Charles Chemel
email: c.chemel@herts.ac.uk

Chantal Staquet

Mixing efficiency and entrainment at an atmospheric inversion layer

Keywords: Atmospheric flows, Mixing and dispersion, Stratified flows

The context is that of the convectively-driven atmospheric boundary layer capped by an inversion layer (i.e. a stably-stratified interface) and we focus on the regime of equilibrium entrainment, i.e. when the boundary-layer evolution is in a quasi-steady state. The parameterization of the entrainment process across the interfacial layer is usually based on the entrainment ratio, namely the ratio of the negative of the heat flux at the interface to the heat flux at the ground surface. Hence the issue is to relate the entrainment ratio to measurable parameters. In this study, we rely on a formulation of convective entrainment in terms of mixing efficiency, which can be computed directly for instance from high-resolution vertical profiles of virtual potential temperature. We discuss the applicability of this parameterization for an explicit treatment of the entrainment process in classical boundarylayer parameterization schemes implemented in meso-scale models.

Résumé :

Dans cette étude, on s'intéresse à la couche limite atmosphérique convective présentant une inversion à son sommet (i.e. une interface stablement stratifiée) et l'on se limite au cas où son évolution est quasi stationnaire. La paramétrisation du processus d'entraînement au sommet de la couche limite est classiquement basée sur le rapport, en valeur absolue, entre et le flux de chaleur à l'interface et le flux de chaleur à la surface du sol. Ainsi l'enjeu est de relier ce rapport quantifiant l'entraînement à des paramètres mesurables. Dans ce travail, on utilise une formulation de l'entraînement exprimée en termes d'efficacité de mélange, laquelle peut être calculée directement par exemple à partir de profils verticaux de température potentielle virtuelle à haute résolution. On discute de l'applicabilité de cette paramétrisation pour un traitement explicite du processus d'entraînement dans les schémas classiques de paramétrisations de la couche limite implémentés dans les modèles à méso échelles.

Introduction

In the convectively-driven atmospheric boundary layer, buoyancy-driven thermals overturn continuously the air, thereby mixing the convective layer (referred to as the mixed layer). Hence, profiles of conserved variables (e.g. the virtual potential temperature) are uniform with height within the mixed layer. Thereafter we consider a horizontally homogeneous boundary layer without large-scale subsidence. Let Θ and F denote the horizontally-averaged virtual potential temperature and heat flux, respectively. The mixed layer is generally capped by an inversion layer (i.e. a stably-stratified interface) at a height z i , which is defined as the level where F is minimum (being negative). The interfacial layer is dominated by both local entrainment effects and the properties of the inversion and stably-stratified region above. As a result of entrainment of air from the free atmosphere above, the mixed layer deepens, the evolution of which is governed by the entrainment velocity w e ≡ żi Several entrainment law formulations have been derived out to date to relate w e to measurable parameters of the mixed and interfacial layers. [START_REF] Betts | Reply to comment on the paper 'Non-precipitating cumulus convection and its parameterization[END_REF]. The strength of the capping inversion is denoted ∆Θ. The interfacial-layer thickness is designated δ and Γ FA is the Θ lapse rate in the free atmosphere.

z i 0 z ↑ z ↑ 0 Θ → F → F s Interfacial F i Γ FA z i + δ layer ∆Θ δ 0 z ↑ z ↑ 0 Θ → F → F s Interfacial F i Γ FA (b) z i

Structure of the convective boundary layer

The structure of the convective boundary-layer can be represented in varying the degree of complexity, especially regarding the structure of the interfacial layer. In the zero-order model (ZOM) proposed by [START_REF] Lilly | Models of cloud-topped mixed layers under a strong inversion[END_REF], the thickness of the interfacial layer is assumed infinitesimal, while the temperature profile exhibits a jump across that interface [see figure 1(a)]. In firstorder models (FOMs), the finite thickness δ of the IL is taken into account (e.g. [START_REF] Betts | Reply to comment on the paper 'Non-precipitating cumulus convection and its parameterization[END_REF][START_REF] Duynkerke | Entrainment parametrization in convective boundary layers[END_REF]. [START_REF] Betts | Reply to comment on the paper 'Non-precipitating cumulus convection and its parameterization[END_REF] assumed that the mixed layer extends up to z i , and that above z i the heat flux increases linearly to zero up to z i + δ [see figure 1(b)]. Both zero-and first-order models are commonly used in many studies of mixed-layer growth and entrainment, though more refined versions that provide a more detailed representation of the interfacial layer do exist (e.g. [START_REF] Fedorovich | A model for a shear-free convective boundary layer with parameterized capping inversion structure[END_REF]. The most important quantity which appears in the model here concerns the entrainment heat flux at z i , denoted by F i (see figure 1). Indeed, the entrainment law is classically based on the ratio R = -F i /F s of the negative of the heat flux at the interface to the heat flux at the ground surface. This ratio is usually taken constant, with value 0.2 (e.g. [START_REF] Stull | The energetics of entrainment across a density interface[END_REF].

Model for R in terms of mixing efficiency

In the regime of equilibrium entrainment (i.e. when the boundary-layer evolution is in a quasisteady state), we proposed a parameterization for R that involves the mixing efficiency, denoted by γ, and the thickness of the interface δ, namely R = [z i / (z i + δ)] γ/ (γ + 1). This parameterization was actually found to match very well the numerical computation of R from a high-resolution large-eddy simulation [START_REF] Chemel | Eulerian-and Lagrangian-based estimates of convective entrainment rate from large-eddy simulation[END_REF]. This result enabled us to derive modified expressions for the classical entrainment laws within the ZOM and FOM frameworks as a function of γ. We showed that within the ZOM framework, the scaling factor in the entrainment law, denoted by A, is the flux Richardson number. This parameterization of A is further improved within the FOM framework, as direct computation of A from the aforementioned large-eddy simulation showed it.

From a practical point of view, the present parameterizations rely on the computation or measurement of γ. In this study, γ is computed directly from the large-eddy simulation results mentioned previously following the method proposed by Winters et al. (1995). The values of γ are between 0.26 ad 0.30. This range is consistent with the one found in turbulent stratified flows (e.g. [START_REF] Staquet | Gravity and inertia-gravity internal waves: breaking processes and induced mixing[END_REF]. The understanding of the approximate 0.2 value found for γ in stably-stratified flows would provide the final closure but this question is still a challenge. Some boundary-layer flow parameterization schemes within meso-scale models do not represent explicitly the entrainment process. This is for instance the case for the K-profile model (Troen & Mahrt, 2003), for which the vertical mixing and the growth of the boundary layer are modelled separately. Hence one may wonder whether our formulation of convective entrainment in terms of mixing efficiency could not be applied to represent the entrainment process in such parameterization schemes. Since Winters & D'Asaro (1996) provided a recipe on how to infer γ from vertical temperature soundings, it should be possible. Further research is under way to take the step forward by testing this parameterization.

Further reading

Figure 1 :

 1 Figure 1: Sketch profiles of typical virtual potential temperature Θ and heat flux F for the convectivelydriven boundary layer. (a) Zero-order model (ZOM) by Lilly (1968); (b) first-order model (FOM) by[START_REF] Betts | Reply to comment on the paper 'Non-precipitating cumulus convection and its parameterization[END_REF]. The strength of the capping inversion is denoted ∆Θ. The interfacial-layer thickness is designated δ and Γ FA is the Θ lapse rate in the free atmosphere.

  Chemel, C. & C. Staquet 2007 A formulation of convective entrainment in terms of mixing efficiency. J. Fluid Mech. In press. A part of this paper is adapted here thanks to CUP.


Jean-Bernard B Lasserre

THE MOMENT-SOS HIERARCHY AND THE CHRISTOFFEL-DARBOUX KERNEL

published or not. The documents may come

Introduction

Consider the Polynomial Optimization Problem (POP):

f * = min x { f (x) : x ∈ B },
where B ⊂ R n is a compact basic semi-algebraic set. For the hierarchy of upper bounds discussed below, B is restricted to be a "simple" set like e.g. a box, an ellipsoid, a simplex, a discrete-hypercube, or their image by an affine transformation. Indeed, to define an SOS-hierarchy of upper bounds converging to the global minimum f * as described in e.g. [START_REF] De Klerk | Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial optimization on the sphere[END_REF][START_REF] Lasserre | A new look at nonnegativity on closed sets and polynomial optimization[END_REF][START_REF] Slot | Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets[END_REF], we use a measure µ whose support is exactly B, and for which all moments µ α := B x α dµ , α ∈ N n , can be obtained numerically or in closed-form. For instance if B is a box, an ellipsoid or a simplex, µ can chosen to be the Lebesgue measure restricted to B. On the hypercube {-1, 1} n µ one may choose for µ the counting measure, etc.

Work partly funded by the AI Interdisciplinary Institute ANITI through the French "Investing for the Future PI3A" program under the Grant agreement ANR-19-PI3A-0004.

1.1. Background. Let B ⊂ R n be the basic semi-algebraic set defined by (1.1) B = { x ∈ R n : g j (x) ≥ 0 j = 1, . . . , m } , for some polynomials g j ∈ R[x], j = 1, . . . , m. Let g 0 (x) = 1 for all x, and let d j := ⌈deg(g j )/2⌉, j = 0, . . . , m. Define Σ[x] t be the set of sums-of-squares (SOS) polynomials of degree at most 2t.

A hierarchy of lower bounds. To approximate f * from below, consider the hierarchy of semidefinite programs indexed by t ∈ N:

(1.2) ρ t = sup λ σ j { λ : f -λ = m j=0 ψ j g j ; ψ j ∈ Σ[x] t-d j , j = 0, . . . , m } ,
where Σ[x] t denotes the space of sum-of-squares (SOS) polynomials of degree at most 2t. Under some Archimedean assumption on the g j 's, ρ t ≤ f * for all t and the sequence of lower bounds (ρ t ) t∈N is monotone non decreasing and converges to f * as t increases. Moreover, by a result of Nie [START_REF] Nie | Optimality Conditions and Finite Convergence of Lasserre's Hierarchy Math[END_REF], its convergence is finite generically, and global minimizers can be extracted from an optimal solution of the semidefinite program which is the dual of (1.2); see e.g. [START_REF] Lasserre | An introduction to Polynomial and Semi-Algebraic Optimization[END_REF]. The sequence of semidefinite programs (1.2) and their duals, both indexed by t, forms what is called the Moment-SOS hierarchy initiated in the early 2000's. For more details on the Moment-SOS hierarchy and its numerous applications in and outside optimization, the interested reader is referred to [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF][START_REF] Lasserre | An introduction to Polynomial and Semi-Algebraic Optimization[END_REF].

A hierarchy of upper bounds. Let µ be a finite Borel measure whose support is exactly B, where now B is a "simple" set as mentioned earlier.

(Hence all moments of µ are available in closed form.) To approximate f * from above, consider the hierarchy of semidefinite programs

(1.3) u t = inf σ { B f σ dµ : B σ dµ = 1 ; σ ∈ Σ[x] t } . That u t ≥ f * is straightforward since f ≥ f * on B ⇒ B f σ dµ ≥ f * B σ dµ = f * ,
for any feasible SOS σ. In [START_REF] Lasserre | A new look at nonnegativity on closed sets and polynomial optimization[END_REF] it was proved that u t ↓ f * as t increases, and in fact solving the dual of (1.3) is solving a generalized eigenvalue problem for a certain pair of real symmetric matrices. In a series of papers, de Klerk, Laurent an co-workers have provided several rates of convergence of u t ↓ f * for several examples of sets B. For more details and results, the interested reader is referred to [START_REF] De Klerk | Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial optimization on the sphere[END_REF][START_REF] Slot | Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets[END_REF][START_REF] Slot | Near-optimal analysis of univariate moment bounds for polynomial optimization[END_REF][START_REF] Slot | Sum-of-squares hierarchies for binary polynomial optimization[END_REF] and references therein.

The meaning of (1.3) is clear if one recalls that

(1.4) f * = inf φ { B f dφ : φ(B) = 1 ; φ ∈ M (B) + },
where M (B) + is the space of all finite Borel measures on B. Indeed in (1.3) one only considers the (restricted) subset of probability measures on B that have a density (an SOS of degree at most 2t) with respect to µ whereas in (1.4) one considers all probability measures on B. In particular, the Dirac measure φ := δ ξ at any global minimiser ξ ∈ B belongs to M (B) + but does not have a density with respect to µ, which explains why the convergence u t ↓ f * as t increases, can be only asymptotic and not finite; an exception is when B is a finite set (e.g. B = {-1, 1} n and µ is the counting measure).

1.2. Contribution. Our contribution is to show that in fact the dual of the semidefinite program (1.2) for computing the lower bound ρ t has also an interpretation of the same flavor as (1.3) where one now considers signed Borel measures φ t with a distinguished polynomial density with respect to µ. Namely, the dual of (1.2) minimizes B f dφ t over signed measures φ t of the form:

(1.5)

dφ t (x) = σ t (x) dµ(x) =   |α|≤2t σ α T α (x)   dµ(x) ,
where :

-(T α ) ⊂ R[x]
is a family of polynomials that are orthonormal with respect to µ, and -the coefficients

σ t = (σ α ) α∈N n 2t of the polynomial σ t ∈ R[x]
2t satisfy the usual semidefinite constraints that are necessary for σ t to be moments of a measure on B.

Eventually for some t ∈ N, σ t satisfies:

(1.6)

σ α = T α (ξ) = B T α (x) δ ξ (dx) , |α| ≤ 2t ,
where ξ is an arbitrary global minimizer and δ ξ is the Dirac measure at ξ ∈ B. Indeed then

B f (x) dφ t (x) := B f (x) |α|≤2t T α (ξ) T α (x) dµ(x) = f (ξ) , because the Christoffel-Darboux Kernel K t (x, y) := |α|≤2t T α (x) T α (y) is a reproducing kernel for R[x] 2t
, considered to be a finite-dimensional subspace of the Hilbert space L 2 (B, µ). Moreover, σ t (ξ) -1 is nothing less than the Christoffel function evaluated at the global minimizer ξ of f on B.

As a take home message and contribution of this paper, it turns out that the dual of the step-t semidefinite relaxation (1.2) is a semidefinite program that computes the coefficients σ t = (σ α ) of the polynomial density σ t in (1.5). In addition, when the relaxation is exact then σ t (ξ) -1 is the Christoffel function of µ, evaluated at a global minimizer ξ of f on B.

Interestingly, in the dual of (1.2) there is no mention of the reference measure µ. Only after we fix some arbitrary reference measure µ on B, we can interpret an optimal solution as coefficients σ t of an appropriate polynomial density with respect to µ.

So in both (1.3) and the dual of (1.2), one searches for a polynomial "density" with respect to µ. In (1.3) one searches for an SOS density (hence a positive density) whereas in the dual of (1.2) one searches for a signed polynomial density whose coefficients (in the basis of orthonormal polynomials) are moments of a measure on B (ideally the Dirac at a global minimizer).

The advantage of the (signed) polynomial density in (1.5) compared to the (positive) SOS density in (1.3), is to be able to obtain the global optimum f * as the integral of f against this density, which is impossible with the SOS density of (1.3).

At last but not least, this interpretation establishes another (and rather surprising) simple link between polynomial optimization (here the Moment-SOS hierarchy), the Christoffel-Darboux kernel and the Christoffel function, fundamental tools in the theory of orthogonal polynomials and the theory of approximation. Previous contributions in this vein include [START_REF] Lasserre | Connecting optimization with spectral analysis of tri-diagonal matrices[END_REF] to characterize upper bounds (1.3), [START_REF] De Klerk | Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial optimization on the sphere[END_REF][START_REF] Slot | Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets[END_REF][START_REF] Slot | Near-optimal analysis of univariate moment bounds for polynomial optimization[END_REF] to analyze their rate of convergence to f * , and the more recent [START_REF] Slot | Sum-of-squares hierarchies for binary polynomial optimization[END_REF] for rate of convergence of both upper and lower bounds on B = {0, 1} n .

Main result

2.1. Notation and definition. Let R[x] = R[x 1 , . . . , x n ] be the ring of real polynomials in the variables x 1 , . . . , x n and let R[x] t ⊂ R[x] be its subspace of polynomials of degree at most t. Let N n t := {α ∈ N n : |α| ≤ t} where |α| = i α i . For an arbitrary Borel subset X of R n , denote by M (X ) + the convex cone of finite Borel measures on X ⊂ R n , and by P(X ) is subset of probability measures on X ..

2.2.

Moment and localizing matrices. Given an sequence y = (y α ) α∈N n and polynomial g ∈ R[x], x → g(x) := γ g γ x γ , the localizing matrix M t (g y) associated with g and y is th real symmetric matrix with rows and columns indexed by α ∈ N n t and with entries (2.1)

M t (g y)(α, β) := γ g γ y α+β+γ , α, β ∈ N n t .
If g(x) = 1 for all x then M t (g y) (= M t (y)) is called the moment matrix.

A sequence y = (y α ) α∈N n has a representing measure if there exists a (positive) finite Borel measure φ on R n such that y α = x α dφ for all α ∈ N n .

If y has a representing measure supported on {x : g(x) ≥ 0} then M t (y) 0 and M t (g y) 0 for all t ∈ N. The converse is not true in general; however, the following important result is at the core of the Moment-SOS hierarchy.

Theorem 2.1. (Putinar [START_REF] Putinar | Positive polynomial on compact semi-algebraic sets[END_REF]) Let g j ∈ R[x], j = 0, . . . , m with g 0 (x) = 1 for all x, and let G := {x ∈ R n : g j (x) ≥ 0, j = 1, . . . , m } be compact. Moreover, assume that for some M > 0, the quadratic polynomial x → Mx 2 can be written in the form m j=0 ψ j g j , for some SOS polynomials ψ 0 , . . . ψ m .

Then a sequence y = (y α ) α∈N n has a representing measure on G if and only if M t (g j y) 0 for all t ∈ N, and all j = 0, . . . , m.

Orthonormal polynomials. Let B ⊂ R n be the compact basic semialgebraic set defined in (1.1) assumed to have a nonempty interior. Let µ be a finite Borel (reference) measure whose support is exactly B and with associated sequence of orthonormal polynomials (T α )

α∈N n ⊂ R[x]. That is B T α T β dµ = δ α=β , ∀α, β ∈ N n .
For instance, if B = [-1, 1] n and µ is the uniform probability distribution on B, one may choose for the family (T α ) the tensorized Legendre polynomials. Namely if (T j ) ⊂ R[x] is the family of univariate Legendre polynomials, then

T α (x) := n j=1 T α j (x j ) , α ∈ N n .
For every t ∈ N, the mapping

K t : B × B → R, (x, y) → K t (x, y) := |α|≤t T α (x) T α (y) , x, y ∈ B
is called the Cristoffel-Darboux kernel associated with µ. An important property of K t is to reproduce polynomials of degree at most t, that is:

(2.2) p(x) = B p(y) K t (x, y) dµ(y) ∀x ∈ B , ∀p ∈ R[x] t .
This is why K t is called a reproducing kernel, and R[x] t viewed as a finitedimensional vector subspace of the Hilbert space L 2 (B, µ), is called a Reproducing Kernel Hilbert Space (RKHS). For more details on the theory of orthogonal polynomials, the interested reader is referred to e.g. [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF] and the many references therein.

Main result.

An observation. Let f ∈ R[x] and let t ≥ deg(f ) = d f be fixed. Let P(B) ⊂ M (B) + be the space of probability measures on B. Then

f * = min φ∈P(B) + B f dφ = min φ∈P(B) + B B f (y) K t (x, y) dµ(y) dφ(x) = min φ∈P(B) + B f (y) ( B K t (x, y) dφ(x) ) dµ(y) = min φ∈P(B) + B f (y)     |α|≤t ( B T α (x) dφ(x) σα ) T α (y)     dµ(y) = min φ∈P(B) + B f (y) ( |α|≤t σ α T α (y) σt(y)∈R[x]t
) dµ(y) ,

where the second equality follows from Fubini-Tonelli interchange theorem valid in this simple setting. In other words, we have proved the following:

Lemma 2.2. Let B ⊂ R n be as in (1.1) and let µ be a finite Borel (reference) measure whose support is exactly B and with associated sequence of orthonormal polynomials (T α ) α∈N n . Let f * = min {f (x) : x ∈ B}. Then for every fixed t ≥ deg(f ):

(2.3)

f * = inf σ∈R[x]t B f (y) σ(y) dµ(y) ,
where the infimum is over all polynomials σ ∈ R[x] t of the form:

σ(x) = |α|≤t σ α T α (x) , ∀x ∈ B with (2.4) σ α = B T α (x) dφ(x) , ∀α ∈ N n
t , for some φ ∈ P(B). (2.5) So solving (2.3) is equivalent to searching for a signed measure σ dµ with polynomial (signed) density σ ∈ R[x] t that satisfies (2.4)-(2.5).

2.4.

A hierarchy of relaxations of (2.3). In this section we show the SOS-hierarchy defined in (1.2) is the dual semidefinite program of a natural SDP-relaxation of (2.3). In fact the only difficult constraint in (2.3) is (2.5) which demands σ to admit a representing probability measure φ on B.

Let D t be the lower triangular matrix for the change of basis of R

[x] 2t from the monomial basis v 2t (x) = (x α ) α∈N n 2t of R[x] 2t to the basis (T α ) α , i.e., (2.6)
    T 0 • • • T α • • •     = D t •     1 • • • x α • • •     = D t • v 2t (x)
and denote D ′ t the transpose of D t . The matrix D t is nonsingular with positive diagonal. Then with σ = (σ α ) α∈N n 2t , (2.5) reads

(2.7) σ = D t • y with y = B v 2t (x) dφ(x).
That is, y = (y α ) α∈N n 2t is required to be a moment sequence as it has a representing probability measure φ ∈ P(B). So in view of Theorem 2.1, the constraint (2.7) can be relaxed to σ = D t • y with y 0 = 1 and M t-d j (g j y)

0 , j = 0, . . . , m .

Therefore, consider the following relaxation of (

ρ 2t = inf σ∈R[x] 2t { B f (x) ( α∈N n 2t σ α T α (x)) dµ(x) : σ = D t • y ; y 0 = 1 ; M t-d j (g j y) 0 , j = 0, . . . , m } . Lemma 2.3. Let B ⊂ R n be as in (1.1 2.3) (2.8)
) and let µ be a finite Borel (reference) measure whose support is exactly B and with associated sequence of orthonormal polynomials (T α ) α∈N n . The semidefinite relaxation (2.8) of (2.3) reads:

(2.9) inf y { f , y : y 0 = 1 ; M t-d j (g j y) 0 , j = 0, . . . , m } , which is the dual of (1.2)

Proof. With f (x) = α f α x α = f , v 2t (x) , write f (x) = α∈N n 2t fα T α (x) in the basis (T α ) α∈N n 2t . Then with f = ( fα ) one obtains f , D t • v 2t (x) = D ′ t f , v 2t (x) = f , v 2t (x) ⇒ f = (D ′ t ) -1 f
. Finally, as the T α 's form an orthonormal basis, the criterion

B f (x) ( α∈N n 2t σ α T α (x)) dµ(x)
to minimize in (2.8) reads:

B f (x) ( α∈N n 2t σ α T α (x)) dµ(x) = f , σ = (D ′ t ) -1 f , D t y = f , y ,
which yields that (2.8) is exactly (2.9). Next, that (2.9) is a dual of (1.2) is a standard result in polynomial optimization [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF][START_REF] Lasserre | An introduction to Polynomial and Semi-Algebraic Optimization[END_REF].

Of course by reverting the process of the above proof, the semidefinite program (2.9) can be transformed to (2.8) once a reference measure µ with support exactly B is defined with its associated orthonormal polynomials (T α ). Indeed, once µ and the T α 's are defined, one may use the change of basis matrix D in (2.6) to pass from (2.9) to (2.8).

Corollary 2.4. Let B ⊂ R n be as in (1.1) and let µ be a finite Borel (reference) measure whose support is exactly B and with associated sequence of orthonormal polynomials (T α ) α∈N n . Let f * be the global minimum of f on B.

Let t be such that the semidefinite relaxation (2.8) (or equivalently (2.9)) is exact, i.e., if 2t = f * . If an optimal solution y * of (2.9) has a representing measure φ * ∈ M (B) + , then an optimal polynomial density σ * ∈ R[x] 2t of (2.8) satisfies:

σ * (ξ) = α∈N n 2t T α (ξ) 2 = K 2t (ξ, ξ) , ∀ξ ∈ supp(φ * ) , that is, σ * (ξ) -1 is the Christoffel function evaluated at the global minimizer ξ ∈ B.
Proof. If y * has a representing measure φ * ∈ M (B) + then necessarily f (ξ) = f * for all ξ ∈ sup(φ * ); see e.g. [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF][START_REF] Lasserre | An introduction to Polynomial and Semi-Algebraic Optimization[END_REF]. In particular, for every ξ ∈ sup(φ * ), the vector ŷ := (ξ α ) α∈N n 2t is also an optimal solution of (2.9). Then

σ * = D t • ŷ = D t • v 2t (ξ) =     T 0 (ξ) • • • T α (ξ) • • •     , i.e., σ * α = T α (ξ) for all α ∈ N n 2t . Therefore, x → σ * (x) = α∈N n 2t T α (ξ) T α (x) = K 2t (ξ, x) ,
and so σ * (ξ) = K 2t (ξ, ξ). In other words, σ * (ξ) -1 is the Christoffel function associated with µ, evaluated at ξ ∈ B.

Discussion. Observe that the formulation (2.8) does not require that the set B is a "simple" set as it is required in (1.3). Indeed the orthonormal polynomials (T α ) are only used to provide an interpretation of the hierarchy of lower bounds (2.9) (and its dual (1.2)). On the other hand, for the hierarchy of upper bounds (1.3), B indeed needs to be a "simple" set for computational purposes. This is because one needs the numerical value of the moments of µ for a practical implementation of (1.3).

Lemma 2.3 shows that the Moment-SOS hierarchy described in [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF][START_REF] Lasserre | An introduction to Polynomial and Semi-Algebraic Optimization[END_REF] amounts to compute a hierarchy of signed polynomial densities with respect to some reference measure µ with support exactly B. When the step-t relaxation is exact (which takes place generically [START_REF] Nie | Optimality Conditions and Finite Convergence of Lasserre's Hierarchy Math[END_REF]) the resulting optimal density σ in (2.8) is nothing less than the polynomial x → K t (ξ, x) where ξ is a global minimizer of f on B, K t (ξ, x) is the celebrated Cristoffel-Darboux kernel in approximation theory, and σ(ξ, ξ) is the reciprocal of the Christoffel function evaluated at a global minimizer ξ.

Conclusion

We have shown that the Moment-SOS hierarchy that provides an increasing sequence of lower bounds on the global minimum of a polynomial f on a compact set B, has a simple interpretation related to orthogonal polynomials associated with an arbitrary reference measure whose support is exactly B. This interpretation strongly relates polynomial optimization (here the Moment-SOS hierarchy) with the Christoffel-Darboux kernel and the Christoffel function, fundamental tools in the theory of orthogonal polynomials and the theory of of approximation.

It is another item in the list of previous contributions [START_REF] Lasserre | Connecting optimization with spectral analysis of tri-diagonal matrices[END_REF][START_REF] De Klerk | Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial optimization on the sphere[END_REF][START_REF] Slot | Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets[END_REF][START_REF] Slot | Near-optimal analysis of univariate moment bounds for polynomial optimization[END_REF] that also link some issues in polynomial optimization with orthogonal polynomials associated with appropriate measures. We hope that such connections will stimulate even further investigations in this direction.