h) = 0, s ζ (h) = 0 et s ′ ζ (h) = 1. When ζ ∈ C \ [-k2
, +∞), these solutions write explicitly as

c ζ (x) = cosh {r ζ (x) (x -h)} and s ζ (x) = sinh {r ζ (x) (x -h)} r ζ (x) ,
but these expressions actually do not depend of the choice of a branch of the complex square root, since they both are even functions of r ζ (x). Hence c ζ et s ζ are entire functions of ζ. Every solution to (3) is a linear combination of these functions. We consider two particular solutions. On one hand,

Φ ζ (x) := c ζ (0) s ζ (x) -s ζ (0) c ζ (x), (4)
is clearly an entire function of ζ, and satisfies the boundary conditions

Φ ζ (0) = 0 and Φ ′ ζ (0) = 1.
On the other hand,

Θ ζ (x) := c ζ (x) -rζ s ζ (x) (5)
is analytic in C \ [-k2 , +∞) but no more entire because of the coefficient rζ . This is the evanescent solution to (3): it decreases exponentially when

x → +∞ since Θ ζ (x) = exp{-r ζ (x -h)} if x ≥ h, (6)
where Re rζ > 0 because of our choice of the principal branch of the complex square root. On the branch cut [-k2 , +∞), this function has one-sided limits

Θ λ±i0 (x) := lim εց0 Θ λ±iε (x) = c λ (x) ± i βλ s λ (x) ∀λ ∈ [-k2 , +∞). ( 7
)
Function Φ λ may be seen as a linear combination of these limits:

Φ λ = Θ λ-i0 (0) Θ λ+i0 -Θ λ+i0 (0) Θ λ-i0 2i βλ = -Im Θ λ+i0 (0) Θ λ+i0 βλ . ( 8
)
We will need the derivative of Θ ζ (x) with respect to ζ, which can be expressed as follows:

∂ ∂ζ Θ ζ (x) = 1 2r ζ {(x -h) Θ ζ (x) + Θζ (x)}, (9)
where Θζ (x) is given by

Θζ (x) :=    k2 -k2 k2 + ζ (s ζ (x) -(x -h) c ζ (x)) if 0 ≤ x < h, 0 if x ≥ h. ( 10
)
3 The generalized Fourier transform associated with A

The results given in this section are classical (see, e.g., [START_REF] Coddington | Theory of Ordinary Differential Equations[END_REF][START_REF] Titchmarsh | Eigenfunction expansions associated with second order differential equations[END_REF][START_REF] Weidmann | Spectral theory of ordinary differential operators[END_REF]), but we recall the proofs for the sake of clarity. The construction of a generalized Fourier transform associated with our operator A is based on the spectral theorem [START_REF] Berezansky | Functional Analysis II[END_REF] which ensures the existence of a spectral measure E (also called resolution of the identity) such that for every measurable function f : R → C, the operator f (A) can be expressed as

f (A)ϕ = R f (λ) dE λ ϕ ∀ϕ ∈ D(f (A)), (11)
where the domain of f (A) is characterized by

D(f (A)) = ψ ∈ L 2 (R + ); R |f (λ)| 2 d E λ ψ 2 R + < ∞ .
Stone's formula offers a convenient expression of E by means of the resolvent R ζ := (A -ζ) -1 of A : for every closed interval I = [λ 1 , λ 2 ] ⊂ R and every ϕ ∈ L 2 (R + ), we have

E I ϕ 2 R + = 1 2iπ lim ηց0 lim εց0 λ2+η λ1-η ((R λ+iε -R λ-iε )ϕ, ϕ) R + dλ. ( 12
)
To apply this formula, we shall use an explicit integral representation of

R ζ ϕ for ζ ∈ C \ R : (R ζ ϕ)(x) = R + γ ζ (x, x ′ ) ϕ(x ′ ) dx ′ ∀x ∈ R + , (13)
where the kernel γ ζ is the Green's function of A, that is, the bounded solution to

-γ ′′ ζ (•, x ′ ) -(k 2 + ζ) γ ζ (•, x ′ ) = δ x ′ in D ′ (R + ), γ ζ (0, x ′ ) = 0,
where δ x ′ denotes the Dirac measure at x = x ′ . Using the function Φ ζ and Θ ζ introduced in (4) and ( 5), this yields

γ ζ (x, x ′ ) = Φ ζ (min{x, x ′ }) Θ ζ (max{x, x ′ }) Θ ζ (0) , (14)
Formula ( 13) is classically obtained by integrations by parts using the equations satisfied by R ζ ϕ and the above equations for γ ζ .

INRIA

Proposition 1 The spectrum Λ of A consists of, on one hand, a finite point spectrum Λ p composed of the zeros of Θ ζ (0) located in (-k2 , -k2 ), that is, the roots of the dispersion equation

tan k2 + λ h = - k2 + λ -k2 -λ with λ ∈ (-k2 , -k2 ), (15)
and on the other hand, an absolutely continuous spectrum

Λ c := [-k2 , +∞).
For every ϕ ∈ L 2 comp (R + ), the spectral measure can be expressed as

d E λ ϕ 2 R + = | ϕ, Φ λ R + | 2 ρ λ dλ| Λc + λ∈Λp |(ϕ, Φ λ ) R + | 2 ρ λ δ λ , (16)
where dλ| Λc is the Lebesgue measure restricted to Λ c , δ λ is the Dirac measure at λ ∈ Λ p , and

ρ λ :=    Φ λ -2 R + = 2 β2 λ h + r-1 λ -1 if λ ∈ Λ p , βλ π|Θ λ+i0 (0)| 2 if λ ∈ Λ c . (17)
Proof. Formula (14) shows that γ ζ is a meromorphic function of ζ in C \ [-k2 , +∞). Its poles are the zeros of Θ ζ (0), which are easily seen to be located in (-k2 , -k2 ]. For such a pole λ, functions Φ λ and Θ λ are proportional: Φ λ = -s λ (0) Θ λ . These are eigenfunctions associated with the eigenvalue λ, apart from the case when λ = -k2 is a zero of Θ ζ (0), that is, when

k2 -k2 = n + 1 2 2 π 2 h 2 for some n ∈ N. (18)
Indeed in this case Φ λ / ∈ L 2 (R + ), so λ = -k2 cannot be an eigenvalue of A. That is why this bound is excluded in the dispersion equation (15) which is equivalent to the equation Θ λ (0) = 0.

As a consequence, E vanishes on (-∞, -k2 ) and is a pure point measure on the interval (-k2 , -k2 ). If I ⊂ (-k2 , -k2 ) contains only one eigenvalue λ, then E I is the spectral projection E {λ} associated with λ, i.e.,

E {λ} ϕ = (ϕ, Φ λ ) R + Φ λ Φ λ 2 R + ,
where one can check that that Φ λ 2

R + = (h + r-1 λ )/(2 β2
λ ). On the other hand, at every point λ ∈ (-k2 , +∞), the Green's function has one-sided limits

γ λ±i0 (x, x ′ ) := lim εց0 γ λ±iε (x, x ′ ) = Φ λ (min{x, x ′ }) Θ λ±i0 (max{x, x ′ }) Θ λ±i0 (0) .
From (8), we see that

Θ λ+i0 (x) Θ λ+i0 (0) - Θ λ-i0 (x) Θ λ-i0 (0) = 2i βλ |Θ λ+i0 (0)| 2 Φ λ (x),
which yields the gap of the Green's function across (-k2 , +∞) :

γ λ+i0 (x, x ′ ) -γ λ-i0 (x, x ′ ) = 2iπ ρ λ Φ λ (x) Φ λ (x ′ ), (19)
where ρ λ is given by (17). We are now ready to use Stone's formula (12) together with the integral representation (13). If we choose ϕ ∈ L 2 comp (R + ), we can pass to the limits η ց 0 and ε ց 0 thanks to the Lebesgue's dominated convergence theorem, which yields

E I ϕ 2 R + = 1 2iπ λ2 λ1 R + ×R + {γ λ+i0 (x, x ′ ) -γ λ-i0 (x, x ′ )} ϕ(x ′ ) ϕ(x) dx dx ′ dλ.
for every interval I = [λ 1 , λ 2 ] ⊂ (-k2 , +∞). Therefore, using (19) and Fubini's theorem, we have

E I ϕ 2 R + = λ2 λ1 | ϕ, Φ λ R + | 2 ρ λ dλ.
The expression (16) of the spectral measure on Λ c follows.

Remark 2

The values of the gap k2 -k2 given by (18) define the thresholds of the problem: when the gap increases and meets one one these values, a new eigenvalue appears in Λ p from the lower bound -k2 of the continuous spectrum.

We can now introduce the generalized Fourier transform associated with A, that is, the operator of 'decomposition' on the familly {Φ λ ; λ ∈ Λ}. We denote

Fϕ(λ) := ϕ, Φ λ R + ∀λ ∈ Λ, ∀ϕ ∈ L 2 comp (R + ). ( 20
)
Let dµ be the measure on R define by

dµ := λ∈Λp ρ λ δ λ + ρ λ dλ| Λc .
Theorem 3 The transformation F defined by (20) extends by density to a unitary operator from L 2 (R + ) to L 2 (Λ; dµ) (still denoted by F), which diagonalizes A in the sense that for every measurable function f : R → C,

f (A)ϕ = F * f (λ) F ϕ, ∀ϕ ∈ D(f (A)) = F * (L 2 (Λ; (1 + |f (•)| 2 )dµ)), ( 21
)
where f (λ) stands for the operator of multiplication by f (λ) in L 2 (Λ; dµ). Its adjoint F * = F -1 appears as the operator of 'recomposition' on the family {Φ λ ; λ ∈ Λ} :

F * ϕ = λ∈Λp ρ λ ϕ(λ) Φ λ + lim M →+∞ M -k2 +M -1 ϕ(λ) Φ λ ρ λ dλ, ( 22
)
for all ϕ ∈ L 2 (Λ; dµ).

INRIA

Proof. The expression of the measure d(E λ ϕ, ψ) for ϕ, ψ ∈ L 2 comp (R + ) follows from (16) by the polarization principle. Hence, using (11) with a bounded function f, the inner product (f (A)ϕ, ψ) R + can be written as

λ∈Λp f (λ) ρ λ (ϕ, Φ λ ) R + (ψ, Φ λ ) R + + Λc f (λ) ϕ, Φ λ R + ψ, Φ λ R + ρ λ dλ, which amounts to the concise form (f (A)ϕ, ψ) R + = (f (λ) Fϕ, Fψ) Λ;dµ . ( 23
) Choosing f (λ) ≡ 1, that is, f (A) = I, we see that F is isometric from L 2 comp (R + ) to L 2 (Λ; dµ).
Therefore it extends by density to an isometry from L 2 (R + ) to L 2 (Λ; dµ), and the above formula holds for every measurable function f and all ϕ ∈ D(f (A)), which yields (21). The expression of the adjoint of F is easily derived by noticing that

(F * ϕ, ψ) R + = ( ϕ, Fψ) Λ;dµ = Λ ϕ(λ) R + ψ(x) Φ λ (x) dx dµ(λ), which yields (F * ϕ, ψ) R + = R + Λ ϕ(λ) Φ λ (x) dµ(λ) ψ(x) dx,
by permuting both integrals. This permutation is justified if ϕ has a compact support which does not contain -k2 (note that ρ λ becomes singular near λ = -k2 for the thresholds defined by (18)): formula (22) then follows by density.

To see that F is unitary, it remains to prove that the range R(F) of F is the whole space L 2 (Λ; dµ). First notice that (21) implies Ff (A)ϕ = P f (λ) F ϕ where P := FF * denotes the orthogonal projection on R(F). Moreover we have Ff (A)ϕ Λ;dµ = f (A)ϕ R + = f (λ) F ϕ Λ;dµ by (23), which shows that we can remove P in the previous relation:

Ff (A)ϕ = f (λ) F ϕ. As a consequence, if ϕ ∈ L 2 (Λ; dµ) is orthogonal to R(F), the same holds for f (λ) ϕ for every bounded function f since (f (λ) ϕ , Fψ) Λ;dµ = ( ϕ , f (λ) Fψ) Λ;dµ = ( ϕ , F f (A)ψ) Λ;dµ = 0. Hence ϕ = 0, thus R(F) = L 2 (Λ; dµ).

Relation between physical decay and spectral regularity

The spectral characterization (21) of D(f (A)) shows how the physical regularity of a given ϕ ∈ L 2 (R + ) is related to the asymptotic behaviour for large λ of Fϕ(λ). Indeed choosing for instance

f (A) = A n with n ∈ N gives ϕ ∈ D(A n ) ⇐⇒ (1 + |λ| n ) Fϕ ∈ L 2 (Λ ; dµ), ( 24
)
and it is easy to see from the definition (1) of D(A) that a function ϕ ∈ D(A n ) is characterized by the following conditions:

ϕ| (0,h) ∈ H 2n (0, h) and ϕ| (h,∞) ∈ H 2n (h, ∞), (25)
d 2ℓ x ϕ(0) = 0 for ℓ = 0, . . . , n -1, (26) [(d 2 x + k 2 ) ℓ ϕ] h = [d x (d 2 x + k 2 ) ℓ ϕ] h = 0 for ℓ = 0, . . . , n -1, ( 27
)
where we use the simplified notation d x instead of d/dx, and [ψ] h stands for the gap of

ψ at x = h, that is, [ψ] h := lim εց0 {ψ(h + ε) -ψ(h -ε)}.
In this section, we address the complementary issue, that is, the relation between physical decay and spectral regularity. For the usual Fourier transform, both issues coalesce, since the inverse transform amounts to a conjugation. The question is not so obvious for the generalized Fourier transform F. A first element of answer is given by the following result which is a straightforward consequence of the fact that Φ ζ is an entire function of ζ.

Proposition 4 If ϕ ∈ L 2 comp (R + )
, then Fϕ(λ) extends to an entire function.

Theorem 5 below provides us a more precise answer. First notice that speaking of spectral regularity actually makes sense only on the continuous spectrum, which leads us to split F into two parts F c and F p which simply denote the restrictions of F to Λ c and Λ p :

F c ϕ := (Fϕ)| Λc and F p ϕ := (Fϕ)| Λp . So F c is defined from L 2 (R + ) to L 2 (Λ c ; ρ λ dλ)
, and its adjoint corresponds to the continuous part of the expression (22) of F * :

F * c ϕ = lim M →+∞ M -k2 +M -1 ϕ(λ) Φ λ ρ λ dλ ∀ ϕ ∈ L 2 (Λ c ; ρ λ dλ). (28)
Similarly, F * p corresponds to the point spectrum contribution. The diagonalization formula (21) then turns into the orthogonal decomposition

f (A)ϕ = F * c f (λ) F c ϕ + F * p f (λ) F p ϕ. ( 29
)
Theorem 5 Suppose that k2 -k2 is not one of the thresholds given by (18).

Then, for every n ∈ N and ϕ ∈ L 2 (R + ), we have

x n ϕ(x) ∈ L 2 (R + ) ⇐⇒ D n λ F c ϕ ∈ L 2 (Λ c ; ρ λ dλ), (30)
where D λ is the spectral derivation operator given by D λ ϕ := d( βλ ϕ)/dλ.

In order to prove this result, we have to understand the link between F c (xϕ) and D λ F c ϕ, which amounts to the link between x Φ λ and D λ Φ λ . But there is no simple relation between the latter quantities. Actually, on both intervals (0, h) and (h, +∞), function Φ λ is a linear combination of exponential functions of the form exp(±iβ λ (x) x). Such a relation exists for each of these functions, but not for their combination. That is why the proof is based on the decomposition (8) of Φ λ and the expression (9) of the spectral derivative of Θ ζ . The former allows us to rewrite the generalized Fourier transform in a more convenient form. Indeed, since Φ λ is real, for all real ϕ ∈ L 2 comp (R + ), we have

F c ϕ = -Im{T ϕ} where T ϕ(λ) := Θ λ+i0 (0) βλ R + ϕ(x) Θ λ+i0 (x) dx. ( 31
)
Lemma 6 The operator T defined in (31) for ϕ ∈ L 2 comp (R + ) extends by density to a bounded operator from L 2 (R + ) to L 2 (Λ c ; ρ λ dλ).

INRIA

Proof. We prove below that there exists a constant C > 0 such that

T ϕ Λc;ρ λ dλ ≤ C ϕ R + , (32)
for all ϕ ∈ L 2 comp (R + ), which can be seen as a perturbed Plancherel identity. The conclusion will then follow from the density of L 2 comp (R + ) in L 2 (R + ). First assume that the support of ϕ ∈ L 2 comp (R + ) is contained in [h, +∞). From ( 6), the expression of T ϕ simplifies in this case as

T ϕ(λ) = Θ λ+i0 (0) e i βλ h βλ +∞ h ϕ(x) e -i βλ x dx,
where we recognize the usual Fourier transform. Hence using the change of variable β = βλ and Plancherel equality, we infer that

+∞ -k2 |T ϕ(λ)| 2 ρ λ dλ = 2 π +∞ 0 R + ϕ(x) e -iβx dx 2 dβ = 2 R + |ϕ(x)| 2 dx,
which shows that for functions whose support is contained in [h, +∞), (32) holds with C = √ 2. Assume now that the support of ϕ is contained in [0, h]. Choosing some fixed λ * > -k2 , we split T ϕ 2 Λc;ρ λ dλ as the sum of two integrals respectively on (-k2 , λ * ) and (λ * , +∞). On one hand, we have by Cauchy-Schwarz inequality

λ * -k2 |T ϕ(λ)| 2 ρ λ dλ = λ * -k2 h 0 ϕ(x) Θ λ+i0 (x) dx 2 dλ π βλ ≤ C ϕ 2 (0,h) , (33)
where

C = 2 βλ * π sup λ∈[-k2 , λ * ] h 0 |Θ λ+i0 (x)| 2 dx.
On the other hand, for the complementary integral on (λ * , +∞), we rewrite the expression (7) of Θ λ+i0 (x) as Θ λ+i0 (x) = e i βλ (x-h) -i k2 -k2 βλ + βλ s λ (x) ∀x ∈ (0, h).

Hence

T ϕ = T 1 ϕ+T 2 ϕ where              T 1 ϕ(λ) := Θ λ+i0 (0) e i βλ h βλ h 0 ϕ(x) e -i βλ x dx, T 2 ϕ(λ) := i ( k2 -k2 ) Θ λ+i0 (0) βλ ( βλ + βλ ) h 0 ϕ(x) s λ (x) dx.
For the former contribution we proceed as above using the change of variable β = βλ and Plancherel equality, which yields

+∞ λ * |T 1 ϕ(λ)| 2 ρ λ dλ ≤ 2 βλ * π βλ * +∞ βλ * h 0 ϕ(x) e -iβx dx 2 dβ ≤ 2 βλ * βλ * h 0 |ϕ(x)| 2 dx.

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For the latter, noticing that s λ 2 (0,h) ≤ h 3 /3, we deduce from Cauchy-Schwarz inequality that

+∞ λ * |T 2 ϕ(λ)| 2 ρ λ dλ ≤ h 3 ( k2 -k2 ) 2 3π +∞ λ * dλ βλ ( βλ + βλ ) 2 h 0 |ϕ(x)| 2 dx.
Both these estimates together with (33) finally show that (32) is also valid if the support of ϕ is contained in [0, h], hence it holds for all ϕ ∈ L 2 comp (R + ). Proof of Theorem 5. It is sufficient to verify (30) for n = 1 and real ϕ : the general case (n > 1 and complex ϕ) follows immediately. For every real ϕ ∈ L 2 comp (R + ), we deduce from (31) and Lebesgue theorem that

D λ F c ϕ(λ) = - R + ϕ(x) ∂ ∂λ Im Θ λ+i0 (0)Θ λ+i0 (x) dx,
We see from (9) that

∂ ∂λ Θ λ+i0 (x) = i 2 βλ (x -h) Θ λ+i0 (x) + Θλ (x) . ( 34
)
Hence

∂ ∂λ Im Θ λ+i0 (0)Θ λ+i0 (x) = -1 2 βλ Re xΘ λ+i0 (0) -Θλ (0) Θ λ+i0 (x) + Θ λ+i0 (0) Θλ (x) .
As a consequence

D λ F c ϕ(λ) = 1 2 Re T {xϕ(x)}(λ) - Θλ (0) Θ λ+i0 (0) T ϕ(λ) + T ϕ(λ) , (35)
where we have denoted

T ϕ(λ) := Θ λ+i0 (0) βλ h 0 ϕ(x) Θλ (x) dx, (36)
which defines a bounded operator from L 2 (R + ) to L 2 (Λ c ; ρ λ dλ). Indeed using the expression (10) of Θλ , we notice that Θλ (0,h) ≤ C/( k2 + λ), and consequently, by Cauchy-Schwarz inequality,

T ϕ Λc;ρ λ dλ ≤ C ϕ (0,h) .
Thanks to the assumption that k2 -k2 is not one of the thresholds given by (18), it is readily seen that Θλ (0)/Θ λ+i0 (0) is bounded on Λ c . Hence, in view of Lemma 6, it follows from (35) that

D λ F c ϕ Λc;ρ λ dλ ≤ C 1 xϕ R + + C 2 ϕ R + .
We have proved this estimate for ϕ ∈ L 2 comp (R + ). By density, it holds for all ϕ ∈ L 2 (R + ) such that xϕ ∈ L 2 (R + ). Thus, for such a ϕ, we know that

D λ F c ϕ ∈ L 2 (Λ c ; ρ λ dλ).
Let us now prove the converse. By virtue of the identity

F * c F c + F * p F p = I (see (29)) and the fact that xF * p F p ϕ ∈ L 2 (R + ) for all ϕ ∈ L 2 (R + ) (since the INRIA eigenfunctions Φ λ , λ ∈ Λ p , decrease exponentially at infinity), it is enough to prove that if ϕ ∈ L 2 (Λ c ; ρ λ dλ) is such that D λ ϕ ∈ L 2 (Λ c ; ρ λ dλ), then x F * c ϕ ∈ L 2 (R + ).
As in (31), we can use the decomposition (8) of Φ λ to rewrite the expression (28) of F * c in the following form, assuming that ϕ is real and has a compact support:

F * c ϕ = Im{T * ϕ} where (T * ϕ)(x) = 1 π Λc ϕ(λ) Θ λ+i0 (x) Θ λ+i0 (0) dλ. ( 37
)
By an integration by parts, we have

(T * D λ ϕ)(x) = 1 π Λc d dλ ( βλ ϕ(λ)) Θ λ+i0 (x) Θ λ+i0 (0) dλ = -1 π Λc ϕ(λ) βλ ∂ ∂λ Θ λ+i0 (x) Θ λ+i0 (0) dλ,
where we deduce from (34) that βλ ∂ ∂λ

Θ λ+i0 (x) Θ λ+i0 (0) = i 2 x Θ λ+i0 (x) Θ λ+i0 (0) - Θλ (0) Θ λ+i0 (x) Θ λ+i0 (0) 2 + Θλ (x) Θ λ+i0 (0) . Hence T * D λ ϕ = -i 2 x T * ϕ -T * Θλ (0) Θ λ+i0 (0) ϕ + T * ϕ ,
where the adjoint T * of T (see (36)) is given by

( T * ϕ)(x) = 1 π Λc ϕ(λ) Θλ (x) Θ λ+i0 (0) dλ.
Going back to (37), we finally obtain that for real ϕ ∈ L 2 comp (Λ c ; ρ λ dλ),

x F * c ϕ = Im 2i T * D λ ϕ + T * Θλ (0) Θ λ+i0 (0) ϕ -T * ϕ .
Using again the fact that Θλ (0)/Θ λ+i0 (0) is bounded on Λ c , we infer that this relation holds by density if ϕ and D λ ϕ belong to L 2 (Λ c ; ρ λ dλ), which completes the proof of Theorem 5.

Extension of F to a space of distributions

One of the basic properties which makes the usual Fourier transform a very powerful tool for solving partial differential equations is the possibility to interpret it in the sense of distributions, that is, to extend it to the Schwartz space S ′ (R N ) of tempered distributions [START_REF] Schwartz | Théorie des Distributions[END_REF]. This property rests essentially on the exchange of regularity and decay between the physical and spectral spaces.

In the previous section, we have studied how such an exchange occurs for the generalized Fourier transform. We have now all the ingredients to construct a space of distributions similar to the Schwartz space to which F can be extended.

Formal construction

In order to generalize the notion of distribution, we use the notion of rigging of a Hilbert space by linear topological spaces, more precisely by nuclear spaces [START_REF] Berezansky | Functional Analysis II[END_REF]. Suppose that we can define a space S A (R + ) such that

S A (R + ) is nuclear, (38) S A (R + ) ⊂ L 2 (R + ) and S A (R + ) is dense in L 2 (R + ), (39)
S A (R + ) ⊂ D(A) and A(S A (R + )) ⊂ S A (R + ), (40)
where the symbol ⊂ stands for continuous embeddings. We choose to identify L 2 (R + ) with its dual space (here we use the term 'dual' to denote the collection of antilinear continuous functionals). Therefore assumption (39) shows that L 2 (R + ) can be interpreted as a subspace of the dual space S ′ A (R + ) of S A (R + ), which yields the following chain of spaces:

S A (R + ) ⊂ L 2 (R + ) = L 2 (R + ) ′ ⊂ S ′ A (R + ), (41)
where the duality product • , • R + between S A (R + ) and S ′ A (R + ) appears as an extension of the inner product of L 2 (R + ) :

ϕ, ψ R + = (ϕ, ψ) R + ∀ϕ ∈ L 2 (R + ), ∀ψ ∈ S A (R + ),
(which is compatible with the initial integral definition of • , • R + , see §1). In the scheme (41), S A (R + ) represents the space of test functions, whereas S ′ A (R + ) plays the role of space of distributions, or generalized functions.

Assumption (40) means that A is a continuous operator in S A (R + ). It allows us to interpret A in the sense of distributions, that is, to extend it to S ′ A (R + ) by setting

Aϕ, ψ R + := ϕ, Aψ R + ∀ϕ ∈ S ′ A (R + ), ∀ψ ∈ S A (R +
). The topological assumption (38) essentially assures that the kernel theorem applies in the functional scheme (41). This theorem, which was initially proved by Schwartz for S ′ (R N ), states that every continuous sesquilinear form a(ϕ, ψ) on S A (R + )×S A (R + ) appears as an 'integral operator' with a distribution kernel in the sense that there exists κ a in the (completed) tensor product

S ′ A (R + ) ⊗ S ′ A (R + ), that is, a distribution of two variables, such that a(ϕ, ψ) = κ a , ϕ ⊗ ψ R + ×R + ∀(ϕ, ψ) ∈ S A (R + ) × S A (R + ).
The rigging (41) of the physical space L 2 (R + ) is naturally converted into a rigging of the spectral space L 2 (Λ; dµ) by the generalized Fourier transform F. Indeed setting S A (Λ) := F(S A (R + )), we can extend F to distributions of S ′ A (R + ) by the formula

Fϕ, ψ Λ := ϕ, F -1 ψ R + ∀ϕ ∈ S ′ A (R + ), ∀ ψ ∈ S A (Λ),
where • , • Λ denotes the duality product between S A (Λ) and its dual space S ′ A (Λ) which defines the space of spectral distributions, and F -1 is simply the restriction of F * to S A (Λ). This can be summarized in the following scheme:

S A (R + ) ⊂ L 2 (R + ) ⊂ S ′ A (R + ) ↓ ↓ ↓ F S A (Λ) ⊂ L 2 (Λ; dµ) ⊂ S ′ A (Λ)

INRIA

where F appears as an isomorphism between each pair of corresponding spaces (vertical arrows). We show below how to construct such spaces. Note that the spectral spaces are derived from the physical spaces via the generalized Fourier transform. Such a construction has a practical interest if we are able to identify S A (Λ) in an intrinsic way, not only as F(S A (R + )). That is where we shall need the results of §4.

Physical distributions

How can we choose a space S A (R + ) which satisfies (38)-( 40)? One could be tempted to consider the restrictions to R + of functions of the usual Schwartz space S(R). But condition (40) would not be satisfied since for a regular ϕ, the function Aϕ is in general discontinuous at x = h. Actually condition (40) implies that S A (R + ) ⊂ D(A n ) for all n ∈ N. Hence a natural way to adapt the definition of S(R) to our operator A is to choose

S A (R + ) := n,m∈N S n,m (42)
where

S n,m := {ϕ ∈ L 2 (R + ); ϕ ∈ D(A n ) and x m A n ϕ ∈ L 2 (R + )}.
Thus S A (R + ) appears as a projective limit of Hilbert spaces, equipped for instance with the countable family of norms ϕ n,m defined by

ϕ 2 n,m := ϕ 2 R + + (1 + x m )A n ϕ 2 R + .
According to (25)-( 27) and the classical embedding theorems of weighted Sobolev spaces [START_REF] Berezansky | Functional Analysis II[END_REF], we have the following characterization of S A (R + ).

Proposition 7 A function ϕ belongs to S A (R + ) if and only if it is infinitely differentiable on both intervals (0, h) and (h, +∞), rapidly decaying at infinity as well as its derivatives (in the sense that lim x→+∞ d n x (x m ϕ(x)) = 0 for all n, m ∈ N) , and satisfies

d 2n x ϕ(0) = 0 and [(d 2 x + k 2 ) n ϕ] h = [d x (d 2 x + k 2 ) n ϕ] h = 0 ∀n ∈ N. ( 43
)
Proposition 8 The space S A (R + ) defined by (42) fulfils assumptions (38)-(40).

Proof. Let us first prove that S A (R + ) is nuclear. Following [START_REF] Berezansky | Functional Analysis II[END_REF], we have to verify that for any (n, m) ∈ N 2 , one can find (n ′ , m ′ ) ∈ N 2 such that S n ′ ,m ′ ⊂ S n,m and the embedding operator S n ′ ,m ′ → S n,m is quasinuclear, i.e., a Hilbert-Schmidt operator. Each S n,m can be seen as a closed subspace of H 2n (0, h) ⊕ H 2n m (h, +∞) (characterized by conditions (26) and ( 27)), where H 2n m (h, +∞) is defined as the usual Sobolev space H 2n (h, +∞) by replacing the Lebesgue measure dx by the weighted measure (1 + x m ) 2 dx. Therefore it is enough to verify that one can find (n ′ , m ′ ) ∈ N 2 such that both embedding operators

H 2n ′ (0, h) → H 2n (0, h) and H 2n ′ m ′ (h, +∞) → H 2n m (h,
(R + ) ⊂ L 2 (R +
) is obvious, and the density of S A (R + ) in L 2 (R + ) follows from the density of D(0, h) and D(h, +∞) respectively in L 2 (0, h) and L 2 (h, +∞) (recall that D(X) denotes the space of infinitely differentiable functions in X with compact support [START_REF] Schwartz | Théorie des Distributions[END_REF]). Hence (39) is satisfied. Finally noticing that Aϕ n,m ≤ C ϕ n+1,m yields (40).

Proposition 7 shows that the dual space S ′ A (R + ) of S A (R + ) is similar to the Schwartz space S ′ (R). It actually contains all restrictions to R + of distributions of S ′ (R) whose order is less than 2 in a vicinity of x = h. For instance, the space of locally integrable tempered functions

L 1 temp (R + ) := f ∈ L 1 loc (R + ); ∃m ∈ N, lim x→+∞ x -m f (x) = 0 (44)
is obviously contained in S ′ A (R + ). Owing to the boundary condition ϕ(0) = 0 in (43), S ′ A (R + ) also contains functions which are not integrable near x = 0, such as x -s for 1 ≤ s < 2. The Dirac distribution δ x at a given point x > 0, defined by δ x , ϕ R + = ϕ(x), belongs to S ′ A (R + ), as well as its derivative δ ′ x . Further derivatives of δ x are also in S ′ A (R + ) if x = h, but not if x = h since the derivatives of order greater than 1 of functions of S A (R + ) are in general discontinuous at this point. On the other hand, A n δ h belongs to S ′ A (R + ) for every n ∈ N : this is the functional

A n δ h , ϕ R + = (-d 2
x -k 2 ) n ϕ(h), which is well defined by virtue of (43).

Spectral distributions

Consider now the space S A (Λ) := F(S A (R + )) of spectral test functions, that is, Spectral distributions of S ′ A (Λ) are then very similar to tempered distributions on (-k2 , +∞). In particular, every distribution of S ′ (R) whose support is contained in (-k2 , +∞) (completed by any scalar values at points λ ∈ Λ p ) belongs to S ′ A (Λ). On the other hand, we see from the proof of Proposition 10 that a function f : Λ → C such that C f ∈ L 1 temp (R + ) (see ( 44)) belongs to S ′ A (Λ). This shows that the space is contained in S ′ A (Λ). We end this section by a simple application of the above results. For a given λ 0 ∈ Λ, function Φ λ0 clearly belongs to L 1 temp (R + ). Its generalized Fourier transform is then defined in the sense of S ′ A (R + ) by FΦ λ0 , ψ Λ = Φ λ0 , F -1 ψ R + = F F -1 ψ(λ 0 ) = ψ(λ 0 ), for all ψ ∈ S A (Λ). This means that FΦ λ0 is the Dirac measure at λ 0 :

S A (Λ) = n,m∈N
FΦ λ0 = δ λ0 .
On the other hand, for a given x 0 ∈ R + , the Dirac measure δ x0 at x 0 belongs to S ′ A (R + ) and

Fδ x0 , ψ Λ = δ x0 , F -1 ψ R + = F -1 ψ(x 0 ) = Φ • (x 0 ), ψ Λ ,
for all ψ ∈ S A (Λ). As a consequence,

Fδ x0 = Φ • (x 0 ).
These relations could have been written formally from the begining. We have now a proper functional context in which they make sense.

Conclusion

We have shown how the generalized Fourier transform associated with the Sturm-Liouville operator A defined in (1) can be interpreted in the sense of distributions by constructing suitable spaces of tempered physical and spectral distributions. This construction is based on the fact that the generalized Fourier transform exchanges regularity and decay between the physical and spectral variables. Thanks to an intensive use of the explicit form of the generalized eigenfunctions of A, the proof of this property requires only very simple tools. On the other hand, it cannot be adapted to more complicated Sturm-Liouville operators for which the generalized eigenfunctions are not known in closed form. The extension of the present results to such situations seems to be an open question.

S

  n,m , where S n,m := F(S n,m ). Theorem 3 (more precisely its consequence (24)) and Theorem 5 provide us a characterization of S n,m :Corollary 9 A function ϕ ∈ L 2 (Λ; dµ) belongs to S n,m if and only if λ n ϕ ∈ L 2 (Λ; dµ) and D m λ (λ n ϕ| Λc ) ∈ L 2 (Λ c ; ρ λ dλ).The following result makes precise what kind of functions S A (Λ) contains.Proposition 10 A function ϕ belongs to S A (Λ) if and only if there exists φ ∈ S(R) such thatϕ(λ) = 1 βλ φ( βλ ) ∀λ ∈ Λ c . Proof. Using the change of variable β = βλ , that is, λ = λ(β) := β 2 -k2 , we see that +∞ -k2 | ϕ(λ)| 2 ρ λ dλ =+∞ 0 |β ϕ(λ(β))| 2 ρβ dβ where ρβ := 2 π|Θ λ(β)+i0 (0)| 2 , INRIA which means that the transformation C defined by(C ϕ)(β) := β ϕ(λ(β)) ∀β ≥ 0 is unitary from L 2 (Λ c ; ρ λ dλ) to L 2 (R + ; ρβ dβ), hence an isomorphism from L 2 (Λ c ; ρ λ dλ) to L 2 (R + ), since M 1 ≤ ρβ ≤ M 2 for some positive constants M 1 and M 2 . Moreover, noticing that C(λ n ϕ) = λ(β) n C ϕ and C(D m λ ϕ) = 1 2 m d m (C ϕ) dβ m ,we infer that ϕ ∈ S A (Λ) if and only if C ϕ is the restriction to R + of a function of S(R). The conclusion follows. Hence a function ϕ of S A (Λ) is infinitely differentiable on (-k2 , +∞) with rapidly decaying derivatives. Near λ = -k2 , it may have a singular behaviour: the above proposition actually tells us that (d m ϕ/dλ m )(λ) = O( β-1-2m λ ) as λ tends to -k2 .

  ) := f : Λ → C; f | Λc ∈ L 1 loc (Λ c ) and ∃m ∈ N, lim λ→+∞ λ -m f (λ) = 0

  +∞) are quasinuclear. For the former, this holds if n ′ > n [1, Theorem 3.2, p.137]. For the latter, we must have n ′ > n and m ′ > m [1, Theorem 4.4, p.147]. This completes the proof.The continuous embedding S A

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