Guillaume Saës
email: guillaume.saes@u-pec.fr

Characterization of p-exponents by continuous wavelet transforms, applications to the multifractal analysis of sum of random pulses

Keywords: Wavelet Analysis, Stochastic processes, Hausdorff dimension, Fractals and multifractals

The theory of orthonormal wavelet bases is a useful tool in multifractal analysis, as it provides a characterization of the different exponents of pointwise regularities (Hölder, p-exponent, lacunarity, oscillation, etc.). However, for some homogeneous self-similar processes, such as sums of random pulses (sums of regular, well-localized functions whose expansions and translations are random), it is easier to estimate the spectrum using continuous wavelet transforms. In this article, we present a new characterization of p-exponents by continuous wavelet transforms and we provide an application to the regularity analysis of sums of random pulses.

Introduction

In the mid-1980s, measurements of the speed of a turbulent flow [START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF][START_REF] Gagne | Etude expérimentale de l'intermittence et des singularités dans le plan complexe en turbulence pleinement développée[END_REF] highlighted signals that seemed irregular everywhere and whose irregularities seemed to vary from one point to another. The notion of pointwise regularity allows to characterize this type of behavior. For a function f ∈ L ∞ loc (R) the pointwise regularity at a point x 0 is most often measured via the Hölder pointwise exponent defined as follows.

Definition 1. Let f ∈ L ∞ loc (R). Let x 0 ∈ R and α ≥ 0. A function f belongs to C α (x 0 ) when there exists a polynomial P f,x0 of degree less than α and C, r > 0 such that ∀x ∈ (x 0 -r, x 0 + r), |f (x) -P f,x0 (x -x 0 )| ≤ C|x -x 0 | α .

The Hölder exponent of f at x 0 is h f (x 0 ) = sup{α : f ∈ C α (x 0 )}.

The Hölder exponent is well defined only when f is locally bounded. Calderón and Zygmund introduced a notion of pointwise regularity for functions that belong to L p loc (R), p ≥ 1, to study the regularity of solutions of certain partial differential equations [START_REF] Calderón | Local properties of solutions of elliptic partial differential equations[END_REF]. Definition 2. Let f ∈ L p loc (R) with p ≥ 1. Let x 0 ∈ R. A function f belongs to T p α (x 0 ) when there exist a polynomial P f,x0 of degree less than α and C, R > 0 such that ∀r ∈ (0, R),

1 r x0+r x0-r |f (x) -P f,x0 (x -x 0 )| p dx 1 p ≤ Cr α .
The p-exponent of f at x 0 is h p f (x 0 ) = sup{α : f ∈ T p α (x 0 )}.

Remark 1. If f ∈ L p (R) then for all x 0 ∈ R, f ∈ T p -1 p (x 0 ). So for all functions f ∈ L p (R), we will be interested only in knowing whether f ∈ T p α (x 0 ) for α ≥ -1/p. The polynomial P f,x0 is unique and note also that, if α ≤ 0, the polynomial is null. The condition for a function to belong to T p α (x 0 ) extends to values p ∈ (0, 1), in this case the spaces L p are replaced by Hardy spaces H p [START_REF] Jaffard | Pointwise regularity associated with function spaces and multifractal analysis[END_REF]. Note that when p = +∞, then h p f (x 0 ) = h f (x 0 ). The wavelet coefficients are known to be convenient tools to compute Hölder exponents and p-exponents [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. However, for some homogeneous self-similar processes, estimating some continuous wavelet transforms (with respect to its wavelet coefficients) turns out to be easier.

In this paper, our goal is to determine a characterization of Hölder pointwise exponents and p-exponents by the continuous wavelet transforms (CWT) and to apply it to determine the pointwise regularity of a class of stochastic processes. For C α (x 0 ) spaces, a characterizations of the Hölder exponent was proposed in [START_REF] Jaffard | Pointwise regularity associated with function spaces and multifractal analysis[END_REF][START_REF] Holschneider | Pointwise analysis of Riemann's "nondifferentiable" function[END_REF] while for T p α (x 0 ) spaces and p-exponents comparable results remain to be established. Such characterizations provide new tools to compute the multifractal p-spectrum of certain functions, processes or signals.

D (p) f (h) = dim H E (p) f (h),
where the set E (p) f (h) is defined as

E (p) f (h) = {x ∈ R : h (p) f (x) = h},
and dim H is the Hausdorff dimension with the convention dim H (∅) = -∞.

In the case p = +∞, since h +∞ f (x 0 ) = h f (x 0 ) we recover the classical definition of the multifractal spectrum. Definition 4. Let f ∈ L ∞ loc (R). The multifractal spectrum D f : [0, +∞] → R + ∪{-∞} of f is the mapping defined for every h ≥ 0 by

D f (h) = dim H E f (h),
where the set E f (h) is defined as E f (h) = {x ∈ R : h f (x) = h}.

A classic example of a multifractal function is the Riemann series defined by

R(x) = +∞ n=1 sin(πn 2 x) n 2 , x ∈ R.
The multifractal spectrum associated with the Hölder exponents was studied by S. Jaffard [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF] and the author already made good use of the continuous wavelet transforms to do so. The spectrum he obtained is the following:

D R (h) =            4h -2 if h ∈ [ 1 2 , 3 4 ] 0 if h = 3 2 -∞ else.
On the other hand, the only known result related to the p-spectrum of the Riemann function concerns its extension

F s (x) = +∞ n=1 e 2iπn 2 x
n s , x ∈ R.

In [START_REF] Seuret | Local l 2 -regularity of Riemann's Fourier series[END_REF], S.

Fs (h) = 4h + 2 -2s, h ∈ 0, s 2 - 1 4 ,
which leaves open the conjecture that the p-spectrum of these functions is independent of p and shifts by s when we operate a fractional derivation of order 1 opened. Another example of not locally bounded multifractal function is the Brjuno function. Its complex version was introduced in 1971 by A. Brjuno to give a condition for the function to be holomorphic in 0. The Brjuno function is defined for every x ∈ R\Q by

B(x) = +∞ n=0 xA(x) . . . A n-1 (x) log 1 A n (x)
,

where A is the Gaussian map which to an irrational x of (0, 1] associates A(x) = {1/x} the fractional part of 1/x. Moreover, S. Marmi, P. Moussa and J-C. Yoccoz showed in [START_REF] Marmi | The brjuno functions and their regularity properties[END_REF] that, B is not locally bounded and B ∈ L p (R) for all p < +∞ (and is even BMO). The multifractal analysis of this function (using p-exponents) was achieved by S. Jaffard and B. Martin in [START_REF] Jaffard | Multifractal analysis of the Brjuno function[END_REF], and they obtained

D (1) B (h) = 2h if h ∈ [0, 1 2 ] -∞ else.
As in the case of the Riemann function, the p-spectrum of the Bjruno function turns out to be independent of p. Note that a continuous wavelet transform technique was already used in the mentioned paper to obtain a lower-bound for the p-exponents of B.

In the two previous cases, the proofs rely on continuous wavelet transform estimates since orthonormal wavelet bases, based on a dyadic grid, are not suitable for the problem.

Continuous wavelet transforms are used to estimate the pointwise regularity of some processes [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF], functions [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF][START_REF] Jaffard | Multifractal analysis of the Brjuno function[END_REF] or signals [START_REF] Arneodo | Wavelet-based multifractal analysis[END_REF]. In the case of sums of random pulses introduced and studied in many papers (see [START_REF] Cioczek-Georges | Stable fractal sums of pulses: the cylindrical case[END_REF][START_REF] Cioczek-Georges | A class of micropulses and antipersistent fractional brownian motion[END_REF][START_REF] Cioczek-Georges | Alternative micropulses and fractional brownian motion[END_REF][START_REF] Lovejoy | Fractal properties of rain, and a fractal model[END_REF][START_REF] Mandelbrot | Introduction to fractal sums of pulses[END_REF]) in order to model rain fields and the volume of water over time [START_REF] Lovejoy | Fractal properties of rain, and a fractal model[END_REF]. In these different cases, the properties of almost sure convergence, continuity, stationarity and self-affinity have been studied. During his PhD., Y. Demichel [START_REF] Demichel | Analyse fractale d'une famille de fonctions aléatoires: les fonctions de bosses[END_REF] computed the uniform Hölder exponent of a large family of sums of pulses and was interested in obtaining some information about the structure of the graphs of such functions and in particular, in estimating their Hausdorff dimension. For the study of the pointwise regularity of such processes, a first multifractal analysis has been performed in [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF][START_REF] Saës | Sommes fractales de pulses : étude dimensionnelle et multifractale des trajectoires et simulations[END_REF], which will be the application case we will consider. This last model is defined as a sum of random dilatations and translations of a function arbitrarily chosen at start. Definition 5. Let (Ω, F, P) a probability space. Let (C n ) n∈N * be a real Poisson point process whose intensity is the Lebesgue measure on R + . Let S be an independent point process of (C n ) n∈N * whose intensity is the Lebesgue measure on R * + × [0, 1]. We write S = (B n , X n ) n∈N * where (B n ) n∈N * is an increasing sequence.

By construction, the three sequences of random variables (C n ) n∈N * , (B n ) n∈N * and (X n ) n∈N * are independent. Let us now recall the definition the sums of random pulses [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF] that we will later study. Definition 6. Let ψ : R → R a non-zero Lipschitz function with support equal to [-1, 1]. The sum of random pulses F α,η : R → R is the stochastic process defined by

F α,η (x) = +∞ n=1 C -α n ψ B 1 η n (x -X n ) , x ∈ R (1)
If α > 0 and η ∈ (0, 1), the sample paths of F α,η are locally bounded and we represent on Figure 1 an example of such a trajectory.

h D Fα,η (h) - - η 1 - - αη α 1 -
Fig. 1 Sample path of Fα,η with α = 0.5, η = 0.9 and ψ :

t → t(1 -t 2 ) 2 if t ∈ [-1, 1], 0 else (left)
and multifractal spectrum of Fα,η with α = 0.5 and η = 0.9 (right)

Unlike in the case of random series or lacunar wavelets series [START_REF] Aubry | Random wavelet series[END_REF][START_REF] Jaffard | On lacunary wavelet series[END_REF], the dyadic network is not privileged in the case of sums of random pulses and it is then not surprising that continuous wavelet transforms are more suited to estimate the multifractal properties in such cases. We emphasize that the proof of the following result obtained in [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF] already partially relied on estimates of some continuous wavelet transforms.

Theorem 1. Let ψ : R → R be a non-zero Lipschitz function supported on [-1, 1], α, η ∈ (0, 1) and let F α,η be the random series defined by [START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF]. With probability 1,

D Fα,η (h) = h α if h ∈ [αη, α] -∞ else.
The multifractal spectrum is plotted in Figure 1. The proof relies on a characterization of the Hölder exponent by continuous wavelet transforms. As it allows to choose wavelets that are "positioned" at any point, this tool is more flexible and convenient to use in the case of sums of random pulses as we can choose a wavelet suited to the location of our pulses. The main difficulty is to obtain a characterization of the p-exponents by continuous wavelet transforms. Section 2 presents a consistent definition of continuous p-leaders in terms of wavelet coefficients and Theorem 4, which together with Theorem 2 is the main result of this paper, yields a characterization of the T p α (x 0 ) spaces based on p-leaders. Section 3 proposes a proof of this theorem.

The only known examples of random processes for which one gets almost surely a p-spectrum are lacunary wavelet series [START_REF] Abry | Multifractal analysis based on p-exponents and lacunarity exponents[END_REF]. We will study the p-spectrum of F α,η processes for α < 0. Those are examples of processes whose sample paths are almost surely non-locally bounded. As such they are an interesting model since many signals cannot be modeled by locally bounded functions [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF][START_REF] Leonarduzzi | p-exponent and p-leaders, part ii: Multifractal analysis. relations to detrended fluctuation analysis[END_REF] and such situations are in general not well understood yet. The section 4 of the article proposes a proof of the following theorem Theorem 2. Let η ∈ (0, 1), α < 0 such that η -1 < αη and p ∈ (1, -1/(αη) + 1/α).

Let ψ : R → R a continuous lipschitzian function with support on [-1, 1]. Let F α,η : R → R the process defined by [START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF]. Then almost certainly, we have

D (p) Fα,η (H) = Hηp + η αηp + 1 if H ∈ αη, α + 1 -η ηp .
Finally, we emphasize also that the p-exponents can differ from the Hölder exponents and thus give additional information on the nature of the irregularities at a point, even in the case of processes with locally bounded sample paths. It is then natural to study new characterizations of T p α (x 0 ) spaces by continuous wavelet transforms. This was achieved by V. Perrier and C. Basdevant in their paper [START_REF] Perrier | Besov norms in terms of the continous wavelet transform. application to structure functions[END_REF] in the case of the space L p (using a result of Stein's book [START_REF] Stein | Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals[END_REF]). We will start from this characterization to obtain the one of L p spaces as it has been done in the discrete case in the proof of S. Jaffard and C. Melot in [START_REF] Jaffard | Wavelet analysis of fractal boundaries. part 1: Local exponents[END_REF].

2 The continuous p-leaders, characterisation of T p α (x 0 )

Let α ∈ R be fixed and r ≥ ⌊max(1, α)⌋. Let ψ ∈ C r (R) be a non-zero even function which is called wavelet, with support included in [-1, 1], and having r + 1 vanishing moments, i.e.

R ψ(x)dx = R xψ(x)dx = • • • = R x r ψ(x)dx = 0. Definition 7.
The continuous wavelet transform of a function f ∈ L 2 (R) is defined in [START_REF] Antoine | Two-Dimensional Wavelets and Their Relatives[END_REF] (see also [START_REF] Meyer | Principe d'incertitude, bases hilbertiennes et algèbres d'opérateurs[END_REF][START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]) by

W f (a, b) = 1 a R f (x)ψ x -b a dx.
There exists a constant c ψ > 0 such that the reconstruction formula is valid for f ∈ L 2 (R) with

f (x) = 1 c ψ R * + R 1 a 2 W f (a, b)ψ x -b a dbda, x ∈ R d ,
if the wavelet ψ verifies the following admissibility condition

R- | ψ(ξ)| 2 |ξ| dξ = R+ | ψ(ξ)| 2 |ξ| dξ < +∞,
which is the case for ψ under our assumption. Additionally it is possible to take another wavelet ϕ of class C N (R) where N > max(0, α) with this time (at least) the first vanishing moment such that the recomposition of f is

f (x) = R * + R 1 a 2 W f (a, b)ϕ x -b a dbda, x ∈ R, (2)
(see for example [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][START_REF] Grossmann | Reading and understanding continuous wavelet transforms[END_REF][START_REF] Jaffard | Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions[END_REF] for possible choices of ϕ). The reconstruction formula (2) holds almost everywhere if f ∈ L p (R) with p > 1. The choice of a reconstruction wavelet that differs from the analyzing wavelet offers an additional flexibility that will prove important in proving Theorem 4. Before stating the characterization of T p α (x 0 ) spaces by continuous wavelet transforms, we recall the definitions of wavelet coefficients and its associated quantities in an orthonormal wavelet basis. Definition 8. Let φ be a non-zero oscillating function, with support in [-1, 1], having a number r φ ≥ 1 of vanishing moments and of class C rφ-1 (R). The function φ is called a mother wavelet when {φ j,k (t) = 2

j 2 φ(2 j t -k)} (j,k)∈N×Z forms an orthonormal basis of L 2 (R). The discrete wavelet coefficients of a function f ∈ L 2 (R) are defined by c j,k = 2 j 2 R f (x)φ j,k (x)dx = 2 j R f (x)φ 2 j x -k dx, (j, k) ∈ N × Z.
For any (j, k) ∈ N × Z, let λ j,k be the dyadic interval defined by λ j,k = [k2 -j , (k + 1)2 -j ]. We will use the notation 3λ j,k for the union of λ j,k and the 2 intervals adjacent to

λ j,k , 3λ j,k = [(k -1)2 -j , (k + 2)2 -j ].
Finally, for all j ∈ N and x 0 ∈ R, let λ j (x 0 ) the unique dyadic interval λ j,k such that x 0 ∈ λ j,k . The wavelet leaders are defined as follows.

Definition 9. Let p ∈ R * + and f ∈ L p loc (R). If p = +∞, then the wavelet leaders of f are ∀(j, k) ∈ N × Z, l f (j, k) = sup λ j ′ ,k ′ ⊆3λ j,k j ′ ≥j {|c j ′ ,k ′ |}. If p < +∞, then the p-leaders of f are ∀(j, k) ∈ N × Z, l (p) f (j, k) =     λ j ′ ,k ′ ⊆3λ j,k j ′ ≥j |c j ′ ,k ′ | p 2 -(j ′ -j)     1 p .
In the case of continuous wavelet transforms, we define a continuous version of the p-leaders as a local L p -norm of coefficients W f (a, b). For this, we rely on a characterization of L p -spaces for p > 1 that V. Perrier and C. Basdevant [START_REF] Perrier | Besov norms in terms of the continous wavelet transform. application to structure functions[END_REF] (Theorem 3.1) have constructed.

Theorem 3. Let f ∈ L p loc (R) with p ∈ (1, +∞). Let α > -1/p. Let ψ be even function with support in [-1, 1] having r ≥ max(α, 1) vanishing moments and ψ ∈ C r (R). There exists C 1 , C 2 > 0, depending only on the wavelet ψ such that if N f = +∞ -∞ +∞ 0 |W f (s, t)| 2 ds s p 2 dt 1 p , then C 2 N f ≤ ∥f ∥ L p ≤ C 1 N f . Definition 10. Let f ∈ L ∞ loc (R). The continuous leaders of f are ∀(a, b) ∈ R * + × R, L f (a, b) = sup (s,t)∈(0,a)×B(b,a) |W f (s, t)|.
where

B(b, a) = (b -a, b + a). Let f ∈ L p loc (R). The continuous p-leaders of f are ∀(a, b) ∈ R * + × R, L (p) f (a, b) = 1 a B(b,a) a 0 |W f (s, t)| 2 ds s p 2 dt 1 p
.

This indicates that the notion of T p α (x 0 ) regularity can be related to p-leaders in the framework of continuous wavelet transforms. However for p = 1, one cannot obtain such a characterization since, for p > 1 their characterization is based on the characterization of L p -spaces, and L 1 cannot be characterized by wavelets (L 1 has no unconditional basis) [START_REF] Meyer | Ondelettes et Opérateurs[END_REF]. The result we will prove is the following.

Theorem 4. Let f ∈ L p loc (R) with p ∈ (1, +∞). Let α > -1/p and x 0 ∈ R. Let ψ be even function with support in [-1, 1] having r ≥ max(α, 1) vanishing moments and ψ ∈ C r (R). 1. For any f ∈ T p α (x 0 ), there exists C f > 0 such that for any a ∈ R * + small enough, |L (p) f (a, x 0 )| ≤ C f a α . (3) 2. Conversely, if α / ∈ N and (3) is verified, then f belongs to T p α (x 0 ).
Moreover, this theorem allows to determine the p-exponent in the following way.

Corollary 2.1. Let f ∈ L p loc (R) with p ∈ (1, +∞). If h p x0 (x 0 ) ≤ 1 then the p-exponent of f in x 0 is h (p) f (x 0 ) = lim inf a→0 + log(|L (p) f (a, x 0 )|) log(a)
.

The section 3 present a proof of theorem 4.

3 Proof of characterization of T p α (x 0 ) spaces

The Definition 10 of p-leaders is a continuous version of the discrete leaders defined using wavelet coefficients. We will check on an example the consistency of this definition with classical results on discrete p-leaders.

Consider the cusp function g α : x → |x| α with α > 0. When α is not an even integer and if we take the wavelet φ supported in [-1, 1], then the wavelet coefficients in (j, k) are given by a change of variable u = 2 j x -k by

c j,k = 2 j R |x| α φ(2 j x -k)dx = 2 -αj 1 -1 |u + k| α φ(u)du = 2 -αj w φ,α (k) where w φ,α (t) = 1 -1 |u + t| α φ(u)du. We deduce the p-leaders of g α by [l (p) gα (j, 0)] p = 2 j λ j ′ ,k ′ ⊆3λj,0 j ′ ≥j 2 -(1+αp)j ′ w φ,α (k ′ ) p = 2 -αpj λ j ′ ,k ′ ⊆3λj,0 j ′ ≥j 2 -(1+αp)(j ′ -j) w φ,α (k ′ ) p = C p α 2 -αjp
where

C α =     λ j ′′ ,k ′ ⊆3λj,0 j ′′ ≥0 2 -(1+αp)j ′′ w φ,α (k ′ ) p     1 p > 0
which is a well defined constant because α ≥ -1/p. As a conclusion, it is known [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF][START_REF] Jaffard | Exposants de hölder en des points donnés et coefficients d'ondelettes[END_REF] that the discrete p-leader l (p) gα (j, k) have a scaling law behavior 2 -αj in 0, i.e., there exists C α > 0 such that for any j ∈ N, l (p) gα (j, 0) = C α 2 -αj and from the discrete characterization of T p α (x 0 ) [START_REF] Jaffard | p-exponent and p-leaders, Part I: Negative pointwise regularity[END_REF], we have h

(p)
gα (0) = α. In the continuous case of p-leaders, it is therefore expected that their behavior follows the same a α scaling invariance. The continuous wavelet transform of the cusp for (a, b

) ∈ R * + × R is W gα (a, b) = R g α (x)ψ x -b a dx a = R g α (au+b)ψ(u)du = a α 1 -1 u + b a α ψ(u)du. Thus, W gα (a, b) = a α w ψ,α (b/a) où w ψ,α (t) = 1 -1 |u + t| α ψ(u)du.
The continuous pleaders of the cusp g α are computed by changing the variable t = t ′ /ad and s = s ′ /a in the following manner

L (p) gα (a, 0) p = 1 a a -a a 0 |s α w ψ,α (t/s)| 2 ds s p 2 dt = a αp 1 -1 1 0 s ′α w ψ,α t ′ s ′ 2 ds ′ s ′ p 2 dt ′ .
We obtain precisely the power law of the continuous p-leader with L (p) gα (a, 0) = Ka α where

K =   1 -1 1 0 s ′α w ψ,α t ′ s ′ 2 ds ′ s ′ p 2 dt   1 p
.

By integration by part, one easily checks that w is a regular and well-localized function so we deduce the same behavior as the p-leaders in the discrete case.

In a first step, we present a proof of part 1. of Theorem 4 and in a second step the proof of part 2.

Proof of point 1. of Theorem 4

We recall the hypothesis of Theorem 4. Let f ∈ L p loc (R) with p ∈ (1, +∞). Let α > -1/p and x 0 ∈ R. Let ψ be even function with support in [-1, 1] having r ≥ max(α, 1) vanishing moments and ψ ∈ C r (R).

In this section, we prove for any f ∈ T p α (x 0 ), there exists C f > 0 such that for any a ∈ R * + small enough, we have (3) holds. Proof. Since f ∈ T p α (x 0 ), there exist two constants C, R ∈ R * + and a polynomial P f,x0 of degree less than α such that (1) holds. Recall that if α ∈ [-1/p, 0], then the polynomial P f,x0 vanishes. Let r ∈ (0, R) and g be the function defined by

g(x) = [f (x) -P f,x0 (x -x 0 )]1 B(x0,r) (x), x ∈ R.
Thus from (1),

∥g∥ L p = x0+r x0-r |f (x) -P f,x0 (x -x 0 )| p dx 1 p ≤ Cr α+ 1 p . (4)
According the Theorem 4, there exist C 1 , C 2 > 0 depending only on the wavelet ψ such that,

C 2 N g ≤ ∥g∥ L p ≤ C 1 N g . (5)
Let r > 0. Since the wavelet

ψ is supported in [-1, 1], x → ψ x-t s is supported in B(t, s) and for all s ∈ 0, r 2 , t ∈ B x 0 , r 2 , W g (s, t) = 1 s B(x0,r)∩B(t,s) (f (x) -P f,x0 (x -x 0 ))ψ x -t s dx. If x ∈ B(t, s) then |x -x 0 | ≤ |x -t| + |t -x 0 | ≤ s + r 2 < r. Hence B(t, s) ⊆ B(x 0 , r) and W g (s, t) = 1 s B(t,s) f (x)ψ x -t s dx - 1 s B(t,s) P f,x0 (x -x 0 )ψ x -t s dx.
The function ψ has r ψ + 1 ≥ α + 1 vanishing moments and the polynomial P f,x0 is of degree less than α, hence

1 s B(t,s) P f,x0 (x -x 0 )ψ x -t s dx = 0.
Therefore,

∀s ∈ 0, r 2 , ∀t ∈ B x 0 , r 2 , W g (s, t) = 1 s R f (x)ψ x -t s dx = W f (s, t).
(6) Applying ( 6) with r = 2a for a small enough, and using (5), we found an upper bound for the L p -leaders (3) by writing

∥g∥ L p ≥ C 2 B(x0,a) a 0 |W g (s, t)| 2 ds s p 2 dt 1 p = C 2 B(x0,a) a 0 |W f (s, t)| 2 ds s p 2 dt 1 p = C 2 L (p) f (a, x 0 )a 1 p .
By applying (4), we deduce that there exists C ′ > 0 such that

L (p) f (a, x 0 ) ≤ 1 C 2 ∥g∥ L 2 a -1 p ≤ C ′ r α+ 1 p a -1 p ≤ C ′ a α .

Proof of point 2. of the Theorem 4

In this section, we proof if α / ∈ N and (3) is verified, then f belongs to T p α (x 0 ). Proof. Suppose that for α / ∈ N, there exists C f > 0 such that for all a ∈ R * + ,

|L (p) f (a, x 0 )| ≤ C f a α .
Let us show that f ∈ T p α (x 0 ). Since ψ is even, the admissibility condition (2) is verified. By the hypothesis (2.4.6) of [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF] or (B.26) of [START_REF] Jaffard | Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions[END_REF], we fix ϕ a wavelet different from ψ of class C N ψ (R) where N ψ > max(α, 0) and compactly supported on [-1, 1] with at least one vanishing moment and thus (2) is valid with this new wavelet.

We separate the cases α > 0 and α ∈ (-1/p, 0].

• For α > 0 : Let r ∈ (0, 1) and x ∈ B(x 0 , r). Let

C(x 0 , r) = {(s, t) ∈ R * + × R : |t -x 0 | < s + 2r} t s B(x 0 , 2r + s) x 0 B(x 0 , 2r) B(x 0 , r) r 1 - - D 1 (x 0 , r) D 2 (x 0 , r) D 3 (x 0 , r) D 4 (x 0 , r) Fig. 2 Cutting into 4 parts For all (s, t) ∈ R * + × R, we define V 1 (s, t) = W f (s, t) if (s, t) ∈ D 1 (x 0 , r) = C(x 0 , r) ∩ ((0, r) × R) 0 else V 2 (s, t) = W f (s, t) if (s, t) ∈ D 2 (x 0 , r) = C(x 0 , r) ∩ ([1, r) × R) 0 else V 3 (s, t) = W f (s, t) if (s, t) ∈ D 3 (x 0 , r) = C(x 0 , r) ∩ ([r, +∞) × R) 0 else V 4 (s, t) = W f (s, t) if (s, t) ∈ D 4 (x 0 , r) = R * + × R \C(x 0 , r) 0 else.
We decompose W f into the four integrals 2). For j = 1, 2, 3, 4, let

W f = V 1 + V 2 + V 3 + V 4 (see Figure
I j (x) = 1 c ϕ +∞ s=0 +∞ t=-∞ V j (s, t)   ϕ x -t s - ⌊α⌋ k=0 ϕ (k) x0-t s k! x -x 0 s k   dtds s 2 .
Thus,

I j (x) = 1 c ϕ Dj (x0,r) W f (s, t)   ϕ x -t s - ⌊α⌋ k=0 ϕ (k) x0-t s k! x -x 0 s k   dtds s 2
We will bound the quantities, ∥I j 1 B(x0,r) ∥ L p for j = 1, 2, 3, 4 in order to show that the Taylor-Lagrange polynomial does exist.

Estimation of I 1 : The proof of ( 5) provides the upper-bound of the norm L p of I 1 only based on the reconstruction formula (2) (indeed, this demonstration does not use that W f is a continuous wavelet transform). We deduce that

∥I 1 1 B(x0,r) ∥ L p ≤ C 1 +∞ 0 +∞ 0 |V 1 (s, t)| 2 ds s p 2 dt 1 p
Since 0 < s < r, one has B(x 0 , 2r + s) ⊆ B(x 0 , 3r) and so

∥I 1 1 B(x0,r) ∥ L p ≤ C 1 B(x0,3r) r 0 |W f (s, t)| 2 ds s p 2 dt 1 p ≤ C 1 (3r) 1 p L (p) f (3r, x 0 ).
From (3), we conclude that there exists C ′ 1 > 0 such that

∥I 1 1 B(x0,r) ∥ L p = C 1 (3r) 1 p (3r) -α ≤ C ′ 1 r α+ 1 p . ( 7
)
Estimation of I 2 : According to the Taylor-Lagrange theorem applied to ϕ ∈ C ⌊α⌋+1 , there exists C > 0 such that

|I 2 (x)| ≤ C 1 s=r t∈B(x0,2r+s) |W f (s, t)| • x -x 0 s ⌊α⌋+1 dtds s 2 .
There exists a unique J ∈ N * such that 2 -J < r ≤ 2 -(J-1) . Thus,

|I 2 (x)| ≤ C|x -x 0 | ⌊α⌋+1 J j=1 B(x0,2r+2 -(j-1) ) 2 -(j-1) 2 -j |W f (s, t)| s 1 2 ds s ⌊α⌋+ 5 2 dt.
By Cauchy-Schwarz,

|I 2 (x)| ≤ C|x -x 0 | ⌊α⌋+1 J j=1 B(x0,2r+2 -(j-1) ) 2 -(j-1) 2 -j |W f (s, t)| 2 ds s 1 2 × 2 -(j-1) 2 -j ds s 2⌊α⌋+5 1 2 dt, so that |I 2 (x)| ≤ C|x -x 0 | ⌊α⌋+1 J j=1 2 (⌊α⌋+2)j B(x0,6•2 -j ) 6×2 -j 0 |W f (s, t)| 2 ds s 1 2
dt.

According to Hölder's inequality, for q > 0 such that 1/p + 1/q = 1,

|I 2 (x)| ≤ C|x -x 0 | ⌊α⌋+1 J j=1 2 (⌊α⌋+2)j   B(x0,6•2 -j ) 6•2 -j 0 |W f (s, t)| 2 ds s p 2 dt   1 p × B(x0,6•2 -j ) 1 q dt 1 q ≤ C|x -x 0 | ⌊α⌋+1 J j=1 2 (⌊α⌋+2)j • (6 • 2 -j ) 1 p • L (p) f (6 • 2 -j , x 0 ) • (2 • 6 • 2 -j ) 1 q .
From (3), there exists C > 0 such that

|L (p) f (6 • 2 -j , x 0 )| =   1 6 • 2 -j B(x0,6•2 -j ) 6•2 -j 0 |W f (s, t)| 2 ds s p 2 dt   1 p ≤ C(6 • 2 -j ) α .
We deduce that

|I 2 (x)| ≤ C|x -x 0 | ⌊α⌋+1 J j=1 2 (⌊α⌋+2)j C(6 • 2 -j ) α+ 1 p (2 • 6 • 2 -j ) 1 q .
Since 1/p+1/q = 1, there exists C ′ > 0 independent of J and x such that for α ∈ (0, 1)

|I 2 (x)| ≤ C ′ |x -x 0 | ⌊α⌋+1 J j=1 2 (1-α+⌊α⌋)j .
We conclude that there exists

C α > 0 such that |I 2 (x)| ≤ C α |x -x 0 | ⌊α⌋+1 2 (1-α+⌊α⌋)J .
But since x ∈ B(x 0 , r) and r ∈ (2 -J , 2 -(J-1) ], we have

|I 2 (x)| ≤ 2C α 2 -αJ ≤ 2C α r α .
So there is a constant C 2 = 2 1+1/p C α > 0 independent of r such that

∥I 2 1 B(x0,r) ∥ L p = x0+r x0-r |I 2 (x)| p dx 1 p ≤ 2C α r α (2r) 1 p = C 2 r α+ 1 p . (8)
Estimation of I 3 : Since ϕ ∈ C α (R) and s > 1, there exists C > 0 such that

ϕ x -t s - ⌊α⌋ k=0 ϕ (k) x0-t s k! x -x 0 s k ≤ C x -x 0 s α ≤ C|x -x 0 | α . 14
We note

C ′ = C c ϕ +∞ s=1 t∈B(x0,3+s) |W f (s, t)| dtds s 2 .
Thus, for all x ∈ B(x 0 , r), we have

|I 3 (x)| ≤ C c ϕ Dj (x0,r) |W f (s, t)||x -x 0 | α dtds s 2 ≤ C c ϕ r α +∞ s=1 t∈B(x0,2r+s) |W f (s, t)| dtds s 2 ≤ C ′ r α
from which, as for I 2 , we deduce directly that there exists C 3 > 0 such that

∥I 3 1 B(x0,r) ∥ L p ≤ C 3 r α+ 1 p . ( 9
)
Estimation of I 4 : For all s ∈ R * + , one knows that supp(ϕ) ⊆ B(0, 1) and so for (s, t) → ϕ ((x -t)/s), supp(ϕ s,t ) ⊆ B(t, s). Thus, if t ∈ R\B(x 0 , 2r + s), then t / ∈ B(x 0 , s). Moreover, since x ∈ B(x 0 , r), it follows that

2r + s ≤ |t -x 0 | ≤ |t -x| + |x -x 0 | ≤ |t -x| + r and so |t -x| ≥ r + s ≥ s. We conclude that if t ∈ R\B(x 0 , 2r + s), then t / ∈ B(x 0 , s) ∪ B(x, s), and ϕ x0-t s = ϕ x-t s = 0. So I 4 (x) = 0. (10)
In conclusion, for i = 1, 2, 3, 4, the functions x → I i (x)1 B(x0,r) belong to T p α (x 0 ) and so according to the wavelet reconstruction formula (2), we have

I 1 + I 2 + I 3 = f (x) -P f,x0 (x -x 0 ) with P f,x0 (x -x 0 ) = ⌊α⌋ k=0 +∞ s=0 +∞ t=-∞ W f (s, t) ϕ (k) x0-t s k! dtds s k+2 (x -x 0 ) k
Combining ( 7), ( 8), ( 9) and ( 10), there exists C > 0 such that

1 r B(x0,r) |f (x) -P f,x0 (x -x 0 )| p dx 1 p ≤ 1 r 1 p 4 k=1 ∥I k 1 B(x0,r) ∥ L p ≤ Cr α .
This shows that f ∈ T p α (x 0 ), i.e. item 1. of Theorem 4 for α > 0.

• For α ∈ (-1/p, 0] : Let r ∈ (0, 1) and x ∈ B(x 0 , r). Since α ≤ 0, we can take P f,x0 = 0, hence

f (x) = 1 c ϕ +∞ s=0 +∞ t=-∞ W f (s, t)ϕ x -t s dtds s 2 .
We have f (x) = I ′ 1 (x) + I ′ 2 (x) + I ′ 3 (x) + I ′ 4 (x), with for j = 1, 2, 3, 4,

I ′ j (x) = 1 c ϕ Dj (x0,r) W f (s, t)ϕ x -t s dtds s 2 .
We need to find an upper-bound for the L p -norm of x → |f (x)|1 B(x0,r) (x).

Estimation of I ′ 1 : The estimate leading to (7) remains valid. Estimation of I ′ 2 : Bounding ϕ by ∥ϕ∥ ∞ since it is continuous compactly supported, we obtain

|I ′ 2 (x)| ≤ ∥ϕ∥ ∞ c ϕ 1 s=r t∈B(x0,2r+s) |W f (s, t)| dtds s 2 .
Let J ∈ N such that 2 -J < r ≤ 2 -(J-1) . Thus, there exists C 2 > 0 such that

|I ′ 2 (x)| ≤ C 2 J j=1 B(x0,2r+2 -(j-1) ) 2 -(j-1) 2 -j |W f (s, t)| s 1 2 ds s 3 2 dt.
By Cauchy-Schwarz inequality,

|I ′ 2 (x)| ≤ C 2 J j=1 B(x0,2r+2 -(j-1) ) 2 -(j-1) 2 -j |W f (s, t)| 2 ds s 1 2 2 -(j-1) 2 -j ds s 3 1 2
dt.

The remaining part of the computation is the same as in the case α ∈ (0, 1) but with α ∈ (-1/p, 0), so there exists C 4 > 0 such that

|I ′ 2 (x)| ≤ C 4 J j=1 2 -αj .
Since α < 0, there exists C α > 0 such that |I ′ 2 (x)| ≤ C α 2 -αJ . Therefore there exists a constant C > 0 independent of r such that

∥I ′ 2 1 B(x0,r) ∥ L p = B(x0,r) |I ′ 2 (x)| p dx 1 p ≤ Cr α+ 1 p .
Estimation of I ′ 3 : The reconstruction formula by the continuous wavelet transform restricted to s ≥ 1 gives a function g defined by

g(x) = 1 c ϕ +∞ s=1 t∈R W f (s, t)ϕ x -t s dtds s 2 .
The function g is of class C N where N is the regularity of the wavelet ϕ [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. Therefore, since α ∈ (-1/p, 0), 1 ≤ r α and therefore there exists C ′ > 0 such that

C ′ = ∥ϕ∥ ∞ c ϕ +∞ s=1 t∈R |W f (s, t)| dtds s 2 < +∞.
Thus, for all x ∈ B(x 0 , r), we have

|I ′ 3 (x)| ≤ ∥ϕ∥ ∞ c ϕ +∞ s=1 t∈R |W f (s, t)| dtds s 2 ≤ C ′ ≤ C ′ r α .
And it follows that there exists C 3 > 0 such that

∥I ′ 3 1 B(x0,r) ∥ L p ≤ C 3 r α+ 1 p .
Estimation of I ′ 4 : It is identical to the α > 0 case. Combining the 4 previous results, gives C > 0 such that

1 r B(x0,r) |f (x) -P f,x0 (x -x 0 )| p dx 1 p ≤ 1 r 1 p 4 k=1 ∥I ′ k 1 B(x0,r) ∥ L p ≤ Cr α .
We conclude that f ∈ T p α (x 0 ).

p-spectrum of sums of random pulses

Definition 6 yields locally bounded sample paths of sums of random pulses for α > 0 and η ∈ (0, 1/α) [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF]. However, in this part, it is also relevant to consider the nonlocally bounded case with α < 0. We will start by giving sufficient conditions for convergence of such series in L p . More precisely, we will show in a first step that, when α < 0, η ∈ (0, 1) and η -1 < αη, the sums of random pulses are in a space L p loc (R) for p ∈ [1, -1/(αη) + 1/α). In a second step, we will compute their p-multifractal spectra.

Recalls

We recall the notations [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF]. Let

A j = {n ∈ N * : 2 j-1 ≤ B 1 η n < 2 j } if j ̸ = 0 A 0 = {n ∈ N * : 0 ≤ B 1 η n < 1}
We also state preliminary lemmas whose proofs can be found in [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF] (see Lemmas 3.1, 3.2 and 3.3) see also [START_REF] Saës | Sommes fractales de pulses : étude dimensionnelle et multifractale des trajectoires et simulations[END_REF]. Lemma 4.1. Almost surely for j large enough and ε j = log 2 (j)/(ηj).

2 ηj(1+εj ) ≤ Card(A j ) ≤ 2 ηj(1+εj ) . Lemma 4.2. Almost surely, there exists K 1 , K 2 > 0 such that for all j ∈ N * and n ∈ A j ,

K 1 2 ηj(1-εj ) ≤ B n , C n ≤ K 2 2 ηj(1+εj ) .
For all x, r ∈ [0, 1], we note

T n (x, r) = 1 if B(X n , B
1/η n )∩B(x, r) ̸ = ∅ and T n (x, r) = 0 else. For r = 0, we write T n (x) = T n (x, 0). Lemma 4.3. Almost surely, there exists K > 0 such that for all x ∈ [0, 1] and J, j ∈ N with j ̸ = 0,

n∈Aj T n (x, 2 -ηJ ) ≤ Kj 2 max{1, 2 η(j-J) } et n∈Aj T n (x) ≤ Kj 2 .
Let p 0 be a sufficiently large integer such that ρ > 3-3α 1-αη . Let gamma ∈ (0, 1 η -1) and consider for all j ∈ N, the sets

A j = ⌊γj⌋ j ′ =⌊(1-ρηεj )j⌋ A j and I j = {n ∈ A j : ∀m ∈ A j , n ̸ = m, B(X n , B -1 η n ) ∩ B(X m , B -1 η m ) = ∅}.
We define the families of sets G δ and G ′ δ by

G δ = lim sup j→+∞ n∈Aj B(X n , B -δ n ) and G ′ δ = lim sup j→+∞ n∈ Aj B(X n , B -δ(1-εj ) n ) and
We prove the Theorem 2 which yields the multifractal analysis of random sums of pulses in the p-exponent framework.

When p = +∞, this result boils down to the results proved in [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF]. Let us notice that the sample paths of F α,η have the same p-spectrum as the random lacunar wavelets series studied [START_REF] Leonarduzzi | p-exponent and p-leaders, part ii: Multifractal analysis. relations to detrended fluctuation analysis[END_REF].

Suitable of p values for p-multifractal analysis

First, let us determine the spaces L p loc (R) to which the function f belongs according to the values of α < 0. Proposition 5. Let ψ : R → R a continuous lipschitzian function with support on [-1, 1]. Let η ∈ (0, 1), α < 0 such that η -1 < αη. Almost surely, for all p ∈ (1, -1/(αη)+1/α), the sample paths of the process F α,η : R → R defined by (1) belongs to L p loc ([0, 1]). Note that the condition η -1 < αη guarantees that -1/(αη) + 1/α > 1.

Proof. Let p ∈ [1, -1/(αη) + 1/α), and consider

F α,η (x) = +∞ j=0 F j (x) where F j (x) = n∈Aj C -α n ψ(B 1 η n (x -X n )).
We estimate separately the L p -norm of each F j . Applying the definition of T n and the Lemma 4.3, there exists K > 0 such that for all x ∈ R,

|F j (x)| ≤ K∥ψ∥ ∞ max n∈Aj {C -α n } n∈Aj T n (x) ≤ Kj 2 max n∈Aj {C -α n }1 n∈A j B(Xn,B -1/η n ) (x)
and

∥F j ∥ L p ≤ Kj 2 max n∈Aj {C -α n }∥1 n∈A j B(Xn,B -1/η n ) ∥ L p . But the Lebesgue measure of the support of n∈Aj 1 B(Xn,B -1/η n ) is bounded by Card(A j ) max n∈Aj {B -1/η n }, we deduce that ∥F j ∥ L p ≤ Kj 2 max n∈Aj {C -α n }Card(A j ) 1/p max n∈Aj {B -1 ηp n }.
By Lemmas 4.1 and 4.2,

∥F j ∥ L p ≤ Kj 2 j α 2 -αηj j 1/p 2 η p j j 1/ηp 2 -1 p j ≤ Kj C(α,η,p) 2 (-αη+ η p -1 p )j .
Finally,

j∈N ∥F j ∥ L p ≤ K j∈N j C(α,η,p) 2 (-αη+ η p -1 p )j .
By hypothesis, -αη + η/p + 1/p < 0. We conclude that almost surely the series j∈N |F j (x)| converges in L p for p ∈ [1, -1/(αη) + 1/α). So F j converges too and it converges to F α,η . Thus, F α,η ∈ L p for p ∈ [1, -1/(αη) + 1/α), as stated.

Study of p-exponents

Proposition 6. Let ψ : R → R be a continuous Lipschitz function with support on [-1, 1]. Let η ∈ (0, 1), α < 0 such that η -1 < αη. Consider the process F α,η defined by (1). Almost surely, for all p ∈ [1, -1/(αη) + 1/α), for all δ ∈ (1, 1 η ) and for all

x 0 / ∈ G δ , h (p)
Fα,η (x 0 ) ≥ α δ + 1 -δη δηp = α + 1/(ηp) δ + 1/p.
Proof. Let x / ∈ G δ . By definition, there exists J x ∈ N depending on x such that for any integer j ≥ J x , for any n

∈ A j , x / ∈ B(X n , B -δ n ). The sum Jx j=1 F j
has a finite number of terms, so its global regularity is that of the wavelet. So, in estimating the regularity of F α,η in x, we can assume that the sum in F α,η is taken to j ≥ J x . We take r > 0 fixed. To estimate ∥F α,η ∥ L p (B(x,r)) , we decompose F α,η into j∈N F j as in ( 11) and we will determine first for which j the supports of the pulses indexed by elements of

A j can intersect B(x, r). Let j ≥ J x , n ∈ A j . Since x / ∈ G δ , |x -X n | ≥ B -δ n ≥ 2 -δηj . If B(x, r) intersects B(X n , B -1/η n ), it implies that r ≥ 1 2 B -δ n ≥ 1 2 2 -ηδj .
Let J ∈ N be the first integer such that 2 -ηJ ≤ (2r) 1/δ .

• If B n ∈ A j for j < J, then B(X n , B -δ n ) ∩ B(x, r) = ∅ and so ∥F j ∥ L p (B(x,r)) = 0. • If B n ∈ A j for j ≥ J, then Lemma 4.3 states that there are at most Kj 2 2 η(j-J) pulses in A j whose support intersects B(x, r). Moreover, at a given point, at most j 2 overlap. A computation similar to the proof of Proposition 5 gives

B(x,r) |F j (x)| p dx ≤ Cj 2p Kj 2 2 η(j-J) max n∈Aj C -α n p max n∈Aj B -1 η n . By Lemma 4.2, B(x,r) |F j (x)| p dx ≤ Cj 2p Kj 2 2 η(j-J) 2 -ηpαj 2 -j 2 -αηpjεj . Since 2 ηjεj = j, ∥F j ∥ L p (B(x,r)) ≤ Cj 2+ 2 p -α 2 (-ηα+ η-1 p )j 2 -η p J .
The series of norms converges since -αη + η-1 p < 0 ⇐⇒ p < 1 α -1 αη and thus

∃c 1 > 0, +∞ j=J ∥F j ∥ L p (B(x,r)) ≤ J c1 2 (-ηα-1 p )J ≤ 2 -(αη+ 1 p )J 2 c1ηJε J . Since 2 -J ≤ Cr 1/(δη) , +∞ j=J ∥F j ∥ L p (B(x,r)) ≤ Cr α δ + 1 pδη w(r)
with w(r) = r ε(r) where lim r→0 ε(r) = 0. Paying attention to the fact that the pexponent is in fact given by

h p Fα,η (x) = lim inf r→0 log 1 r B(x,r) |F α,η (t) -P x (t -x)dt| p 1 p log 1 r
and that for α < 0, we have P x = 0, we obtain by taking into account the factor 1/r before the integral

h p Fα,η (x) ≥ α δ + 1 pδη + 1 p = α + 1 ηp δ + 1 p . Proposition 7. Let ψ : R → R be a continuous Lipschitz function with support on [-1, 1]. Let η ∈ (0, 1), α < 0 such that η -1 < αη.
Let F α,η be defined by [START_REF] Frisch | On the singularity structure of fully developed turbulence[END_REF]. Almost surely, for all p ∈ [1, -1/(αη) + 1/α), for all δ ∈ (1, 1 η ) and for all

x 0 / ∈ G ′ δ , h (p)
Fα,η (x 0 ) ≤ α δ + 1 -δη δηp = α + 1/(ηp) δ + 1/p.
Proof. We choose the wavelet function ϕ to compute the continuous wavelet transform to belong to C 1 (R), even and supported in [-1, 1] with at least one vanishing moment. We also impose that

1 -1 ϕ(u)ψ(u)du ̸ = 0
and by continuity of the continuous wavelet transform, there exists c ∈ (0, 1) such that for all ε ∈ [-c, c],

1 -1 ϕ u -ε 1 + ε ψ(u)du ̸ = 0. ( 11
)
The existence of such a ϕ function is straightforward.

The continuous wavelet transform of the sums of random pulses F α,η , is, for any

(s, t) ∈ R * + × R W Fα,η (s, t) = 1 s R F α,η (x)ϕ x -t s dx = n∈N * C -α n d n (s, t),
where

d n (s, t) = 1 s R ψ n (x)ϕ x -t s dx with ψ n (x) = ψ(B 1 η n (x -X n )).
The lemma below, proved in [START_REF] Demichel | Analyse fractale d'une famille de fonctions aléatoires: les fonctions de bosses[END_REF] (Proposition 2.2.1), allows to obtain a upper-bound for the wavelet coefficients of F α,η .

Lemma 4.4. If ϕ has zero integral, then there exists K > 0 such that

∀(s, t) ∈ R * + × R, |d n (s, t)| ≤ K min{sB 1 η n , s -1 B -1 η n }T n (t, s). Let x 0 ∈ G ′ δ .
There exists sequences of integers (n k ) k∈N and (j k ) k∈N such that

n k ∈ I j k and x 0 ∈ n∈Ij k B(X n , B -δ(1-εj k ) n ). Let k ∈ N * with n k ∈ I j k and ε ∈ [-c, c]. We estimate the continuous wavelet transforms W Fα,η (B -1 η n k + εB -1 η n k , X n k + εB -1 η n k ). Let J k = ⌊(1 -ρηε j k )j k ⌋ and J k = ⌊γj k ⌋, where γ ∈ (0, 1/η) has been set. We decompose W F (B -1 η n k + εB -1 η n k , X n k + εB -1 η n k ) in W F (B -1 η n k + εB -1 η n k , X n k + εB -1 η n k ) = S 1 + S 2 + S 3 .
with

S 1 = J k -1 j=0 n∈Aj C -α n d n (B -1 η n k (1 + ε), X n k + εB -1 η n k ), S 2 = J k j=J k n∈Aj C -α n d n (B -1 η n k (1 + ε), X n k + εB -1 η n k ), S 3 = +∞ j= J k +1 n∈Aj C -α n d n (B -1 η n k (1 + ε), X n k + εB -1 η n k ).
The idea is find a lower-bound for W F . To do this, we show that the term S 2 is much larger than the two other terms S 1 and S 3 (which correspond to the high and low frequency terms).

Let us first consider the sum S 2 . By the definition of I j in [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF], there exists a

unique n k ∈ A j k such that x 0 ∈ B(X n k , B -1 η n k ). Thus n k = n k and S 2 = C -α n k d n k (B -1 η n k (1 + ε), X n k + εB -1 η n ).
We have

d n k (B -1 η n k (1 + ε), X n k + εB -1 η n k ) = B 1 η n k R ψ(B 1 η n k (x -X n k ))ϕ   x -X n k -εB -1 η n k (1 + ε)B -1 η n k   dx = R ψ(u)ϕ   B -1 η n k u -εB -1 η n k (1 + ε)B -1 η n k   du = R ψ(u)ϕ u -ε 1 + ε du.
The condition [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF] implies that for a constant K 2 > 0 (depending on ψ and ϕ only), we have according to the Lemma 4.2 with α < 0,

|S 2 | ≥ K 2 C -α n k ≥ K 2 2 -αη(1-εj k )j k ≥ 2 α(1-εj k ) K 2 B -α(1-εj k ) n k . So ∃K ′ 2 > 0, |S 2 | ≥ K ′ 2 B -α(1-εj k ) n k . (12)
Then,by Lemmas 4.2 and 4.4 with α < 0, one has

|S 1 | ≤ J k -1 j=0 n∈Aj C -α n |d n (B -1 η n k (1 + ε), X n k + εB -1 η n k )| ≤ J k -1 j=0 n∈Aj C -α n min{(1 + ε)B -1 η n k B 1 η n , (1 + ε) -1 B 1 η n k B -1 η n } × T n (X n k + εB -1 η n k , (1 + ε)B -1 η n k ).
Since ε ∈ [-c, c] with c ∈ (0, 1),

T n (X n k + εB -1 η n k , (1 + ε)B -1 η n k ) ≤ T n (X n k , 2 -j k +2 ).
Therefore,

|S 1 | ≤ J k -1 j=0 2 -αηj(1+εj ) min{(1 + ε)B -1 η n k 2 j , (1 + ε) -1 B 1 η n k 2 -j+1 } n∈Aj T n (X n k , 2 -j k +2 ).
According to Lemma 4.3 and since j < (1 -ρηε j k )j k ,

∃K 1 , K 2 > 0, |S 1 | ≤ K(1+ε)B -1 η n k J k -1 j=0 j 2-α 2 (1-αη)j ≤ K 1 B -1 η n k j 3-α k 2 (1-αη)(1-ρηεj k )j k . Therefore j k = 2 ηεj k j k and that n k ∈ I j k , 2 j k ≤ 2B 1 η n k , ∃K ′ 1 > 0, |S 1 | ≤ K 1 B -1 η n k B (3-α)εj k n k B ( 1 η -α)(1-ρηεj k ) n k ≤ K ′ 1 B -α-(ρ-3+α-αηρ)εj k n k
.

By our choice of the integer ρ > 3-3α 1-αη , ρ -3 + α -αηρ > -2α. Thus

|S 1 | ≤ K ′ 1 B -α(1+2εj k ) n k . ( 13
)
Now we bound S 3 in the same way as S 1 with Lemmas 4.2 and 4.4. Thus

|S 3 | ≤ +∞ j= J k +1 n∈Aj C -α n |d n (B -1 η n k (1 + ε), X n k + εB -1 η n k )| ≤ +∞ j= J k +1 n∈Aj C -α n min{(1 + ε)B -1 η n k B 1 η n , (1 + ε) -1 B 1 η n k B -1 η n } × T n (X n k + εB -1 η n k , (1 + ε)B -1 η n k ) ≤ +∞ j= J k +1 2 -αηj(1-εj ) B -1 2η n k min{B -1 η n k 2 j , B 1 η n k 2 -j+1 } n∈Aj T n (X n k , 2 -j k +2 ).
Therefore j > J k = ⌊γj k ⌋, the above minimum is reached at B 1 η n k 2 -j+1 . Then, according to Lemma 4.3, there exists a constant K > 0 such that the sum n∈Aj T n (X n , 2 -j k +2 ) is bounded by Kj 2 when j ≤ (j k -2)/η, and by Kj 2 2 η(j-j k /η) when j > (j k -2)/η. Thus by Lemma 4.2, there exists a constant K 3 > 0 (which can change value at each line but does not depend on k or other parameters) such that

|S 3 | ≤ K 3 1 + ε   ⌊j k /η⌋ j=⌊γj k ⌋ j 2-α 2 -αηj B 1 η n k 2 -j + +∞ j=⌊j k /η⌋+1 j 2-α 2 -αηj B 1 η n k 2 -j 2 η j-(j k -2) η   ≤ K 3 1 + ε B 1 η n k   ⌊j k /η⌋ j=⌊γj k ⌋ j 2-α 2 -(1+αη)j + 2 -j k +2 +∞ j=⌊j k /η⌋+1 j 2-α 2 (η-1-αη)j   .
Since (1 + αη) > η > 0 by hypothesis, the first sum above is bounded by

⌊j k /η⌋ j=⌊γj k ⌋ j 2-α 2 -(1+αη)j ≤ K 3 j 2-α k 2 -(1+αη)γj k
and the second sum by

2 -j k +2 +∞ j=⌊j k /η⌋+1 j 2+α 2 (η-1-αη)j ≤ K 3 2 -j k j 2-α k 2 (η-1-αη) j k -2 η ≤ K 3 j 2-α k 2 -j k η (1+αη) . As B 1 η n k ∼ 2 j k and j k = 2 j k ηεj k ∼ B εj k
n k and γ < 1/η, we obtain that

|S 3 | ≤ K 3 B 1 η n k j 2-α k 2 -(1+αη)γj k + K 3 B 1 η n k j 2-α k 2 -j k η (1+αη) ≤ 2K 3 B (2-α)εj k + 1 η -( 1 η +α)γ n k ≤ 2K 3 B (2-α)εj k -αηγ+γ-1 η n k . We notice that αηγ+γ-1 η -(2 -α)ε j k > α(1 + 2ε j k ). Indeed, γ(αη + 1) -1 η > α + (2 + α)ε j k . Thus, |S 3 | ≤ K 3 B -α(1+2εj k ) n k . (14)
By comparing ( 12), ( 13) and ( 14), we obtain

|W Fα,η (B -1 η n k (1 + ε), X n k + εB -1 η n k )| ≥ KB -α(1-2εj k ) n k . ( 15
)
So for all ε ∈ [-c, c], the inequality (15) holds. In particular, there exists K > 0 such that for all (s, t)

∈ [(1 -c)B -1 η n k , (1 + c)B -1 η n k ] × [X n k -cB -1 η n k , X n k + cB -1 η n k ], |W Fα,η (s, t)| ≥ KB -α(1-2εj k ) n k . (16)
In order to conclude the proof, it remains to estimate

L (p) Fα,η (B -1 η n k , X n k ) =   B 1 η n k Xn k +B -1 η n k Xn k -Bn k   B -1 η n k 0 |W Fα,η (s, t)| 2 ds s   p 2 dt    1 p . Time t Scale s X n k B -1 η n k • X n k -cB -1 η n k X n k + cB -1 η n k (1 + c)B -1 η n k (1 -c)B -1 η n k X n k -B -1 η n k X n k -B -1 η n k On this domain |W Fα,η(s,t) | ≥ KB -α(1-2ε j,k ) n k Domain use to compute L (p) Fα,η (B -1 η n k , X n k ) Domain use to obtain a lower-bound of L (p) Fα,η (B -1 η n k , X n k ) Fig. 3 Decomposition time/scale to lower-bound L (p) Fα,η (B -1 η n k , Xn k )
From ( 16) (which can be visualized using Figure 3), we have

L (p) Fα,η (B -1 η n k , X n k ) ≥   B 1 η n k Xn k +cB -1 η n k Xn k -cBn k   B -1 η n k (1-c)B -1 η n k |W Fα,η (s, t)| 2 ds s   p 2 dt    1 p ≥ KB -α(1-2εj k )+ 1 ηp n k    Xn k +cB -1 η n k Xn k -cBn k   B -1 η n k (1-c)B -1 η n k ds s   p 2 dt    1 p = KB -α(1-2εj k )+ 1 ηp n k   Xn k +cB -1 η n k Xn k -cBn k (-ln(1 -c)) p 2 dt   1 p = Kc ln 1 1 -c 1/2 B -α(1-2εj k ) n k (17) Conclusion : Let ε ∈ R * + . Contradiction, we suppose that F ∈ T p α δ + 1-δη
δηp +ε (x 0 ). According to the characterization supplied by Theorem 4, there exists

K ′ > 0 such that |L (p) Fα,η (B -1 η n k , X n k )| ≤ K ′ B -1 η ( α δ + 1-δη δηp +ε) n k . ( 18
)
According to the ( 17) and ( 18),

KB -α(1-2εj k ) n k ≤ |L (p) Fα,η (B -1 η n k , X n k )| ≤ K ′ B -1 η ( α δ + 1-δη δηp +ε) n k . ( 19

Computation of the multifractal p-spectrum

According to Theorem 4.1 of [START_REF] Saës | Multifractal analysis of sums of random pulses[END_REF] applied to the set G ′ 1 with this time ρ > (3 -3α)/(1 -αη) and α < 0, we have almost surely that G ′ 1 is a covering of [0, 1]. And so to conclude with the proof of the Theorem 2, we just have to apply the propositions to determine inclusions on p-isohölderian sets. According to the Propositions 6 and 7, almost surely, for all x ∈ [0, 1] and δ ∈ 1, 1 η :

• If h We deduce that almost surely, for all H ∈ R, According to the Definition 1 with the choice of the parameters η ∈ (0, 1), α ∈ (-∞, 0) such that η -1 < αη and p ∈ (1, -1/(αη) + 1/α ∈ (1/(αη)) in the Theorem 2, we necessarily have the support of the multifractal spectrum which is a non empty segment containing αη < 0 and α + 1-η ηp > 0. We represent on Figure 4, an example of p-multifractal spectrum.

Conclusion

In conclusion, we presented a characterization of p-exponents by continuous wavelet transforms and then gave an example with the sums of random pulses. On the probabilistic side, a model extensively used and studied is supplied by Lévy processes whose sample paths have locally bounded jumps and whose multifractal analysis has been performed in [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF]. The multifractal spectrum of fractional integrals of Lévy processes were computed by P. Balança in [START_REF] Balança | Fine regularity of Lévy processes and linear (multi)fractional stable motion[END_REF]. A subsequent natural problem is then to determine a multifractal spectrum of fractional derivatives of Lévy processes. Fractional Lévy derivatives have been considered and studied in several papers [START_REF] Jacob | Lévy-Type Processes and Pseudodifferential Operators[END_REF][START_REF] Schilling | On Feller processes with sample paths in Besov spaces[END_REF][START_REF] Fageot | On the Besov regularity of periodic Lévy noises[END_REF][START_REF] Fageot | Multidimensional Lévy white noise in weighted Besov spaces[END_REF][START_REF] Aziznejad | Wavelet analysis of the Besov regularity of Lévy white noise[END_REF] and [START_REF] Unser | An Introduction to Sparse Stochastic Processes[END_REF]. Since these processes have non-locally bounded sample paths, in order to study their pointwise regularities, it is natural to use continuous wavelets transforms.

The study of continuous wavelet transform was also originally motivated by the study of the p-spectrum of Davenport series, which, given a sequence (a n ) n∈N , is defined for every x ∈ R as D(x) = When a n = 1/n β with β > 2, S. Jaffard established that if s is a real in (0, 1) then the fractional derivative of order s belongs to L p (R) as soon as p < 1/s and that the result is optimal in the sense that there are examples where D is not in L p (R) when p > 1/s. These fractional derivatives are an interesting example of non-locally bounded functions for which the multifractal analysis remains open.

Definition 3 .

 3 Let f ∈ L p loc (R) with p ≥ 1. The multifractal p-spectrum D (p) f : [-1/p, +∞] → [0, 1] ∪ {-∞} of f is the mapping

1 p

 1 then x ∈ G δ . Thus, almost surely for all H ∈ R h (p) Fα,η (x) = H =⇒ x ∈

Fig. 4 2 D

 42 Fig. 4 Multifractal 1-spectrum of the sum of random pulses forα = -0.7, η = 0.5

n∈N a n

  {nx}where {x} = x -⌊x⌋ -1/2 if x / ∈ Z 0 otherwise.

  ≥ αη + ε ≥ αη because 1 -δη ∈ [0, 1 -η].But the last inequality implies that which contradicts[START_REF] Saës | Sommes fractales de pulses : étude dimensionnelle et multifractale des trajectoires et simulations[END_REF] since (ε j ) j∈N * tends to 0. Therefore, F α,η / ∈ T

	But δ ∈ [1, 1/η), so
	α δ	+ + ε 1 1 -δη δηp η α δ + 1 -δη δηp	+ ε > α,
						(p) α δ + 1-δη δηp +ε (x 0 ) for
	all ε > 0 which proves that
		h (p) Fα,η (x 0 ) ≤	α δ	+	1 -δηp δηp	.
						)

  Conversely, according to the Propositions 6 and 7, almost surely for any H ∈ R,

			G δ \		G ′ δ .
			αηp+1		αηp+1
		δ<	ηp H+ 1 p	δ>	ηp H+ 1 p
	G ′ α H \		G
		αηp+1
	δ>	ηp H+ 1 p

E Fα,η (H) ⊂ δ ⊆ E Fα,η (H).

Acknowledgments

The author thanks Stéphane Jaffard and Stéphane Seuret for all the enriching discussions around this article.


Vikram Plomp

Jolijn Onvlee

Francois Lique

Sebastiaan Y T Van De Meerakker

Low-Energy Collisions of Zeeman-Decelerated NH Radicals with He Atoms Published as part of The Journal of Physical Chemistry virtual special issue "Cold Chemistry"

We report an experimental study of state-to-state inelastic scattering of NH (X 3 Σ -, N = 0, j = 1) radicals with He atoms. Using a crossed molecular beam apparatus that combines a Zeeman decelerator and velocity map imaging, we study both integral and differential cross sections in the N = 0, j = 1 → N = 2, j = 3 inelastic channel. We developed various new REMPI schemes to state-selectively detect NH radicals, and tested their performance in terms of sensitivity and ion recoil velocity. We found a 1 + 2′ + 1′ REMPI scheme using the A 3 Π ← X 3 Σ -resonant transition, which yields acceptable recoil velocities and is more than an order of magnitude more sensitive than conventional onecolor REMPI schemes to detect NH. We used this REMPI scheme to probe state-to-state integral and differential cross sections around the channel opening at 97.7 cm -1 , as well as at higher energies where structure in the scattering images could be resolved. The experimental results are in excellent agreement with the predictions from quantum scattering calculations which are based on an ab initio NH-He potential energy surface.

I. INTRODUCTION

In recent years, Stark deceleration of neutral polar molecules has emerged as a powerful method to precisely study molecular collision processes in crossed beam experiments, and at low collision energies in particular. The molecular packets that emerge from the decelerator have a tunable velocity, a high state purity, narrow transverse and longitudinal velocity spreads, and a narrow spatial spread. [START_REF] Van De Meerakker | Manipulation and Control of Molecular Beams[END_REF] They serve as excellent starting points in scattering experiments, and in combination with velocity map imaging (VMI) detection techniques, they have been essential to probe scattering processes in unprecedented detail. [START_REF] Onvlee | Molecular collisions coming into focus[END_REF] Examples include the observation of diffraction oscillations, [START_REF] Von Zastrow | State-resolved diffraction oscillations imaged for inelastic collisions of NO radicals with He[END_REF][START_REF] Onvlee | Imaging quantum stereodynamics through Fraunhofer scattering of NO radicals with rare-gas atoms[END_REF] partial wave scattering resonances at low collision energies, [START_REF] Vogels | Imaging resonances in low-energy NO-He inelastic collisions[END_REF][START_REF] De Jongh | Imaging the onset of the resonance regime in low-energy NO-He collisions[END_REF][START_REF] De Jongh | Mapping partial wave dynamics in scattering resonances by rotational de-excitation collisions[END_REF] and rotational product pair correlations in bimolecular inelastic scattering processes. [START_REF] Gao | Observation of correlated excitations in bimolecular collisions[END_REF][START_REF] Besemer | Glory scattering in deeply inelastic molecular collisions[END_REF] By its nature, the Stark deceleration technique is only applicable to molecules that have an electric dipole moment, excluding an important class of species that are inert to electric fields. Yet, these species often have a magnetic dipole moment and can be manipulated by magnetic fields instead. Zeeman decelerators, the magnetic analogue of Stark decelerators, have been developed as well and have successfully been used to slow a variety of species and to load them into traps; see reviews in refs 10-13 and references contained within. Recently, the first successful crossed beam scattering experiments using a Zeeman decelerator have been performed, focusing on inelastic processes involving NO radicals or C atoms. [START_REF] Plomp | High-resolution imaging of molecular collisions using a Zeeman decelerator[END_REF][START_REF] Plomp | High-Resolution Imaging of C + He Collisions using Zeeman Deceleration and Vacuum-Ultraviolet Detection[END_REF] Despite these important proof-of-principle advances, crossed beam scattering using Zeeman decelerators is still in its infancy, and the full potential of the technique to probe scattering processes has yet to be unlocked. In particular, it is important to investigate whether the technique can be used for sensitive measurements on a large variety of species and whether low collision energies are achievable.

Recently, we reported the direct Zeeman deceleration of NH (X 3 Σ -) 16 using a 2.2 m long Zeeman decelerator that was specifically designed for applications in crossed beam experiments. [START_REF] Cremers | Design and construction of a multistage Zeeman decelerator for crossed molecular beams scattering experiments[END_REF] Here, we describe the first measurements of state-tostate inelastic integral and differential cross sections for NH radicals and He atoms using a crossed molecular beam apparatus that uses a Zeeman decelerator to prepare the initial packet of NH radicals and velocity map imaging to detect the products, extending the chemical diversity of this type of experiment to the realm of molecular radicals. Furthermore, we tested various resonance-enhanced multiphoton ionization (REMPI) schemes to (i) state-selectively detect NH radicals with high sensitivity and (ii) limit the recoil velocity imparted to the ion upon ionization to minimize blurring of scattering images measured using VMI. We identified a new two-color 1 + 2′ + 1′ REMPI scheme employing the A 3 Π ← X 3 Σ - resonant transition and subsequent ionization through a 2photon resonant transition to a yet unidentified electronic state. This REMPI scheme was found to offer reduced ion recoil velocities and to be more than an order of magnitude more sensitive than the more commonly used 2 + 1 REMPI scheme via the D 3 Π state. Using this detection scheme, we studied the N = 0, j = 1 → N = 2, j = 3 inelastic scattering channel for NH + He collisions at energies between 90 and 427 cm -1 and probed the threshold behavior of the scattering cross sections around the energetic opening at 97.7 cm -1 . Our experimental findings are in excellent agreement with the predictions from quantum scattering calculations which are based on the recent ab initio NH-He potential energy surface by Ramachandran et al. [START_REF] Ramachandran | A new ab initio potential energy surface for the NH-He complex[END_REF] The NH radical has been identified as a species of primary interest in the physical chemistry and cold molecules research communities. [START_REF] Janssen | Ab initio potential energy surfaces for NH( 3 Σ -)-NH( 3 Σ -) with analytical long range[END_REF][START_REF] Janssen | Cold and ultracold NH-NH collisions: The field-free case[END_REF][START_REF] Janssen | On the role of the magnetic dipolar interaction in cold and ultracold collisions: numerical and analytical results for NH( 3 Σ-) + NH( 3 Σ-)[END_REF][START_REF] Janssen | Cold and ultracold NH-NH collisions in magnetic fields[END_REF][START_REF] Janssen | Quantum Reactive Scattering of Ultracold NH(X 3 Σ -) Radicals in a Magnetic Trap[END_REF][START_REF] Wallis | The prospects of sympathetic cooling of NH molecules with Li atoms[END_REF][START_REF] Hutson | Cold collisions of N ( 4 S) atoms and NH ( 3 Σ) molecules in magnetic fields[END_REF][START_REF] Haxton | Theoretical study of the quenching of NH ( 1 Δ) molecules via collisions with Rb atoms[END_REF] At high energies, a large body of state-tostate scattering experiments have been conducted involving NH and a variety of scattering partners. [START_REF] Dagdigian | Scaling relations in the rotational excitation of NH(X 3 Σ -) N = 0 by argon[END_REF][START_REF] Rinnenthal | State-to-state studies of ground state NH(X 3 Σ -,v = 0,J,N)+Ne[END_REF][START_REF] Rinnenthal | State-to-state energy transfer of NH(X 3 Σ -,v = 0,J,N) in collisions with He and N 2[END_REF] Its 2 μ B magnetic moment in the X 3 Σ -electronic ground state offers a prominent Zeeman effect suitable for magnetic deceleration and trapping experiments. Trapping in the X 3 Σ -state has already been achieved via the buffer gas cooling technique [START_REF] Campbell | Magnetic trapping and Zeeman relaxation of NH X 3 Σ[END_REF][START_REF] Hummon | Cold N plus NH Collisions in a Magnetic Trap[END_REF][START_REF] Campbell | Time-Domain Measurement of Spontaneous Vibrational Decay of Magnetically Trapped NH[END_REF] and after Stark deceleration in the metastable a 1 Δ state prior to optical transfer into the X 3 Σ -electronic ground state. [START_REF] Riedel | Accumulation of Stark-decelerated NH molecules in a magnetic trap[END_REF] Furthermore, NH is an important species in astrochemistry, and pronounced scattering resonances at temperatures below 150 K are theoretically predicted for collisions between NH and He atoms or H 2 molecules. [START_REF] Ramachandran | A new ab initio potential energy surface for the NH-He complex[END_REF][START_REF] Bouhafs | Collisional excitation of NH(X 3 Σ -) by Ne: Potential energy surface, scattering calculations, and comparison with experiments[END_REF][START_REF] Prudenzano | Collisional excitation of NH 3 ( 3 Σ -) by Ar: A new ab initio 3D potential energy surface and scattering calculations[END_REF][START_REF] Pirlot | Collisional excitation of NH by H 2 : Potential energy surface and scattering calculations[END_REF] Finally, the NH molecule provides interesting prospects to control collisions or chemical reactions using externally applied electric or magnetic fields. [START_REF] Hutson | Cold collisions of N ( 4 S) atoms and NH ( 3 Σ) molecules in magnetic fields[END_REF][START_REF] Krems | Cold controlled chemistry[END_REF]

II. METHODS

II.A. Experimental Setup.

The experiments are performed in the molecular beam apparatus schematically depicted in Figure 1, which was recently used to decelerate beams of O atoms, O 2 molecules, [START_REF] Cremers | Multistage Zeeman deceleration of atomic and molecular oxygen[END_REF] NO (X 2 Π 3/2 ) radicals, [START_REF] Plomp | High-resolution imaging of molecular collisions using a Zeeman decelerator[END_REF] NH radicals, [START_REF] Plomp | Multistage Zeeman deceleration of NH X 3 Σradicals[END_REF] and C atoms. [START_REF] Plomp | High-Resolution Imaging of C + He Collisions using Zeeman Deceleration and Vacuum-Ultraviolet Detection[END_REF] The experiment was operated at a repetition rate of 20 Hz. A detailed description of the mechanical and electronic implementation of the decelerator, as well as details on the production and deceleration of NH radicals, are given elsewhere. [START_REF] Plomp | Multistage Zeeman deceleration of NH X 3 Σradicals[END_REF][START_REF] Cremers | Design and construction of a multistage Zeeman decelerator for crossed molecular beams scattering experiments[END_REF] Briefly, a molecular beam of NH (X 3 Σ -) molecules with a forward velocity centered around 550 m/s is formed by an electric discharge of 2% NH 3 seeded in krypton, using a Nijmegen pulsed valve with a discharge assembly. [START_REF] Ploenes | Cold and intense OH radical beam sources[END_REF] After the supersonic expansion, most NH radicals in the X 3 Σ -electronic ground state reside in the v = 0, N = 0, j = 1 rovibrational ground state (see Figure 2 for a rotational energy level scheme of the NH radical together with the Hund case (b) quantum numbers used to label the individual states). This state has a magnetic moment of 2 μ B and splits into m j = 1, m j = 0, and m j = -1 components in the presence of a magnetic field. Only NH radicals in the low field seeking the m j = 1 component are selected by the Zeeman decelerator in the experiments. Approximately 90 mm downstream from the nozzle orifice, the molecular beam passes a 3 mm diameter skimmer and enters the Zeeman decelerator, which consists of an alternating array of 100 solenoids and 101 hexapoles. [START_REF] Cremers | Design and construction of a multistage Zeeman decelerator for crossed molecular beams scattering experiments[END_REF] The decelerator consists of 5 modules of 20 solenoids and 19 hexapoles connected to each other using a hexapole positioned at the interface between the modules. The solenoids are made of a copper capillary through which currents up to 5 kA are pulsed. Cooling liquid is passed through the solenoids. Each solenoid is connected to an individual printed circuit board to provide the current pulses using field effect transistor (FET)-based electronics. The hexapoles consist of six commercially available arc-shaped permanent magnets. The velocity and spatial spreads of the NH packet exiting the decelerator strongly depend on the settings of the decelerator, as discussed by Plomp et al. [START_REF] Plomp | Multistage Zeeman deceleration of NH X 3 Σradicals[END_REF] The NH (X 3 Σ -) radicals that exit the Zeeman decelerator are intersected by a secondary beam of He atoms at an angle of The Journal of Physical Chemistry A 90°. The He beam was produced by a cryogenically cooled Even-Lavie valve (ELV) that was temperature controlled to tune the mean velocity of the He beam. In all cases, the temporal duration of the He beam largely exceeded the temporal duration of the NH packet in the collision region. The flight from the exit of the decelerator to the interaction region amounted to about 364 mm. To reduce the loss of density during this flight, the Zeeman decelerator was extended with a hexapole array that transversally guides the molecules toward the interaction region.

The scattered NH radicals are state-selectively detected using REMPI. Depending on the experiment, different single and multicolor REMPI schemes are used (see section III.A). After the REMPI process, the NH ions are detected with a high-resolution VMI detector using a design as described by Plomp et al., to allow for accurate mapping of large ionization volumes. [START_REF] Plomp | A velocity map imaging apparatus optimized for high-resolution crossed molecular beam experiments[END_REF] A repeller voltage of 2000 V is used to accelerate the ions toward a mass-gated microchannel plate (MCP) detector. Impact positions of impinging ions are recorded by a phosphor screen in combination with a CMOS camera and home-written acquisition and analysis software. For the measurement of spectra, the VMI detector is operated out of velocity focus, and the integral signal recorded by the camera is used. For the measurements of scattering images, the VMI detector is operated in velocity focus, such that 2D images that reflect the differential cross sections (DCSs) of the scattering processes are generated. For the measurement of integral cross sections (ICSs), the VMI detector is also operated in velocity focus to be able to better subtract the contribution of the initial beam (shot to shot) as well as other background signals. The measure of the ICS is then obtained by integrating the scattering signal within the relevant area of the detector. Saturation of the detector does not occur as the scattering signal is quite weak.

III. RESULTS AND DISCUSSION

III.A. Different REMPI Schemes.

To detect the scattered NH radicals using velocity map imaging, a REMPI scheme is desired that (i) allows for the state-selective detection of NH with (N, j) resolution, (ii) is sufficiently sensitive to be used in experiments employing a Zeeman decelerator, especially when considering the predicted low integral cross sections (≲1 Å 2 ) of, for example, inelastic NH + He collisions, [START_REF] Ramachandran | A new ab initio potential energy surface for the NH-He complex[END_REF] and (iii) imparts near-zero recoil velocity to the ions to optimally exploit the high inherent velocity resolution when using a Zeeman decelerator.

A 2 + 1 REMPI scheme via the D state using photons at a wavelength around 224 nm is an established and often used REMPI scheme to state-selectively detect NH radicals. [START_REF] Clement | Triplet Rydberg states of the imidogen radical characterized via two-photon resonance-enhanced multiphoton ionization spectroscopy[END_REF] Although rather efficient, this scheme imparts a recoil velocity of about 38 m/s to the ions, significantly blurring the recorded images when VMI is used. This is inconsequential for the measurement of ICS curves in a crossed-beam experiment, but it is detrimental in measurements of scattering images. We therefore developed various alternative REMPI schemes, which we will describe in detail below, and evaluated their performance in view of the desiderata listed above. Figure 3 illustrates all schemes, whereas Table 1 summarizes the most important findings in terms of sensitivity and ion recoil velocity. The relative sensitivities of the different schemes are derived from the number of ions detected when probing the packet of NH radicals exiting the decelerator, which we list as a figure of merit in Table 1. In the following, we label spectral lines with ΔNΔj using the standard spectroscopic nomenclature P, Q, R, etc. to indicate the value for ΔN and Δj, where we use small and capitalized letters to label ΔN and Δj, respectively.

To probe the performance of the well-known 2 + 1 REMPI scheme via the D state, we measured a REMPI spectrum of the packet of NH radicals that exit the decelerator with the He beam switched off. The 224 nm light is generated by frequency tripling the output of a dye laser (LiopTec), pumped using the second harmonic of a Nd:YAG laser (Innolas SL 1500). Typically, a 7 mm diameter laser beam with a pulse energy of 2.3 mJ in a 5 ns pulse is used that is focused in the interaction region using a spherical lens with 450 mm focal length. The resulting spectrum is shown in Figure 4(a). Clearly, the vast majority of the NH radicals resides in the N = 0, j = 1 rotational state, and we estimate the population in higher rotational states with a typical ratio (N = 0):(N = 1):(N = 2) ∼ 1:0.03:0.002. When the laser is tuned to the strongest peak, typically 100 ions/shot are detected. a The 3 + 1 REMPI scheme involves a first color wavelength around 336 nm that differs slightly from the wavelength used to drive the A ← X transition in the 1 + 2′ and 1 + 2′ + 1′ schemes, as indicated with an asterisk. The number of ions detected when probing the packet of NH radicals exiting the decelerator, using the experimental arrangements as specified in the text, is listed as a figure of merit to appreciate the experimental feasibility of the schemes.

The Journal of Physical Chemistry A

Recently, we reported an alternative 1 + 2′ REMPI scheme. [START_REF] Plomp | Multistage Zeeman deceleration of NH X 3 Σradicals[END_REF] In this scheme, we first resonantly excite NH to the A 3 Π state via the strong A ← X transition around 336 nm. The NH radicals are then ionized through the absorption of 2 photons at a wavelength around 254 nm that are provided by a second tunable dye laser. The wavelength of this ionization laser can be tuned to the ionization threshold, offering a direct route to recoil-free detection of NH. The light of the first color is generated by frequency doubling the output of a dye laser (LiopTec), pumped using the second harmonic of a Nd:YAG laser (Innolas SL 600), yielding a 5 mm diameter laser beam with 5 ns pulse width. About 10 mJ pulse energy (unfocused) was used to ensure saturation of the A ← X transition. The second color is produced by pumping a LiopTec dye laser by the second harmonic of an Innolas SL 1500 Nd:YAG laser. The output of this laser is frequency tripled to generate laser radiation at wavelengths around 254 nm. Typically, a 7 mm diameter laser beam with a pulse energy of 6.5 mJ in a 5 ns pulse is used that is focused in the interaction region using a spherical lens with 470 mm focal length. The delay between the two pump lasers was adjusted to make sure that both lasers intercept the interaction region within the 449 ns lifetime of the A 3 Π state. [START_REF] Song | Accurate spectroscopic calculations of the 12 Λ-S and 25 Ω states of the NH radical including the spin-orbit coupling effect[END_REF] In agreement with our earlier findings, [START_REF] Plomp | Multistage Zeeman deceleration of NH X 3 Σradicals[END_REF] although recoil-free detection of NH is possible, the ionization step is found to be rather inefficient. The ionization efficiency can be enhanced by more than an order of magnitude if the frequency of the ionization laser is blue-detuned by about 600 cm -1 with respect to the ionization threshold, which we speculatively attribute to autoionization of a Rydberg state. [START_REF] Plomp | Multistage Zeeman deceleration of NH X 3 Σradicals[END_REF] This more efficient ionization pathway is accompanied by a small ion recoil velocity of a few m/s, such that this 1 + 2′ REMPI scheme offers a trade-off between ionization efficiency and ion recoil velocity, simply by changing the wavelength of the ionization laser. While the 1 + 2′ REMPI scheme provides interesting prospects for high-resolution detection of NH scattering products, the ionization efficiency was found to be about a factor 20 lower than the 2 + 1 REMPI scheme results via the D state. Considering the low cross sections predicted for the NH-He system, the relatively low ionization efficiency was deemed insufficient to measure a large body of scattering data across a large collision energy window.

An alternative, remarkably efficient, ionization pathway was found in 3 + 1 REMPI detection using photons at a wavelength of around 336 nm. The resonant excited state could not be identified (and is marked with "?" in Figure 3) but lies just below the D state two-photon resonances in energy, and likely is of 3 Σ character. The high ionization efficiency was attributed to the exceptionally strong A 3 Π, N′ = 1, j′ = 2 ← X 3 Σ -, N″ = 0, j″ = 1 transition, which is nearly resonant with a single 336 nm photon. [START_REF] Brazier | Fourier transform spectroscopy of the A 3 Π -X 3 Σtransition of NH[END_REF] To avoid confusion, we add an asterisk to the wavelength (336 *nm) to indicate that this 3 + 1 scheme involves a first color wavelength that differs slightly from the center frequency of the A ← X transition. For this scheme, a 7 mm diameter laser beam with a pulse energy of 10 mJ in a 5 ns pulse was produced using a frequency doubled dye laser (LiopTec) pumped by the second harmonic of an Nd:YAG laser (Innolas SL 1500). The laser beam was partially focused in the interaction region using a spherical lens with 470 mm focal length which was put 3 cm out of focus. The high power density used completely saturates the A ← X transition when on resonance and causes strong power broadening. The resulting spectrum is shown in Figure 4(b), in which the 2 + 1 REMPI spectrum from Figure 4(a) is shown again as an overlay. In order to facilitate the comparison, fundamental wavelengths for the involved lasers are used on the horizontal axis. The fundamental wavelength for the rR line of the onephoton A ← X resonance that addresses the population in the N = 0, j = 1 level is indicated by the vertical dashed line. Clearly, the main 3 + 1 REMPI line is slightly red-shifted with respect to the A ← X (rR) line. Moreover, the 3 + 1 REMPI scheme offers much higher signal levels than the 2 + 1 REMPI scheme with the available lasers. It is noted that, without identification of the excited electronic state, one cannot be sure

The Journal of Physical Chemistry A

about the state selectivity of this REMPI scheme at the full (N, j) level and its suitability for state-selective collision experiments. A proper spectroscopic investigation of this state is beyond the scope of our investigations.

To ensure state selectivity, a 1 + 2′ + 1′ REMPI scheme was developed as an extension to this new 3 + 1 scheme, where the first color laser (as before around 336 nm, 14.5 mJ) is now unfocused and tuned resonantly to the rR (N″ = 0, j″ = 1) line of the A ← X transition, [START_REF] Brazier | Fourier transform spectroscopy of the A 3 Π -X 3 Σtransition of NH[END_REF] and a second color (also around 336 nm, but slightly detuned from the first color wavelength) is used to further excite to the unidentified intermediate state and to ultimately ionize the molecule. For the second color excitation and ionization, a 7 mm diameter laser beam with a typical pulse energy of 15 mJ in a 5 ns pulse was employed, produced by the same laser system used in the (3 + 1) REMPI scheme described above. The second color laser was partially focused in the interaction region by a spherical lens with a 470 mm focal length that was put 3 cm out of focus. The arrivaltime delay of the second laser was set to a few tens of nanoseconds with respect to the first laser to ensure proper overlap within the ∼450 ns lifetime of the A 3 Π state. [START_REF] Song | Accurate spectroscopic calculations of the 12 Λ-S and 25 Ω states of the NH radical including the spin-orbit coupling effect[END_REF] The resulting two-color spectrum was then used to obtain a difference spectrum by subtracting the spectrum observed when only the second color was used. This difference spectrum is shown in Figure 4(c), together with the one-color spectrum recorded with the second color laser only. With only the second color present, the NH radicals can only ionize through the 3 + 1 REMPI process mentioned before, and we retrieve a similar spectrum to the one presented in panel (b). By using two colors, and by fixing the first color to the A 3 Π transition while scanning the second, only a single quantum state is probed. Note that, when the first color is present, the second color can still ionize NH radicals directly from the ground state via the 3 + 1 process. However, part of the NH radicals is already excited to the A-state by the first color, depleting the initial state, and thus resulting in a dip (negative signal) in the difference spectrum. All rotational transitions from the wellknown A ← X electronic transition in NH can be used to probe individual (N, j) quantum states in the X 3 Σ -electronic ground state. This was verified by measuring individual spectra using the rR, rQ, rP, sR, and sQ branches as first color. As an example, the observed difference spectrum using the sR (N″ = 0, j″ = 1) line is shown in Figure 4(c) as well. Furthermore, He-seeded beams were used to increase the rotational temperature of the beam and to probe excited rotational levels relevant to inelastic scattering experiments, again investigating the use of various A ← X branches and corresponding second color resonances (data not shown).

The 1 + 2′ + 1′ scheme described above offers an estimated factor of 20 increase in ion yield with respect to 2 + 1 ionization via the D state. Unfortunately, the ion recoil of around 25 m/s using this scheme is not zero, but it is much smaller than the 38 m/s recoil imparted by the 2 + 1 REMPI scheme. Furthermore, the use of longer wavelengths, the enlarged ionization volumes, the possibility to cross the two laser beams in a well-defined volume, and the ability to tune the powers of both lasers independently together allow for an order of magnitude decrease in ionization of background gas, further improving the overall sensitivity. Efforts to alter this REMPI scheme to obtain nearly recoil-free detection using 1 + 3′ + 1′ or 1 + 2′ + 1″ variants remained unsuccessful.

III.B. Scattering Results. The 1 + 2′ + 1′ REMPI scheme was used to conduct state-to-state scattering experiments between NH (X 3 Σ -, N = 0, j = 1) radicals and He atoms at various collision energies.

Detection of the NH (N = 2, j = 3) final state was chosen as the NH (N = 1, j = 1,2) channels suffered a significant contribution of elastic background signal, which could not be fully separated from the inelastic signal due to the significant ion recoil. The strong elastic contribution for these channels arises from a small amount of initial N = 1 molecules that is codecelerated with the N = 0 ground state, in combination with an unfavorable ratio of high elastic and low inelastic cross sections. The NH (N = 1, j = 0) exit channel should have a very low elastic background signal, as NH (N = 1, j = 0) molecules are defocused by the decelerator. However, the predicted inelastic cross section for the N = 0, j = 1 →N = 1, j = 0 channel is strongly reduced with respect to the other channels considered here, [START_REF] Ramachandran | A new ab initio potential energy surface for the NH-He complex[END_REF] and very low scattering signals would be expected in our experiment. The NH (X 3 Σ -, N = 2, j = 3) scattering products were probed using as the first color the A 3 Π ← X 3 Σ -(sR) transition at 29991.34 cm -1 (∼14 mJ, unfocused), which ensures the selection of only the j = 3 spinrotation level through the electric dipole transition selection rules.

Two types of collision experiments were conducted. In the first, we measured scattering images at selected collision energies ranging from ∼118 to ∼427 cm -1 . For these measurements, the initial NH (X 3 Σ -, N = 0, j = 1) velocity was kept constant (530 m/s guiding), while the mean He velocity was tuned by adjusting the ELV temperature between 50 and 300 K. In the second, we probed both the integral and differential cross sections in a relatively narrow range of collision energies around the opening of the N = 2, j = 3 channel at 97.7 cm -1 . In these measurements, the temperature of the ELV (45 K) and resulting velocity of the He beam were kept constant. Instead, to obtain precise control over the collision energy, the pulsing sequence of the Zeeman decelerator was adjusted to vary the NH velocity from 300 to 650 m/s in steps of 25 m/s.

The images recorded for energies at and above ∼118 cm -1 are shown in Figure 5. The images are consistently presented such that the relative velocity vector is directed horizontally, with forward scattering angles positioned at the right side. The laboratory zero velocity point is located in the lower half of each image. Small segments of the images are masked where the initial beam gives a contribution to the signal. The collision energy was calibrated from the diameter of the recorded images. It is apparent that fine structures, like diffraction oscillations, cannot be discerned in these images due to the 25 m/s recoil velocity imparted to the ions during the ionization step. Yet, larger structures can be clearly identified, like the appearance of strong forward scattering at low collision energies and enhanced scattering intensity at side-scattered angles. The angular scattering distributions were extracted from the experimental image intensities within a narrow annulus around the observed rings. The experimentally obtained distributions can be directly compared to those obtained from simulated images. These image simulations are based on theoretical state-to-state differential cross sections acquired from ab initio quantum coupled-channels calculations that use state-of-the-art NH-He potential energy surfaces, [START_REF] Ramachandran | A new ab initio potential energy surface for the NH-He complex[END_REF] in combination with the particle trajectory simulations on our Zeeman decelerator apparatus. In these simulations, we assume a constant DCS over the energy distribution sampled for a given experimental setting; this assumption is valid as the DCS The Journal of Physical Chemistry A is not expected to abruptly change within the collision energy distribution for a given setting of the experiment. Rapid changes in the DCS are only expected around pronounced scattering resonances and at lower collision energies. The simulated images take into account the spatial, temporal, and velocity spreads of the used molecular beams, [START_REF] Von Zastrow | State-resolved diffraction oscillations imaged for inelastic collisions of NO radicals with He[END_REF] as well as kinematic effects and the experimental ion recoil. The simulated images are analyzed analogously to their experimental counterparts to acquire predicted angular scattering distributions, of which the coarse shape may differ from the theoretical DCSs mainly due to more efficient detection of particles with low laboratory frame velocities after scattering. [START_REF] Von Zastrow | State-resolved diffraction oscillations imaged for inelastic collisions of NO radicals with He[END_REF] The measurements are found to be in good agreement with the simulated distributions.

The scattering images recorded at four energies around the channel opening are depicted in Figure 6. From these images, it is apparent that the ring size and signal intensity quickly drop with decreasing collision energy. The recorded images around the channel opening qualitatively agree with the simulated images based on theoretical DCS predictions. The behavior of the ICS around the channel opening was probed by rapidly cycling through the decelerator sequences corresponding to various NH velocities (300-650 m/s in steps of 25 m/s) while measuring the integral signal recorded in the relevant area of the camera. Figure 6 shows the relative integral cross section as a function of the collision energy, referred to as the excitation function, that is obtained after correction for background signals, variation of initial beam density, and a simplified density-to-flux transformation that is valid in the limit of small laser ionization volume in relation to beam overlap volume. [START_REF] Sonnenfroh | Number density-to-flux transformation revisited: kinematic effects in the use of laser-induced fluorescence for scattering experiments[END_REF] For these near-threshold measurements, the images are too small to reliably calibrate the collision energies from the diameter of the images. The collision energy was more accurately calibrated by fitting the measured excitation function to the theoretical ICS prediction around the channel opening, using the He velocity (opening energy) and collision energy spread (opening shape) as fitting parameters. The measured ICS curves and corresponding fit parameters only slightly varied between measurement series, indicating a high stability of the experiment. A mean helium velocity of 750 ± 5 m/s and a collision energy spread of approximately 10 cm -1 (FWHM) were determined. The measured scattering probabilities are found to be in excellent agreement with the theoretical prediction after convolution with the deduced energy spread.

IV. CONCLUSION

We have presented the first state-to-state inelastic collision experiment using beams of Zeeman decelerated NH (X 3 Σ -, N = 0, j = 1) radicals and tested the performance of various new ) and simulated (Sim.) scattering images at selected collision energies (E col ) for the NH (X 3 Σ -, N = 0, j = 1) + He → NH (X 3 Σ -, N = 2, j = 3) + He process, corresponding to an excitation energy of 97.7 cm -1 . One image pixel corresponds to a velocity of 2.16 m/s. The angular scattering distributions extracted from the experimental (solid red lines) and simulated images (dashed blue lines) are shown in the right panels, together with DCSs from theory (dashed green lines). The different curves were normalized on the maximum intensity for clarity.

The Journal of Physical Chemistry A

REMPI schemes for NH in terms of sensitivity and ion recoil velocities compared to a conventional REMPI scheme's performance. We found a highly efficient 1 + 2′ + 1′ REMPI scheme for the state-selective detection of NH and used it to perform the first crossed beam scattering experiment using Zeeman-decelerated NH, focusing on the NH-He system at relatively low collision energies. Although the 25 m/s ion recoil associated with this scheme does not allow for the observation of fine structures in the scattering images, it is ideally suited for ICS measurements and still allows for the observation of somewhat less delicate features in the scattering distributions. We probed differential cross sections for inelastic scattering in the 118-427 cm -1 energy range and probed both integral and differential cross sections at energies around the N = 0 → N = 2 channel opening at ∼98 cm -1 . Excellent agreement was found with predicted cross sections from quantum scattering calculations using an accurate NH-He interaction potential.

Our search for new REMPI schemes was primarily aimed at the use of NH in controlled crossed-beam scattering experiments, but the efficient REMPI scheme presented here may, for example, find applications in experiments that aim for the trapping of NH after Zeeman deceleration as well. The scattering results on the NH-He system presented here demonstrate that our Zeeman decelerator can be applied in crossed-beam experiments to accurately probe both integral and differential cross sections, even for challenging scattering systems that involve low cross sections, inefficient discharge beam production, and intricate detection schemes.

■ AUTHOR INFORMATION Corresponding Author

Sebastiaan Y. T. van de Meerakker -Institute for Molecules and Materials, Radboud University, 6525 AJ, Nijmegen, The Netherlands; orcid.org/0000-0003-3540-6476; Email: basvdm@science.ru.nl

Figure 1 .

 1 Figure 1. Schematic depiction of the experimental setup. NH molecules in the X 3 Σ -electronic ground state are created by an electric discharge of NH 3 seeded in Kr. The beam is collimated by a skimmer and then passed through a Zeeman decelerator consisting of an alternating array of 100 pulsed solenoids and 101 permanent magnetic hexapoles. The NH radicals exiting the Zeeman decelerator are guided into the interaction region by a number of additional hexapoles and intercepted by a beam of helium atoms under 90 deg angle of incidence. The scattered NH radicals are detected using velocity map imaging and a variety of different REMPI schemes.

Figure 2 .

 2 Figure 2. Rotational energy level diagram of NH radicals in the X 3 Σ - electronic ground state. Each rotational level is labeled by the quantum numbers N, j. The collision induced excitation channel studied in this paper is indicated by the red arrow.

Figure 3 .

 3 Figure 3. Schematic illustration of the various REMPI schemes investigated in this work to state-selectively detect NH radicals. The X 3 Σ -, A 3 Π, and D 3 Π electronic states are indicated. The 3 + 1 and 1 + 2′ + 1′ schemes make use of a yet unidentified intermediate state, just below the energy of the D 3 Π state. The 3 + 1 and 1 + 2′ + 1′ REMPI schemes involve a first color wavelength around 336 nm that differs slightly for the two schemes, as indicated with an asterisk. Energies of the electronic states are not drawn to scale.

Figure 4 .

 4 Figure 4. REMPI spectra probing NH (X 3 Σ -) radicals exiting the Zeeman decelerator. (a) 2 + 1 REMPI spectrum using the D ← X transition. (b) 3 + 1 REMPI spectrum using an unidentified electronic transition (black), together with the 2 + 1 REMPI spectrum from panel (a). On the horizontal axis the fundamental dye laser wavelength is given before either frequency doubling or tripling. The resonance position of the A 3 Π, N′ = 1, j′ = 2 ← X 3 Σ -, N″ = 0, j″ = 1 rR transition before frequency doubling is given by the vertical dashed line. (c) 1 + 2′ + 1′ REMPI difference spectra using the rR (red) or sR (blue) branches of the A ← X transition as first color. The difference spectra are obtained by recording the two-color spectra and subsequently subtracting the spectrum recorded with the second color only (black).

Figure 5 .

 5 Figure 5. Experimental (Exp.) and simulated (Sim.) scattering images at selected collision energies (E col ) for the NH (X 3 Σ -, N = 0, j = 1) + He → NH (X 3 Σ -, N = 2, j = 3) + He process, corresponding to an excitation energy of 97.7 cm -1 . One image pixel corresponds to a velocity of 2.16 m/s. The angular scattering distributions extracted from the experimental (solid red lines) and simulated images (dashed blue lines) are shown in the right panels, together with DCSs from theory (dashed green lines). The different curves were normalized on the maximum intensity for clarity.

Figure 6 .

 6 Figure 6. Experimental excitation function for the NH (X 3 Σ -, N = 0, j = 1) + He → NH (X 3 Σ -, N = 2, j = 3) + He process, corresponding to an excitation energy of 97.7 cm -1 , together with the theoretical ICS (blue line) and the theoretical ICS convoluted with a collision energy spread of 10 cm -1 FWHM. The colored data points correspond to six different measurement sets that were each fitted to convoluted theory through the mean He velocity, collision energy spread, and a uniform intensity scaling parameter. The error bars represent the experimental uncertainty at 95% confidence interval. The vertical arrows indicate the collision energies at which scattering images were measured, which are shown together with simulated images in the lower panel. One image pixel corresponds to a velocity of 2.16 m/s.

Table 1 .

 1 Summary of Relevant Parameters of the Various REMPI Schemes That Were Investigated to State-Selectively Probe NH (X 3 Σ -) Radicals a

	scheme	resonant step	λ (nm)	ions/shot recoil (m/s)
	2 + 1	D ← X	224	∼100	38
	1 + 2′	A ← X	336 + 254	∼5	0-7.5
	3 + 1	? ← X	336*	∼2000	25
	1 + 2′ + 1′	A ← X	336 + 336.4	∼2000	25

https://doi.org/10.1021/acs.jpca.2c08712 J. Phys. Chem. A 2023, 127, 2306-2313 Downloaded via UNIVERSITE DE RENNES 1 on May 30, 2023 at 13:19:46 (UTC).See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

https://doi.org/10.1021/acs.jpca.2c08712 J. Phys. Chem. A 2023, 127, 2306-2313

Scientific Research (NWO). The expert technical support by Niek Janssen and André van Roij is gratefully acknowledged.

■ ACKNOWLEDGMENTS F.L. acknowledges financial support from the European Research Council (Consolidator Grant COLLEXISM, Grant Agreement No. 811363) and of the Institut Universitaire de France. S.Y.T.M. acknowledges financial support from the European Research Council (Consolidator Grant FICOMOL, Grant Agreement No. 817947). J.O. acknowledges support from the European Unions Marie Skodowska-Curie Actions (Grant Agreement No. 886046). This work is part of the research program of The Netherlands Organization for

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