Anton Baranov

Philippe Jaming

Karim Kellay
email: kkellay@math.u-bordeaux.fr

Michael Speckbacher
email: michael.speckbacher@univie.ac.at

Michael Speckbacher Oversampling

Donoho- Logan Type

Oversampling and Donoho-Logan type theorems in model spaces

Keywords: 2000 Mathematics Subject Classification. Primary 30H05; Secondary 30D10, 30E05 Carleson measure, model space, oversampling formula, Bernstein's inequality

come

Introduction

The aim of this paper is to extend two classical results on the Paley-Wiener space to more general model spaces. The first result is the sampling theorem, or more precisely the oversampling formula that improves the convergence in Shannon sampling. The second result is the Donoho-Logan Large Sieve Principle which can be seen as one of the first results on sparsity in signal processing. In the context of complex analysis, this result is a result on Carleson measures for the Paley-Wiener space.

Let us now be more precise. Recall that the Paley-Wiener space is the subspace of L 2 signals that are band-limited to [-c, c]. It is a very common space used to model signal encountered in natural sciences. If we normalize the Fourier transform through f (ξ) = R f (t)e -itξ dt, the Paley-Wiener space P W p c (R), c > 0, 1 ≤ p < ∞, is defined as

P W p c (R) = f ∈ L p (R) : supp f ⊂ [-c, c] ,
where, for p > 2, f is understood in the distributional sense. Two well known properties of the Paley-Wiener spaces are that they consist of entire functions and that every function f ∈ P W 2 c (R) can be reconstructed from samples {f (πk/b)} k∈Z via the Kotelnikov-Nyquist-Shannon Formula

f (x) = k∈Z f k π b 1 2b γ x -k π b , (1.1
f (x) = k∈Z f k π b sinc b x -k π b , (1.2)
where sinc t = sin t/t. However, when b > c one can do better by taking γ smooth in which case γ decreases faster than the sinc function. Most authors take γ to be C ∞ so that, for every N , | γ x -k π b | = O(k -N ) when k → ±∞ and this estimate is even uniform when x stays in a compact set. As a consequence (1.1) converges much better than (1.2), a fact that is well known in signal processing. The drawback of this is that, to the best of our knowledge, there is no example of a function γ as above for which γ is explicitly known.

One way to overcome this is to give up on arbitrarily fast decay and to fix N . One then fixes a parameter a > 0, and takes ψ (1) = ψ = 1 2a 1 [-a,a] and ψ (k+1) = ψ (k) * ψ. Then ψ (N ) is supported in [-N a, N a] and

The second result we are dealing with in this paper is the Donoho-Logan Large Sieve theorem. The analytic large sieve principle is a classical inequality for trigonometric polynomials which finds many applications in analytic number theory (see, e.g., [START_REF] Montgomery | The analytic principle of the large sieve[END_REF] and references therein). It was extended from trigonometric polynomials to their near cousins the band-limited functions by Donoho and Logan [START_REF] Donoho | Signal recovery and the large sieve[END_REF] in the early 90s (after earlier work by Boas [START_REF] Boas | Entire functions bounded on a line[END_REF]), applying it to reconstruction of missing data in signal processing.

For I an interval, we denote by |I| the length of I. When µ is a positive σ-finite measure and δ > 0, we write

D µ (δ) = sup µ(I) |I| : I closed interval, |I| = δ = sup x∈R µ([x, x + δ]) δ . (1.4)
Donoho and Logan proved the following:

Theorem 1.1 (Donoho-Logan). Let c, δ > 0 and µ be a positive σ-finite measure. Then for every

f ∈ P W 2 c (R), R |f (x)| 2 dµ(x) ≤ 1 + cδ π D µ (δ) R |f (x)| 2 dx. (1.5)
Moreover, for every

f ∈ P W 1 c (R), R |f (x)| dµ(x) ≤ D µ (δ) sinc cδ 2 R |f (x)| dx. (1.6)
This theorem is one of the founding results of the use of sparsity constraints in signal recovery problems and has led to considerable research in the decades following it. However, until recently, it seems that this result did not get the attention it deserves in other mathematical communities. For instance, in complex analysis, the Donoho-Logan Large Sieve theorem provides a quantitative estimate of the fact that a positive measure is a Carleson measure of the Paley-Wiener space, that is, a bound on the norm of the injection P W 2 c (R) → L 2 (µ). Note that a first estimate of this type was given by Lin [START_REF] Lin | On equivalent norms in the space of square integrable entire functions of exponential type[END_REF]. Recently, Husain and Littmann extended the result for p = 1 to higher dimensions [START_REF] Husain | Concentration inequalities for Paley-Wiener spaces[END_REF]. This inadvertence may be due to the fact that the strategy of proof has some rigidities that are difficult to overcome. Indeed, the proof relies heavily on the interplay between convolution and the Fourier transform, and on a construction by Selberg (based on a previous construction by Beurling) of an entire function majorizing the sign function, see [START_REF] Montgomery | The analytic principle of the large sieve[END_REF][START_REF] Vaaler | Some extremal functions in Fourier Analysis[END_REF]. Despite its rigidity, this strategy of proof has recently been extended to more general setting, see, e.g., [START_REF] Abreu | Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces[END_REF][START_REF] Abreu | Donoho-Logan large sieve principles for the continuous wavelet transform[END_REF][START_REF] Ph | Planar sampling sets for the short-time Fourier transform[END_REF]. Our first aim in this paper is to provide two new proof strategies, one based on the oversampling formula, the second on Bernstein's inequality. In particular, those proofs rely on real variable techniques only. The price to be paid is that the numerical constants we obtain are slightly worse. The proofs however offer more flexibility so that they apply in larger settings.

To illustrate this, we will extend the oversampling formula and the large sieve inequality to the setting of model spaces on the upper half-plane, an important family of spaces of holomorphic functions which contains the Paley-Wiener space P W 2 c (R). Let us now describe this family. First, let H 2 be the Hardy space on the upper half-plane

C + := {z ∈ C : Im z > 0}. H 2 := f ∈ Hol(C + ) : sup y>0 R |f (x + iy)| 2 dx < ∞ .
Note that we may identify H 2 = {f ∈ L 2 (R) : supp f ⊂ [0, +∞)}. Let Θ be an inner function in C + , that is, a bounded analytic function on C + with unimodular boundary values almost everywhere on R. The corresponding model space is defined by

K 2 Θ := H 2 ∩ (ΘH 2 ) ⊥ . As a fundamental example, if Θ c (z) = exp(icz) for some c > 0, then K 2 Θ 2c = {f ∈ L 2 (R) : supp f ⊂ [0, 2c]} = {Θ c f : f ∈ P W 2 c (R)} or, equivalently, P W 2 c (R) = {e -icz f : f ∈ K 2 Θ 2c }.

Moreover, one can define K p

Θ model spaces in H p , 1 ≤ p ≤ +∞, so as to also cover the P W p c spaces, see Section 5. In the context of model spaces, an analogue of Shannon's sampling formula was established by de Branges [START_REF] De Branges | Hilbert Spaces of Entire Functions[END_REF] in the case of meromorphic inner functions and later by Clark [START_REF] Clark | One-dimensional perturbations of restricted shifts[END_REF] in the general case, see (3.5). However, the oversampling formula for model spaces seems to be unknown. Previously, oversampling results were obtained in [START_REF] Silva | Oversampling and undersampling in de Branges spaces arising from regular Schrödinger operators[END_REF][START_REF] Toloza | Oversampling and aliasing in de Branges spaces arising from Bessel operators[END_REF] for specific model spaces (or closely related de Branges spaces) associated with certain differential operators.

We start by establishing an oversampling formula for the class of model spaces with meromorphic inner functions. This condition is equivalent to Θ ∈ L ∞ (R). Once this is done, we propose a first proof of Donoho and Logan's theorem for model spaces associated with inner functions with bounded derivative, an interesting class of model spaces that share many properties with the Paley-Wiener spaces (see, e.g., [START_REF] Dyakonov | Entire functions of exponential type and model subspaces in H p[END_REF][START_REF] Nikolski | Treatise on the Shift Operator[END_REF]).

Theorem 1.2. Let Θ be an inner function such that Θ ∈ L ∞ (R), and µ be a σ-finite Borel measure on R. For δ > 0, D µ (δ) defined in (1.4) and 1 ≤ p < +∞ there exists C p > 0 such that for every

f ∈ K p Θ R |f (x)| p dµ(x) ≤ C p (1 + Θ ∞ δ) p D µ (δ) R |f (x)| p dx. (1.7)
The proof relying on oversampling gives this result for p = 2 and a rather bad constant C but gives the right behavior with respect to δ and Θ ∞ . We thus give a second proof, based on Bernstein's inequality which allows to obtain the constant C p = 1, falling only short of Donoho and Logan's result in the case of the Paley-Wiener space. The particular constants are discussed in more detail in Remark 5.4. We think that both proofs are interesting due to the fact that they only rely on real analytic arguments and are rather flexible. Finally, we end this article with a version of Theorem 1.2 that applies to a different class of inner functions (not necessarily meromorphic), the so-called one-component inner functions, but requires a modification in the definition of D µ (δ), see Theorem 6.1. This provides a quantitative converse version of the Logvinenko-Sereda Theorem for model spaces established in [START_REF] Hartmann | Quantitative estimates of sampling constants in model spaces[END_REF].

The rest of this paper is organized as follows: Section 2 is devoted to a technical lemma. In Section 3, we introduce the necessary background on model spaces and prove the oversampling formula. In Section 4 we give an extension of the large sieve inequality to model spaces using the oversampling techniques. The approach based on Bernstein type inequalities is considered in Section 5, while in Section 6 a certain analog of the large sieve is given for a class of model spaces generated by the so-called one-component inner functions.

Preliminary lemma

Let Ξ be a continuous function such that

|Ξ(x)| ≤ min(1, |x| -1 ).
(2.1)

As an example, one can take Ξ(x) = sinc x := sin x x .

Lemma 2.1. Let Ξ be a function satisfying the bound (2.1). Let a, b ∈ R, α, δ > 0 and µ be a σ-finite Borel measure on R. Then we have

(1) R Ξ(x -a) 2 Ξ(x -b) 2 dx ≤ 8π 4 + (a -b) 2 , (2) R Ξ(x-a) 2 Ξ(x-b) 2 dµ(x) ≤ 8π C 2 δ δ sup x∈R µ([x, x + δ]) 4 + (b -a) 2 , with C δ = max(4, 1+9δ 2 ).
Proof. The first estimate follows immediately from the simple estimate

|Ξ(x)| 2 ≤ min(1, |x| -2 ) ≤ 2 1 + x 2 ,
and the computation of the integral

R dt (1 + (t -a) 2 )(1 + (t -b) 2 ) = 2π 4 + (a -b) 2 , via the residue formula. Next, for ∈ Z, let I = [ δ, ( + 1)δ[, we have R Ξ(x -a) 2 Ξ(x -b) 2 dµ(x) = ∈Z I Ξ(x -a) 2 Ξ(x -b) 2 dµ(x) ≤ sup ∈Z µ(I ) • ∈Z 4 (1 + d(a, I ) 2 )(1 + d(b, I ) 2 ) = 4 δ sup ∈Z µ(I ) • ∈Z I dx 1 + d(a, I ) 2 1 + d(b, I ) 2 . Now, let x ∈ I . If a ≤ ( -2)δ then x -a ≥ 2δ so that d(a, I ) = δ -a = ( + 1)δ -a -δ ≥ x -a -δ ≥ x -a 2 thus 1 1 + d(a, I ) 2 ≤ 4 1 + (x -a) 2 . The same holds if a ≥ ( + 3)δ. Finally, if ( -2)δ ≤ a ≤ ( + 3)δ, x -a ≤ 3δ thus 1 1 + d(a, I ) 2 ≤ 1 ≤ 1 + 9δ 2 1 + (x -a) 2 . It follows that R Ξ(x -a) 2 Ξ(x -b) 2 dµ(x) ≤ 4 δ C 2 δ sup ∈Z µ(I ) • R dx (1 + (x -a) 2 )(1 + (x -b) 2 ) ≤ 8π C 2 δ δ sup x∈R µ [x, x + δ] 4 + (b -a) 2 , with C δ = max(4, 1 + 9δ 2 ). Applying Lemma 2.1(2) to the dilated measure µ α (A) = µ(A/α), α > 0 we get R Ξ(αx -a) 2 Ξ(αx -b) 2 dµ(x) ≤ 8π C 2 δ δ sup x∈R µ([x, x + δ/α]) 4 + (b -a) 2 , ( 2.2)
with C δ = max(4, 1 + 9δ 2 ).

A slightly more evolved version of the lemma is easily available: for m ≥ 2 and δ > 0 there is a constant C m,δ such that

R Ξ(x -a) 2m Ξ(x -b) 2m dµ(x) ≤ C m,δ 1 + (b -a) 2 m sup x∈R µ([x, x + δ]) δ . (2.3)
This follows from the easily established inequality

R |Ξ(x -a)| 2m |Ξ(x -b)| 2m dx ≤ √ π2 2m+1 Γ(m -1/2) Γ(m) 1 1 + (b -a) 2 m .
3. Oversampling and the Large Sieve in model spaces 3.1. Background on model spaces. The Hardy space on the upper-half plane

C + := {z ∈ C : Im z > 0}, H 2 = H 2 (C + ) contains all holomorphic functions on C + for which sup y>0 R |f (x + iy)| 2 dx < ∞.
Every function f ∈ H 2 has an almost everywhere defined "vertical" boundary function f (x) := lim y→0 f (x + iy), and f ∈ L 2 (R) which may be used to define an inner product on

H 2 f, g := R f (x)g(x) dx.
We say that an analytic function Θ on C + is inner if |Θ| ≤ 1 on C + and if the almost everywhere defined boundary function Θ(x), x ∈ R, has modulus one. If Θ is an inner function, then the corresponding model space is defined by

K 2 Θ := H 2 ΘH 2 = (ΘH 2 ) ⊥ . (3.1)
Recall that the reproducing kernel for functions K 2 Θ is defined by

k Θ z (w) = i 2π 1 -Θ(z)Θ(w) w -z , z, w ∈ C + . (3.2) For every f ∈ K 2 Θ , f (z) = f, k Θ z , and
k Θ z (z) = k Θ z 2 = 1 -|Θ(z)| 2 4πIm z .
Each inner function can be factored as

Θ(z) = e iτ Θ c (z)B λ (z)S µ (z), z ∈ C + ,
where τ is a real constant, Θ c (z) = e icz , c ≥ 0,

B Λ (z) = λ∈Λ e iα λ z -λ z -λ ,
is the Blaschke product with zeros λ ∈ Λ ⊂ C + , repeated according to multiplicity, satisfying the Blaschke condition

λ∈Λ Im λ 1 + |λ| 2 < ∞, (3.3)
and

α λ ∈ R, S µ (z) = exp i R 1 x -z - t x 2 + 1 dµ(x) ,
where µ is a singular measure with respect to the Lebesgue measure such that

R dµ(x) 1 + x 2 < ∞. The spectrum of Θ is the closed set ρ(Θ) := ζ ∈ C + ∪ ∞ : lim inf z→ζ z∈C + |Θ(z)| = 0 . (3.4)
Note that Θ, along with every function in K 2 Θ , has an analytic extension across any interval of R \ ρ(Θ).

By the Ahern-Clark theorem [START_REF] Ahern | Radial limits and invariant subspaces[END_REF], k Θ x ∈ K 2 Θ , x ∈ R, if and only if the modulus of the angular derivative of Θ is finite. This means that

|Θ (x)| = a + 2 λ∈Λ Im λ |x -λ| 2 + R dµ(t) (t -x) 2 < ∞.
For any α ∈ C , |α| = 1 the function (α + Θ)/(α -Θ) has a positive real part in the upper half plane, and hence, by the Herglotz-Riesz representation theorem, there exist c α > 0 and a non-negative measure σ α Θ , called the Clark measure, such that

Re α + Θ(z) α -Θ(z) = c α Im z + Im z π R dσ α Θ (x) |x -z| 2 , z ∈ C + .
The Clark measure σ α Θ is carried by the set {x ∈ R : lim y→0+ Θ(x + iy) = α}. Clark [START_REF] Clark | One-dimensional perturbations of restricted shifts[END_REF] showed that if c α = 0 and σ α Θ is purely atomic, that is

σ α Θ = a n δ xn ,
where δ x denotes the Dirac measure at the point x, then the system {k Θ xn } is an orthogonal basis in K 2 Θ . In particular, one has

f (z) = n∈Z f (x n ) k Θ xn (z) k Θ xn 2 , f ∈ K 2 Θ , (3.5)
and

f 2 = n∈Z |f (x n )| 2 k Θ xn 2 . (3.6)
We end this section with a discussion of a special class of inner functions which will be considered throughout the paper (with exception of Section 6). A meromorphic inner function on C + is an inner function on C + with a meromorphic continuation to C. Any meromorphic inner function Θ can be represented as

Θ = Θ c B Λ ,
where c ≥ 0 and B Λ is the Blaschke product associated with the zeroes Λ = {λ n } of Θ which satisfy |λ n | → ∞ as well as the Blaschke condition (3.3). All elements of the corresponding model space K 2 Θ are also meromorphic in C, and there is a canonical isomorphism of such model spaces with de Branges' Hilbert spaces of entire functions [START_REF] De Branges | Hilbert Spaces of Entire Functions[END_REF].

By the Riesz-Smirnov factorization there exists an increasing, real analytic function ϕ : R → R such that Θ(x) = exp(iϕ(x)).

In that case |Θ | = ϕ and

k Θ x 2 = ϕ (x) 2π . (3.7)
Then, by the Cauchy-Schwarz inequality one has

|k Θ x (t)| = | k Θ x , k Θ t | ≤ ϕ (x)ϕ (t) 2π , x, t ∈ R. (3.8)
In the case of meromorphic inner functions the Clark measure construction becomes much more transparent. In this setting such measures were introduced by de Branges (see, e.g., [START_REF] De Branges | Hilbert Spaces of Entire Functions[END_REF]) long before the work of Clark. For model spaces associated with meromorphic inner functions orthogonal bases of reproducing kernels can be constructed as follows. For γ ∈ [0, 2π) define the set of points {x n } n∈Z by

ϕ(x n ) = γ + 2πn, n ∈ Z. (3.9)
(Note that the points x n may not exist for all n ∈ Z.) Then the family of normalized reproducing kernels {k Θ xn / k Θ xn } n∈Z , with the points {x n } n∈Z ∈ R given by (3.9), forms an orthonormal basis for K 2 Θ for each γ ∈ [0, 2π), except, maybe, one (in the case that Θ -e iγ ∈ L 2 (R) ).

In what follows we will consider the class of inner functions such that Θ ∈ H ∞ (C + ). This condition implies that Θ is meromorphic and is equivalent to Θ = ϕ ∈ L ∞ (R). It was noticed already in [START_REF] Dyakonov | Entire functions of exponential type and model subspaces in H p[END_REF] that the model spaces K 2 Θ with Θ ∈ L ∞ (R) have many properties analogous to the properties of the Paley-Wiener spaces P W 2 c (R).

3.2.

Enlarging model spaces and oversampling. We need the following classical result [START_REF] Nikolski | Treatise on the Shift Operator[END_REF]. However, for the sake of completeness, we give the complete proof here.

Lemma 3.1. Let Θ be an inner function, let

Θ 1 = e iτ 1 Θ c 1 B Λ 1 , and Θ 2 = e iτ 2 Θ c 2 B Λ 2 where Λ 1 , Λ 2 ⊂ C + , i = 1, 2, are two Blaschke sequences. Then (1) K 2 Θ 1 ⊆ K 2 Θ 2 if and only if c 1 ≤ c 2 and Λ 1 ⊆ Λ 2 . (2) K 2 Θ ⊆ K 2 ΘcB Λ Θ
, where c ≥ 0 and Λ ⊂ C + a Blaschke sequence.

(3) If 0 ≤ c 1 ≤ c 2 , Λ 1 ⊆ Λ 2 , and f ∈ K 2 Θ , then Θ c 1 B Λ 1 f ∈ K 2 Θc 2 B Λ 2 Θ . Proof. Let Θ 1 , Θ 2 be inner functions. It is known that K 2 Θ 1 ⊆ K 2 Θ 2 if and only if Θ 2 /Θ 1 is an inner function. Hence, Θ 2 /Θ 1 is inner if and only if Θ 2 (z) Θ 1 (z) = e i(τ 2 -τ 1 ) e i(c 2 -c 1 )z B Λ 2 (z) B Λ 1 (z) = Θ(z), for some Θ inner. If c 2 ≥ c 1 and Λ 1 ⊂ Λ 2 , then B Λ 2 /B Λ 1 = B Λ 2 \Λ
1 is a Blaschke product and Θ is therefore inner. Now assume that Θ 2 /Θ 1 is inner and that there exists λ * ∈ Λ 1 that is not contained in Λ 2 (in the sense that for Λ i = {λ i n } n∈N , i ∈ {1, 2}, there is no injection ϕ : N → N that satisfies

λ 1 n = λ 2 ϕ(n) for every n ∈ N), then #{n ∈ N : λ * = λ 2 n } < #{n ∈ N : λ * = λ 1 n }
, which creates a pole at λ * for Θ 2 /Θ 1 , as the singular inner part is always nonzero, a contradiction to Θ 2 /Θ 1 being inner. Then, as e i(c 2 -c 1 )z is bounded on C + if and only if c 2 ≥ c 1 , the first assertion follows. It only remains to show the last statement. By definition f ∈ K 2 Θ if and only if f ⊥Θg for every g ∈ H 2 . Hence, as Θ c 1 B Λ 1 is an inner function we have

Θ c 1 B Λ 1 f, S c 1 B Λ 1 Θg = R Θ c 1 (x)B Λ 1 (x)f (x)Θ c 1 (x)B Λ 1 (x)Θ(x)g(x) dx = R f (x)Θ(x)g(x) dx = 0, which shows Θ c 1 B Λ 1 f ∈ K 2 Θc 1 B Λ 1 Θ
. The result then follows from the first part of the corollary as

Θ c 2 B Λ 2 Θ = (Θ c 2 -c 1 B Λ 2 \Λ 1 )(Θ c 1 B Λ 1 Θ).
We are now ready to prove an oversampling theorem for model spaces.

Theorem 3.2. Let c > 0, and Θ be an inner function such that some Clark measure for Θ c Θ is purely atomic and let {k ΘcΘ xn } n∈Z be the corresponding basis of K 2 ΘcΘ . Then for every integer m ≥ 1 and for every f ∈ K 2 Θ , the following sampling formula holds:

f (x) = n∈Z f (x n )e -ic(x-xn) 2 sinc c(x -x n ) 2m m k ΘcΘ xn (x) k ΘcΘ xn 2 , x ∈ R. (3.10)
In particular, one can reconstruct f ∈ K 2 Θ from its samples using an expansion with respect to functions of arbitrary polynomial decay.

Further,

f 2 = n∈Z |f (x n )| 2 k ΘcΘ xn 2 , f ∈ K 2 Θ . (3.11) Proof. If 0 ≤ c k ≤ c/m, k = 1, . . . , m, then 0 ≤ k c k ≤ c, and Θ k c k f ∈ K 2
ΘcΘ by Lemma 3.1. Hence, using the sampling formula (3.5) for K 2 ΘcΘ , one has

e i k c k z f (z) = n∈Z f (x n )e i k c k xn k ΘcΘ xn (z) k ΘcΘ xn 2 , z ∈ C + ,
and (3.11) follows directly from (3.6). Further

f (z) = m c m c/m 0 . . . c/m 0 f (z) dc 1 • • • dc m = m c m c/m 0 . . . c/m 0 n∈Z f (x n )e -i k c k (z-xn) k ΘcΘ xn (z) k ΘcΘ xn 2 dc 1 • • • dc m = n∈Z f (x n ) m ic(z -x n ) m 1 -e -ic(z-xn) m m k ΘcΘ xn (z) k ΘcΘ xn 2 .
Moreover, we find that for almost every x ∈ R,

f (x) = n∈Z f (x n )e -ic(x-xn) 2 m ic(x -x n ) e ic(x-xn) 2m -e -ic(x-xn) 2m m k ΘcΘ xn (x) k ΘcΘ xn 2 = n∈Z f (x n )e -ic(x-xn) 2 sinc c(x -x n ) 2m m k ΘcΘ xn (x) k ΘcΘ xn 2 .
The oversampling formula is thus established.

Proof of Theorem 1.2 based on oversampling

We now have everything in place to prove Theorem 1.2. Let Θ ∈ L ∞ (R). Then (3.7) and (3.8) imply

|k Θ x (t)| k Θ x ≤ Θ ∞ 2π , x, t ∈ R. (4.1) Let k ΘcΘ z ∈ K 2
ΘcΘ be the reproducing kernel functions in K 2 ΘcΘ . Denote by {x n } n∈Z ⊂ R the sequence associated in (3.9) with Θ c Θ, that is,

cx n + ϕ(x n ) = γ + 2πn, γ ∈ [0, 2π), n ∈ Z.
For c > 0 to be chosen later, set α = c 4 and Ξ(x) = sinc(x). First, from Theorem 3.2 (applied to m = 2) Fubini's theorem, and the bound (4.1) we get

R |f (x)| 2 dµ(x) = R n∈Z f (x n )e -ic(x-xn) 2 sinc c(x -x n ) 4 2 k ΘcΘ xn (x) k ΘcΘ xn 2 2 dµ(x) ≤ R n,k∈Z |f (x n )f (x k )| k ΘcΘ xn k ΘcΘ x k Ξ α(x -x n ) Ξ α(x -x k ) 2 |k ΘcΘ xn (x)k ΘcΘ x k (x)| k Θ Θ xn k ΘcΘ x k dµ(x) ≤ c + Θ ∞ 2π n,k∈Z |f (x n )f (x k )| k ΘcΘ xn k ΘcΘ x k R Ξ α(x -x n ) 2 Ξ α(x -x k ) 2 dµ(x).
Applying the estimate (2.2) with ηα (instead of δ), η to be chosen, we get

R |f (x)| 2 dµ(x) ≤ 4 C 2 ηα ηα (c + Θ ∞ ) n,k∈Z |f (x n )f (x k )| k ΘcΘ xn k ΘcΘ x k sup x∈R µ([x, x + η]) 4 + α 2 (x n -x k ) 2 (4.2)
with C ηα = max(4, 1 + 9η 2 α 2 ). We will choose η ≤ δ with ηα ≤ 1/ √ 3, i.e., η ≤ 4 √ 3c , so that C ηα = 4. Now, by (4.1),

2π|n -k| = |cx n + ϕ(x n ) -cx k -ϕ(x k )| ≤ (c + ϕ ∞ )|x n -x k | = (c + Θ ∞ )|x n -x k |. Set u = (u k ) k with u k = |f (x k )| k ΘcΘ x k so that u 2 2 (Z) = f 2 with (3.11). Moreover, we set λ = π 2 α 2 (c + Θ ∞ ) 2 = πc 4(c + Θ ∞ ) 2 , v = (v k ) k with v k = 1 1+λk 2 and notice that v 1 (Z) = 1 + 2 k≥1 1 1 + λk 2 ≤ 1 + 2 ∞ 0 dx 1 + λx 2 = 1 + π √ λ = 1 + 4(c + Θ ∞ ) c .
Then

n,k∈Z |f (x n )f (x k )| k ΘcΘ xn k ΘcΘ x k 1 4 + α 2 (x n -x k ) 2 ≤ 1 4 n∈Z u n k∈Z u k 1 + λ(n -k) 2 = 1 4 u, u * v ≤ 1 4 u 2 2 (Z) v 1 (Z) .
All in one, we get

R |f (x)| 2 dµ(x) ≤ 16 ηα M (η) c + Θ ∞ 1 + 4 c + Θ ∞ c f 2 ,
where M (η) := sup x∈R µ([x, x + η]). It remains to chose c, η. To do so, we distinguish two cases.

Case 1. If Θ ∞ ≥ 4 √ 3δ , we take c = Θ ∞ and η = 4 √ 3c ≤ δ thus ηα = 1 √ 3 . Then R |f (x)| 2 dµ(x) ≤ 288 √ 3 M (η) Θ ∞ f 2 ≤ 500 1 δ + Θ ∞ M (δ) f 2 . Case 2. If Θ ∞ ≤ 4 √ 3δ , we take c = 4 √ 3δ and η = 4 √ 3c = δ so that R |f (x)| 2 dµ(x) ≤ 576 1 δ + Θ ∞ M (δ) f 2 ,
as desired.

Remark 4.1. In the proof, we have used the oversampling formula with m = 2. One may of course increase m and this would lead to slightly better numerical constants, at the price of much higher technicality. This requires in particular to replace Lemma 2.1 with (2.3).

As the ultimate constants would anyway be worse than those we find in the next section, we did not follow this path further.

5. An L p -version of Theorem 1.2 and Bernstein-type inequalities

The previous proof gives a rather large constant C in the inequality (1.7). We present another proof that gives a better estimate; in particular, we have the constant 1 in front of 1 δ , as in the classical Donoho-Logan theorem. Also, this proof applies to L p analogs of the model spaces, the subspaces K p Θ of the Hardy space H p , where 1 ≤ p < ∞. Recall that for 1 ≤ p ≤ ∞ the subspace K p Θ is defined as

K p Θ = H p ∩ ΘH p
, where H p is understood as a closed subspace of L p (R). This definition agrees with the one given for K 2 Θ earlier. The properties of the spaces K p Θ are very much similar to the properties of K 2 Θ . In particular, if Θ is a meromorphic inner function, then all elements of K p Θ are meromorphic functions in C, while for Θ(z) = e 2icz we have K p Θ = e icz P W p c (R), where P W p c (R) is the space of all entire functions of exponential type at most c whose restriction to R belongs to L p (R).

Theorem 5.1. Let 1 ≤ p < ∞, Θ ∈ L ∞ (R), µ be a Borel measure on R, and let δ > 0 and D µ (δ) defined in (1.4). Then for any f ∈ K p Θ we have

f p L p (µ) ≤ 1 + δ Θ ∞ p D µ (δ) f p p . (5.1)
The proof is based on a Bernstein-type inequality for model spaces K p Θ with Θ ∈ L ∞ (R) which is due to Dyakonov [START_REF] Dyakonov | Entire functions of exponential type and model subspaces in H p[END_REF][START_REF] Dyakonov | Differentiation in star-invariant subspaces. I: Boundedness and compactness[END_REF].

Theorem 5.2 (Dyakonov). If Θ ∈ L ∞ (R), 1 ≤ p ≤ ∞, and f ∈ K p Θ , then f p ≤ Θ ∞ f p .
Dyakonov proved this inequality up to some constant C(p) on the right-hand side. Here we present a very simple proof with the constant 1 which follows the method from [START_REF] Baranov | Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings[END_REF] and is based on the formula

f (x) = 2πi R f (t) k Θ t (x) 2 dt, f ∈ K p Θ , x ∈ R. (5.2)
Proof of Dyakonov's theorem. For 1 < p < ∞, it follows from (5.2), (3.7) and the Hölder inequality that

|f (x)| p ≤ 2π R |f (t)| p |k Θ x (t)| 2 dt 2π |k Θ x (t)| 2 dt p/q = 2π|Θ (x)| p/q R |f (t)| p |k Θ x (t)| 2 dt ≤ 2π Θ p/q ∞ R |f (t)| p |k Θ x (t)| 2 dt. Since R |k Θ x (t)| 2 dx = |Θ (t)|/(2π)
, we get the result integrating over x. The cases p = 1 or p = ∞ follow trivially from (5.2).

We will also need the following elementary lemma.

Lemma 5.3. Let f, f ∈ L p (R), 1 ≤ p < ∞. Then for any t k ∈ [kδ, (k + 1)δ] =: I k , k ∈ Z, one has k∈Z |f (t k )| p 1/p ≤ δ -1/p f p + δ 1-1/p f p .
(5.3)

Proof. From the triangle inequality,

k∈Z |f (t k )| p 1/p = 1 δ k∈Z I k |f (t k )| p dt 1/p ≤ 1 δ k∈Z I k |f (t)| p dt 1/p + 1 δ k∈Z I k |f (t) -f (t k )| p dt 1/p .
Note that by Hölder's inequality

I k |f (t k ) -f (t)| p dt = I k t k t f (s) ds p dt ≤ δ p/q I k t k t |f (s)| p ds dt ≤ δ p I k |f (s)| p ds.
Summing these estimates we obtain (5.3).

Proof of Theorem 5.1.

Put M (δ) = sup x∈R µ [x, x + δ) = δD µ (δ). Let t k ∈ I k be such that |f (t k )| = sup I k |f |. Then R |f (x)| p dµ(x) ≤ k∈Z µ [kδ, (k + 1)δ) • |f (t k )| p ≤ M (δ) δ -1/p f p + δ 1-1/p f p p .
It follows from Dyakonov's theorem that

R |f (x)| p dµ(x) ≤ M (δ) δ (1 + δ Θ ∞ ) p f p p as claimed.
Remark 5.4. The dependence on the parameter δ and on Θ ∞ in (5.1) is sharp up to the numerical constant. Indeed, for the classical Paley-Wiener space P W 2 c (R), our method (applied to P W 2 c (R) in place of K 2 Θ with Θ(z) = e 2icz ) gives only a factor (1 + 2δc), while the Donoho-Logan bound is 1 + cδ π .

If µ is given by dµ(x) = 1 T (x)dx and one considers the Paley-Wiener spaces, it is common to choose δ to be the reciprocal of the bandwidth 2c and write the estimates in terms of the maximum Nyquist density

D max (T, c) := sup x∈R 2c • T ∩ [x, x + 1/(2c)] .

Donoho and Logan's results then read s

f • 1 T p p ≤ C p • D max (T, c) • f p p , for p = 1, 2, with constants C 1 = sinc(1/4) -1 ≈ 1.

A Donoho-Logan theorem for one component inner functions

For general inner functions, even if we confine ourselves with meromorphic ones, one cannot expect to obtain estimates of Donoho-Logan type considering intervals of a fixed length. The reason is that the behaviour of inner functions on R can be very irregular. On the other hand, for meromorphic inner functions with a sublinear growth of the argument the assumption that the measure is uniformly bounded on intervals of a fixed length can be too strong. It looks more natural then to consider the intervals where the change of the argument of Θ is fixed, i.e., consider the intervals I = [a, b] such that ϕ(b) -ϕ(a) = δ, where ϕ is an increasing continuous argument for Θ on R. There exists a class of inner functions for which such a generalization is possible. These are the so-called one-component inner functions, that is, those inner functions for which the sublevel set

Ω(Θ, ε) = {z ∈ C + : |Θ(z)| < ε}
is connected for some ε ∈ (0, 1). One-component inner functions were introduced by W. S. Cohn [START_REF] Cohn | Carleson measures for functions orthogonal to invariant subspaces[END_REF] in connection to Carleson-type embeddings of model spaces and were subsequently studied by many authors (see, e.g., [START_REF] Aleksandrov | Embedding theorems for coinvariant subspaces of the shift operator[END_REF][START_REF] Aleksandrov | On embedding theorems for coinvariant subspaces of the shift operator[END_REF][START_REF] Baranov | Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings[END_REF][START_REF] Baranov | Embeddings of model subspaces of the Hardy class: compactness and Schatten-von Neumann ideals[END_REF][START_REF] Nicolau | A characterization of one-component inner functions[END_REF]).

Several important properties of one-component inner functions were obtained by A. B. Aleksandrov [START_REF] Aleksandrov | Embedding theorems for coinvariant subspaces of the shift operator[END_REF]. Recall that the spectrum ρ(Θ) of Θ (see (3.4)) is a closed set, and if we write R \ ρ(Θ) = ∪ n J n , where J n are disjoint open intervals, then Θ admits an analytic continuation through any J n . Hence, Θ has an increasing C ∞ branch of the argument ϕ on each J n . It is shown in [START_REF] Aleksandrov | Embedding theorems for coinvariant subspaces of the shift operator[END_REF] that ρ(Θ) has zero Lebesgue measure and, if we additionally assume that ∞ ∈ ρ(Θ), the function

1 ϕ = 1 |Θ |
(defined as 0 on ρ(Θ)) is a Lipschitz function on R. Moreover, it is shown in [START_REF] Baranov | Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings[END_REF] that in this case for any ε ∈ (0, 1) there exist positive constants c 1 , c 2 depending on ε only such that These properties essentially characterize the class of one-component inner functions.

c 1 |Θ (x)| -1 ≤ dist (x, Ω(Θ, ε)) ≤ c 2 |Θ (x)| -1 , x ∈ R. ( 6
Also in [START_REF] Baranov | Bernstein-type inequalities for shift-coinvariant subspaces and their applications to Carleson embeddings[END_REF] the following Bernstein type inequality was proved: if Θ is a one-component inner function and 1 < p < ∞, then f /Θ p ≤ C(Θ, p) f p , f ∈ K p Θ . (6.

3)

The next theorem applies to general one-component (not necessarily meromorphic) functions.

  ) where b ≥ c and γ is any even function supported in [-b, b] with γ(ξ) = 1 for ξ ∈ [-c, c]. Taking b ≥ c and γ = 1 [-b,b] we get the classical Shannon Formula

  R ψ (N ) = 1. It follows that γ = ψ (N ) * 1 [-c-N a,c+N a] is even, supported in [-b, b] with b = c + 2N a and that γ(ξ) = R ψ (N ) = 1 for ξ ∈ [-c, c]. Computing γ, we get f (x) = 2(c + N a)

  0105 and C 2 = (1 + 1/2π) ≈ 1.1592, while our estimate (5.1) gives C 1 = 2 and C 2 = 4.

Theorem 6 . 1 .

 61 Let Θ be a one-component inner function such that ∞ ∈ ρ(Θ), 1 < p < ∞. Let ϕ denote the branch of the argument of Θ which is an increasing C ∞ function on each subinterval of R \ ρ(Θ). For a Borel measure µ on R and δ > 0 putD Θ µ (δ) = sup µ(I) |I| : I = [a, b], ϕ(b) -ϕ(a) = δ .Then for any f ∈ K p Θ we haveR |f | p dµ ≤ D Θ µ (δ)(1 + Cδ) p f p p ,(6.4)where the constant C > 0 depends on Θ and p only.Proof. The proof is analogous to the proof of Theorem 5.1. It follows from (6.1) that ϕ is unbounded on any connected component of R \ ρ(Θ). Therefore, we can writeR \ ρ(Θ) = ∪ k I k , where I k = [a k , b k ] are intervals with disjoint interiors such that ϕ(b k ) -ϕ(a k ) = δ. Let D = D Θ µ (δ). Any function f ∈ K p Θ admits an analytic continuation through R \ ρ(Θ). Choose t k ∈ I k such that |f (t k )| = max I k |f |. Then we have f L p (µ) ≤ k |f (t k )| p µ(I k )

  For any k there existss k ∈ I k such that δ = ϕ(b k ) -ϕ(a k ) = ϕ (s k ) • |I k |. In view of (6.2) we have |Θ (s)| • |I k | ≤ C 1 δ, s ∈ I k , whence |Θ (s)| p ds dt ≤ C 3 δ p f p p by(6.3). We have shown thatf L p (µ) ≤ D 1/p (1 + C 4 δ) f p ,which proves the theorem.

									1/p			1/p
			≤ M 1/p			|f (t)| p dt	+ D 1/p	|f (t) -f (t k )| p dt	.
					k	I k						k	I k
	Furthermore, if p is the dual index to p,	1 p	+	1 p	= 1, then
											t	p
											f (s) ds	dt
		k	I k					k	I k	t k
							≤	k	I k	t t k	|f (s)| p |Θ (s)| p ds	t t k	|Θ (s)| p ds	p/p	dt.
				t t k	|Θ (s)| q ds	p/p	≤ C 2		δ p |I k | p -1	p/p	= C 2	δ p |I k |	.
	for some constants C 1 , C 2 > 0. Thus,
	k	I k	t t k	|f (s)| p |Θ (s)| p ds	t t k	|Θ (s)| p ds	p/p	dt ≤ C 2	|I k | δ p	k	I k I k	|f (s)| p

|f (t) -f (t k )| p dt =

Acknowledgements. M.S. would like to acknowledge the support of the Austrian Science Fund FWF through the projects J-4254, and Y-1199.

The work of A.B. was supported by the Russian Science Foundation project 19-71-30002. The research of K.K. is supported by the project ANR-18-CE40-0035. Ce travail a bénéficié d'une aide de l' État attribué à l'Université de Bordeaux en tant qu'Initiative d'excellence, au titre du plan France 2030.


T Dairay

E Lamballais
email: eric.lamballais@univ-poitiers.fr

S Laizet
email: s.laizet@imperial.ac.uk

J C Vassilicos
email: j.c.vassilicos@imperial.ac.uk

Physical scaling of numerical dissipation for LES

Introduction

In this work, we are interested in an alternative way to perform LES using a numerical substitute of a subgrid-scale model with a calibration based on physical inputs. This approach can be seen as falling within Implicit LES (ILES) because it is based on an artificial dissipation operator where the numerical error is the source of the regularization. However, the purpose here is to control explicitly this error to enable it to play the role of an explicit subgrid-scale model.

The most popular ILES are based on differentiation operators that introduce artificial dissipation in order to mimic the dissipative mechanisms of the subgrid-scales. For this ILES family, the term "implicit" can be understood in the sense that the actual equations to be solved are the Navier-Stokes equations without any extra explicit modelling as in a DNS. In practice, the numerical dissipation provided by some specific schemes (typically upwind schemes) can act at small-scales in a manner similar to an explicit subgrid-scale model. This favourable property, sometimes referred to as the "convenient conspiracy", does not hold systematically because a scheme that provides an acceptable artificial dissipation at one spatial resolution can become sub-or over-dissipative at another resolution. That is why some authors have developed different versions of their schemes depending on their ILES resolution. The goal of the present study is to examine how this weakness can be overcome through flexible and accurate numerics with physical inputs.

General methodology and goal

Here, we present an ILES modelling where the artificial dissipation is introduced by the molecular dissipation operator, namely the viscous term [5]. As for standard ILES, the source of artificial dissipation is the numerical error. However, an originality of the approach is that it is based on high-order centred finite-difference schemes for the computation of the second derivatives in the equations. Importantly, these schemes are implicit in space (they are often referred to as "compact", see [6] for a comprehensive presentation) thereby enabling flexibility for the control of their artificial dissipative features. It can be easily shown that the use of a finite difference scheme with a modified square wavenumber k ′′ that overestimates its exact counterpart k 2 is equivalent to the introduction of a spectral viscosity ν s (k) given by ν s

(k) = ν k ′′ (k)-k 2 k 2
where ν is the kinematic molecular viscosity. The flexibility to shape k ′′ makes it possible to prescribe any value ν 0 to the spectral viscosity at the cutoff wavenumber k c = π/∆ x while at the same time controlling the kernel ν s (k)/ν 0 (∆ x is the mesh size). For instance, [5] have used this flexibility to mimic the Spectral Vanishing Viscosity (SVV) or hyperviscous kernels while ensuring high-order accuracy.

The first calculations based on this approach have been successful for DNS as well as for LES. For the former, a moderate value of ν 0 with a very selective kernel concentrated near k c is helpful to control spurious oscillations due to aliasing. For the latter, after the successful calculation of a turbulent channel flow by [5], [2] have shown that this kind of implicit subgrid-modelling drastically improves the prediction of heat transfer for an impinging jet flow compared to eddy viscosity subgrid-scale closures such as the dynamic Smagorinsky or the WALE models.

Even if these first LES results are encouraging, the flexibility in the prescription of ν s (k) can also be seen as a drawback because of the arbitrariness introduced by the need to chose both ν 0 and the kernel ν s (k)/ν 0 appropriately case by case. The aim of this study is therefore to provide ways to constrain this choice by appropriate physical scaling of the numerical dissipation. To predict the value of ν 0 required for a given LES spatial resolution, we propose an approach based on the solution of a Pao-like equation. The method is validated on DNS and LES of the 3D Taylor-Green vortex problem.

Pao-like solution

In the framework of homogeneous and isotropic turbulence, and assuming that a spectral range exists where the energy spectrum is stationary 1 , the Lin equation leads to 1 Note, however, that evidence has been accumulating over the past 8 years which shows that there is no significant such range except perhaps in the directly viscous dissipation range, see [3] and references therein, but the present work is only a starting point which will evolve accordingly

T (k) = 2νk 2 E(k) (1)
where E(k) is the kinetic energy spectrum whereas T (k) is the transfer term related to the energy flux Π (k) by -dΠ (k)/dk = T (k). As in the Kolmogorov theory, this flux can be assumed to depend only on the dissipation ε and the wavenumber k.

Then, for dimensional reasons, equation (1) becomes

d dk 1/C k ε 1/3 k 5/3 E(k) + 2νk 2 E(k) = 0 (2)
with the analytical solution

E(k) = C k ε 2/3 k -5/3 exp -3/2C k (k/k η ) 4/3 ( 3
)
where k η = ε 1/4 /ν 3/4 is the Kolmogorov wavenumber (see [7] for more details).

As a tool to predict the spectrum that should be obtained by the present type of LES, a similar approach can be used by replacing k 2 on the right hand side of equation ( 2) with k ′′ . Then, the resulting differential equation can be solved numerically in a spectral range between a wavenumber k s (below which ν s (k) is assumed to vanish) and infinity. This computationally inexpensive solution can be done for any value of ν 0 leading to a set of Pao-like solutions. To illustrate the behaviour of such Pao-like solutions, the case k η /k c = 5 is considered with a SVV-like kernel (plotted in figure 2-left). First, it is interesting to investigate how the increase of ν 0 /ν modifies the kinetic energy spectrum and its related dissipation spectrum (including the numerical contribution). Six kinetic energy spectra are presented in figure 1-left. The reference spectrum corresponds to the case ν 0 /ν = 0 for which the Pao spectrum (3) is virtually recovered. Then, ν 0 /ν is increased by steps of 8 up to the value 48. As ν 0 /ν is increased, the overall range of the predicted spectrum is reduced. For the highest values of ν 0 /ν, this reduction corresponds to a strong fall at the cutoff wavenumber k c = 8k s considered for this ex-ample. The resulting low levels of E(k c ) suggest that the corresponding LES could be done using a mesh such as ∆ x = π/k c . For instance, to have the same kinetic energy at k c as at k η for the reference DNS (for which ν 0 /ν = 0), a simple dichotomic search provides the value ν 0 /ν = 42.37. Using the Pao-like solver, this dichotomic search ensuring E LES (k c ) = E DNS (k η ) can be performed extensively for any ratio k η /k c . Figure 2-right presents the resulting behaviour for the range 1 ≤ k η /k c ≤ 30. As expected, the Sharp SVV kernel (plotted in figure 2-left) requires to use significantly higher values of ν 0 /ν (see figure 2-right). The corresponding dissipation spectra (figure 1-right) clearly show how the use of high-order numerical viscosity can concentrate an extra-dissipation near k c in order to compensate the lack of dissipation for k > k c (which could not be taken into account using a discretization based on mesh size with ∆ x = π/k c ).

✠ ν 0 ν ր ✠ ν 0 ν ր k/k s k/k s E(k) k ′′ E(k) k c k η k c k η

DNS and LES results

The suitability of the present prediction of ν 0 /ν is examined by performing LES of the 3D Taylor-Green vortex problem while using filtered DNS results as reference for comparison (case "DNS f "). Both DNS and LES are carried out using the highorder flow solver "Incompact3d" which is based on sixth-order finite-difference schemes [4]. For the sake of clarity, only the Reynolds number Re = 5000 is presented here. For this fairly high value, accurate DNS results can be obtained with a regular Cartesian mesh of 1280 3 mesh odes on a triperiodic domain 2π 3 . For the present calculations, using the symmetries of the problem, only the flow inside the impermeable box π 3 is explicitly calculated with a reduction of the computational cost by a factor 8. Then, comparable LES are performed using 160 3 mesh nodes with k η /k c = 8 which represents a reduction of the computational cost by a factor of 4096 by comparison with DNS. The Pao-like solver provides the values ν 0 /ν ≈ 95 and 232 for the SVV-like and Sharp SVV kernels respectively (see figure 2-right).

When these values are used in the present type of LES, satisfactory results are obtained with a clear improvement compared to a calculation performed at the same spatial resolution but free from subgrid-scale modelling and artificial dissipation. This improvement can be illustrated by considering the time evolution of the kinetic energy E k and its dissipation ε = -dE k /dt (that includes the numerical contribution) as presented in figure 3 (left). The filtered DNS data are obtained after the application of the filter T f (k) = E LES (k)/E DNS (k) in the Fourier space where E DNS (k) is the Pao kinetic energy spectrum whereas E LES (k) is its Pao-like counterpart for a given set of ν 0 and kernel ν s (k)/ν 0 . Here, for the sake of brevity, only the Sharp SVV kernel is considered with ν 0 /ν = 232.

The calculation free from extra dissipation, referred to as "LES no model" case, means that a conventional compact finite difference scheme is used to compute the viscous term. Interestingly, although this configuration is a priori sub-dissipative, an over-dissipative behaviour seems to be observed a posteriori throughout the calculation as shown in figure 3 (left). A close examination of the time evolution of E k and ε reveals that in fact, the "LES no model" case is very slightly sub-dissipative just a the beginning of the simulation. This lack of dissipation is responsible of a pile-up of energy near the cutoff wave number k c at the early transition. As a direct consequence of this pile-up, the dissipation is overestimated despite the nature of the numerical operator that is essentially sub-dissipative. At this coarse resolution, this erroneous dissipation eventually overdamps E k for the rest of the calculation without being able to suppress the spurious energy at small scale. This behaviour is a very simple illustration of the contradictory conclusions that can be provided by a priori and a posteriori analysis of the influence of any subgrid-scale model. Then, when the second derivatives of the Navier-Stokes equations are computed using an over-dissipative scheme with the level of numerical viscosity predicted by the Pao-like solver ("LES ν 0 /ν = 232" case), the agreement with the filtered DNS is clearly improved as shown in figure 3 (left). The a priori/a posteriori paradox is recovered in reverse, namely the extra dissipation is found to slow down the time decrease of E k by comparison with the "LES no model" case while matching remarkably well the "DNS f " data. This matching suggests that the artificial dissipation provided by the choice ν 0 /ν = 232 is able to compensate consistently the small-scale dissipation that is missed by the coarse discretization.

To better understand the role played by the artificial viscosity, it is insightful to consider the time evolution of the kinetic energy spectrum E(k,t). For the "LES no model", from the start of the early transition until the end of the calculation, an unrealistic pile-up of energy can be observed in the vicinity of the cutoff wave number k c . This production of small-scale oscillations leads to very unrealistic vortical structures subjected to a spurious background noise in instantaneous visualisations of, for instance, the enstrophy and the Q-criterion (not shown for reasons of space). Conversely, the use of a calibrated numerical viscosity for the "LES ν 0 /ν = 232" case enables a strong decrease of E(k,t) at k ≈ k c throughout the simulation. These behaviours are illustrated in figure 3 (right) for spectra obtained at t = 14, a time for which the turbulence is fully developed. The spectral range where the artificial dissipation is active can be easily guessed by comparison to non-filtered DNS results. The resulting damping of the small-scale energy ensures the regularity of the LES solution (free from spurious oscillations) as for its numerically converged DNS counterpart computed at high resolution. This feature enables the development of realistic vortices that compare very well, at least qualitatively, to those associated with the filtered DNS results (not shown for reasons of space).

It can be noticed that the damping of the kinetic energy near k c is stronger for the "LES ν 0 /ν = 232" case compared to the "DNS f " data. This discrepancy means that the Pao-like solver cannot predict exactly the full dissipation as it is actually computed in the LES. The assumption of turbulence in equilibrium to simplify the Lin equation can be suspected to be responsible of this inability. Another reason could be connected to the discretization errors in the convective terms that are not taken into account. The improvement of the present Pao-like model is in progress, in particular by taking into account the pervasive non-equilibrium nature of the turbulence. The potential benefit of non-vanishing spectral viscosity as in the subgrid-scale model of [1] is also under investigation.

Fig. 1

 1 Fig. 1 Kinetic energy E(k) and dissipation k ′′ E(k) spectra obtained by solving the Pao-like equation from ν 0 /ν = 0 (reference Pao solution) to ν 0 /ν = 48 by steps of 8.

Fig. 2

 2 Fig. 2 Left: SVV-like and Sharp SVV kernels. Right: evolution of ν 0 /ν with k η /k c to ensure E LES (k c ) = E DNS (k η ) as predicted by the Pao-like solution.

3 Fig. 3

 33 Fig. 3 Left: time evolution of the kinetic energy E k and its dissipation ε. Right: 1D spectra E(k,t) at t = 24.

Acknowledgements This work was performed using HPC resources from GENCI-TGCC/IDRIS (Grant 2014-2a0912).


Pierre-Louis Regazzoni
email: pierre-louis.regazzoni@etu.univ-nantes.fr

Gregory Hanson
email: greg.hanson@ars.usda.gov

Tony Wahl
email: twahl@do.usbr.gov

Didier Marot
email: didier.marot@univ-nantes.fr

Jean-Robert Courivaud
email: jean-robert.courivaud@edf.fr

Jean-Jacques Fry
email: jean-jacques.fry@edf.fr

THE INFLUENCE OF SOME ENGINEERING PARAMETERS ON THE EROSION OF SOILS

Keywords: Erodibility, Jet Erosion Test, Saturation, Water Content, Compaction

An experimental investigation is carried out concerning the effects of engineering parameters on fine soil erosion. The chosen parameters characterize the density and the saturation of the soil. The influence on erosion resistance of energy, density, water content at compaction, and presence of a saturation stage or not are investigated. The soil erosion behavior is evaluated with a Jet Erosion Test device. The interpretation of the performed data is made according to a linear relationship between an excess hydraulic shear stress and the rate of erosion. The effects of the compaction and of the saturation are indicated by the observed variation of the erosion law parameters. The results underline the effects of the soil fabric and the saturation on the soil erodibility.

INTRODUCTION

From the literature concerning soil erosion, water content is one of the key factors affecting erosion behavior, as is the compaction (Hanson and Hunt [START_REF] Hanson | Lessons learned using laboratory JET method to measure soil erodibility of compacted soils[END_REF] ). The objective of this study is to test the influence of these two engineering parameters along with the influence of saturation on the erosion behavior. The coupled water content and energy at the time of compaction defines the primary state of the soil, and the saturation history after compaction defines an altered state for the soil.

A protocol is defined for preparing and testing soil specimens to study the variation of erodibility as a function of the compaction conditions and saturation history. The test results are interpreted using a linear erosion law which represents the erodibility in terms of an erosion rate coefficient and the critical shear stress. The effect of the engineering parameters on the erodibility parameters is presented.

EXPERIMENTAL PROTOCOL (1) Preparation of the specimens

It involves three different operations:

• initial soil preparation;

• compaction;

• curing in one of two environments designed to produce either a saturated specimen or a specimen that can be tested at its original compaction moisture content.

Initially, the soil is air dried, sieved to eliminate material larger than the U.S. No. 4 sieve (A.S.T.M.), and then stockpiled in plastic buckets. Then, the following procedure is used to moisturize the material.

Initial water content is determined for the stockpiled soil. Additional water is added and mixed with the soil to reach a target water content. The soil is placed into a sealed plastic bag, which is then stored in a plastic container with a humidity source for at least 36 h.

Following the initial conditioning of the soil, compacted test specimens are prepared and saturated if desired. The compaction is made in a standard Proctor mould (101.6 mm diameter, 944 cm3, volume) with a standard Proctor plate. Two rammers are used: the "normal" (2.49-kg, 305 mm drop) and the "modified" (4.54-kg, 457 mm drop). The compaction is always made in 3 layers and 25 blows per layer. Water content is determined from uncompacted material. If saturation is required, it is produced in an upward direction under a constant hydraulic gradient of 10 m/m with a permeameter built for a Proctor mould. The specimen is confined to maintain a constant volume during the saturation process. The degree of saturation is evaluated by checking the specimen weight. Samples are kept in the saturation chamber for a minimum of 48 h.

Five different compaction water contents were targeted for each tested soil. After compaction, some specimens were saturated, while others were held at their compaction water content until erosion testing could be performed.

(

2) Description of the Jet Erosion Test

The apparatus used to evaluate erodibility is the submerged jet erosion test (JET) device (Hanson and Cook 2) ). It applies a water jet to a submerged soil surface and the scour depth beneath the jet is measured over time. The JET is composed of 3 parts as seen in figure 1. The erosive stress applied by the jet is adjusted by varying the pressure head applied to the nozzle, and the initial distance of the nozzle to the soil-water interface.

METHOD OF INTERPRETATION (1) The erosion law

Erodibility can be modelled with an erosive law whose parameters are determined from the experimental data (the scour depth [J] versus time). Stein and Nett 3) and Hanson and Cook [START_REF] Hanson | Apparatus, test procedures, and analytical methods to measure soil erodibility in-situ[END_REF] propose a linear relationship between the hydraulic excess shear stress τ-τ c and the rate of erosion ε :

( ) C d * k dt dJ τ - τ = ε = (1)
The erosion law is built on 2 parameters, an erosion coefficient k d [m3/(N s)],and a critical shear stress τ C [Pa].

(2) Description of the jet hydrodynamics Shear stresses applied during a test are estimated from an analysis of the jet hydrodynamics on the centerline. The water velocity U 0 [m/s] is deduced from the head difference ∆H [m] applied on the nozzle.

H * g * 2 U 0 ∆ = (2)
Diffusion of the jet causes the water velocity U at the soil-water interface to be inversely proportional to the distance J from the nozzle, for distances greater than the length of the potential core of the jet, J P . By rewriting our equation of erosion using equation 1, 3 and, using the notion of the equilibrium depth (τ = τ C (no erosion)), a nondimensional equation is built for J > J P (equation 5).

                  - + + - = * * i * * i J J * * J J * R J 1 J 1 Ln 2 1 J T t (5) with e * C d e R J J J k J T = τ = .
To correlate our experimental data set with the theoretical development, a two-step method is used. First, the depth of equilibrium Je is obtained by using the Blaisdell [START_REF] Blaisdell | Ultimate dimension of local scour[END_REF] analysis. The value is used to estimate the critical shear stress by analyzing the hydraulic conditions that would exist at this equilibrium depth (equation 3). Second, the k d value is adjusted to fit the experimental time series of scour depths to the nondimensional model (equation 4).

(4) The experimental conditions

An initial elevation of the jet orifice is set at a distance J(0) > Jp and this elevation is maintained throughout the test. Therefore during the test, as scour of the soil beneath the jet increases with time, the distance from the jet orifice and soil surface increases. This initial distance and the pressure head applied to the nozzle are set prior to testing to produce a desired initial stress in accordance with equations ( 2) and (4). The pressure head was typically kept between 75 and 150 cm (30 to 60 inches). Once a head was chosen, it was kept constant during the duration of the test. The rates of erosion produced led to test durations ranging from 10 min to 4 h.

TESTED SOILS AND RESULTS (1) Soil description

The two soils chosen for the tests are clayey soils. They were classified according to the Unified Soil Classification 5) (L.L. : Liquid Limit, P.I. : Plastic Index). One soil is a CL-ML (P.I.=4, L.L.=21) named P2, the other is a CL (P.I.=15, L.L.=31) soil named P3. The main difference between these 2 soils is the amount of clay.

The compaction curves obtained for P2 are presented in figure 2. The optimum water content for standard Proctor compaction is roughly 11.5% -12% with a density of 1900 kg/m3. Concerning P3 (refer figure 3), the optimal water content is 13.7% for a dry density of 1860 kg/m3. (

2) Results concerning the erosion behavior

In table 1, the values corresponding to the minimal erodibility (k d and τ C ) for the soil P2 and P3 are summarized for a given preparation (water content at compaction, compaction energy and saturation or not). It can be seen that the P3 soil is less erodible in the case of optimal conditions. This can be explained partly by the clay content and the Plasticity Index. On the figure 4 and6, it is apparent that the condition of minimum erodibility corresponds to the optimum water content for compaction. On the dry side (water content less than the optimum), the erosion coefficient is 100 times higher than the measured value for the optimum water content. On the wet side (water content higher than the optimum), the erosion coefficient is only 10 times higher than the measured value for the optimum water content. The erosion rate coefficient is quite dependent on the water content at compaction and the compaction energy. The measurements of the critical shear stress show the same evolution (figure 5 and7). The values for the minimal erosion (at optimum water content) are roughly the same for the normal and the modified compaction but the additional compaction energy reduces the optimum water content and shifts the entire erodibility relationship to the left (figure 4 and5).

Saturation appears to have a similar effect as increased compaction energy by shifting the entire erodibility relationship (i.e. curve) to the left (figure 4 and6). It also seems to reduce erodibility on the dry side and increase erodibility on the wet side. This phenomenon was also observed for dry side compaction of soil P3.

Moreover the curve representing the erosion coefficient kd, according to the water content seems linked to the curves representing the permeability versus the water content at compaction, presented by Lambe and Whitman [START_REF] Lambe | Soil Mechanics[END_REF] .

These results seem to underline an effect of the soil fabric that depends on the saturation and on the compaction process. The effectiveness of this compaction can be defined by the dry density, the water in the pores, and the water adsorbed by the clay particles.

It could be concluded from these results that the erodibility is a function of the applied hydraulic stress and the soil fabric. This emphasizes the difficulties of estimating soil erodibility from correlation to engineering parameters without knowledge of the compaction process and saturation history. For clayey soils it appears that understanding the compaction process leads to a better understanding of the erosion process and the behavior of erodibility parameters. The erodibility seems to be linked closely to the fabric of the soil.

CONCLUSION

The erodibility of a soil impacted by a jet is the result of the interaction between the water and the fabric of this soil. This study shows the influence of the compaction effort and the saturation (at constant volume) on the soil erodibility.

A remaining work is to establish the same kind of curve with other device to quantify erodibility, as Hole Erosion Test. Moreover, consideration should be given to characterize the length scale of the soil fabric and its impact on the erodibility.

AKNOWLEDGEMENT : These tests were made possible with the help of the EDF-CIH, Bureau of Reclamation-Denver, USDA ARS-Stillwater.

Figure 1

 1 Figure 1 Schematic view of the JET.

( 3 )

 3 [m] : nozzle diameter.Then, the water velocity U is related to the shear stress with the Chezy equation (equation 4). Back analysis of the experimental data

Figure 2 Figure 3

 23 Figure 2 Dry density verswater content at compaction for the P2 soil.

Figures 4 ,

 4 Figures 4, 5, 6 and 7 present the values of the erodibility parameters relative to the value referred in the table 1 for a normal compaction of the soil P2 (k d ref and τ C ref). The described parameters are the erosion coefficient and the critical shear stress.

Figure 4 Figure 5

 45 Figure 4 Relative variation of the erosion coefficient k d for the P2 soil according to the water content at compaction.

Figure 6 Figure 7

 67 Figure 6Relative variation of the erosion coefficient kd for the P3 soil according to the water content at compaction.

Table 1

 1 Measured erosion law parameters of the different soils at the water content which minimizes the erosion for a given preparation.

	Soil	Type of preparation		Water content [%]	τ τ τ τ c [m 3 /(N.s)] [Pa] k d
			Normal	11.10 -12	7.3*10 -08	8
	P2		Normal saturated		12.35	2.2*10 -07	4
			Modified	9.6 -10.9	7.6*10 -08	13
			Normal		13.7	1.9*10 -08	4
	P3		Normal saturated	11.7 -13.7	1.8*10 -08	23
	1000				Normal saturated
						Normal
	100				Modified compaction
	k d /k d ref	10			Optimal water content for normal compaction (11.5%)
		1
		6%	8%	10%	12%	14%	16%	18%
					Water content