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

Y Omari
email: omariysf@gmail.com

RIESZ BASES OF REPRODUCING KERNELS IN SMALL FOCK SPACES

Keywords: 2010 Mathematics Subject Classification. primary 42C15; secondary 30C40, 30D10, 30H20, 42B35 Riesz bases, complete interpolating sequences, small Fock spaces

We give a complete characterization of Riesz bases of normalized reproducing kernels in the small Fock spaces F 2 ϕ , the spaces of entire functions f such that f e -ϕ ∈ L 2 (C), where ϕ(z) = (log + |z|) β+1 , 0 < β ≤ 1. The first results in this direction are due to Borichev-Lyubarskii who showed that ϕ with β = 1 is the largest weight for which the corresponding Fock space admits Riesz bases of reproducing kernels. Later, such bases were characterized by Baranov-Dumont-Hartman-Kellay in the case when β = 1. The present paper answers a question in Baranov et al. by extending their results for all parameters β ∈ (0, 1). Our results are analogous to those obtained for the case β = 1 and those proved for Riesz bases of complex exponentials for the Paley-Wiener spaces. We also obtain a description of complete interpolating sequences in small Fock spaces with corresponding uniform norm.

Introduction and statement of main results

For a subharmonic function ϕ tending to infinity when |z| → ∞, define the weighted Fock spaces F p ϕ , p = 2, ∞, the spaces of all entire functions f for which f e -ϕ belongs to L p (C). Seip and Wallstén [START_REF] Seip | Density theorems for sampling and interpolation in the Bargmann-Fock space[END_REF][START_REF] Seip | Density theorems for sampling and interpolation in the Bargmann-Fock space[END_REF] characterized interpolating and sampling sequences in these spaces when ϕ(z) = |z| 2 . They showed that there are no sequences which are simultaneously interpolating and sampling, and hence there are no Riesz bases of reproducing kernels in this case. However, the situation changes in small Fock spaces when the weight increases slowly. In [8, Theorem 2.8, Theorem 2.10], Borichev and Lyubarskii proved that the spaces F 2 ϕ with ϕ(z) = ϕ β (z) = log + |z| β+1 , 0 < β ≤ 1, possess Riesz bases of normalized reproducing kernels at real points. Another proof was given by Baranov, Belov and Borichev in [START_REF] Baranov | Fock type spaces with Riesz Bases of reproducing kernels and de Branges spaces[END_REF]Theorem 1.2] for the case where ϕ(z) = O(log + |z|) 2 with some regularity conditions. In this case, the Fock spaces F 2 ϕ can be viewed as de Branges spaces [3, Theorem 1.2], see also [START_REF] Baranov | Spectral synthesis in de Branges spaces[END_REF]. A more general approach based on the boundedness and invertibility of a discrete Hilbert transform was given by Belov, Mengestie and Seip in [START_REF] Mengestie | Discrete Hilbert transforms on sparse sequences[END_REF]. In a recent work, Baranov, Dumond, Hartmann and Kellay [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF] gave a characterization of complete interpolating sequences (or Riesz bases) in F 2 ϕ , when ϕ(z) = (log + |z|) 2 , in the spirit of Ingham-Kadets 1/4 theorem for the Paley-Wiener space (see [START_REF] Omari | Complete interpolating sequences small Fock spaces[END_REF] for the case of

F p ϕ , 1 ≤ p < ∞).
If the weight grows more rapidly than (log + |z|) 2 , then the associated Fock type space has no Riesz bases of (normalized) reproducing kernels, see [START_REF] Baranov | Fock type spaces with Riesz Bases of reproducing kernels and de Branges spaces[END_REF][START_REF] Borichev | Sampling and interpolation in large Bergman and Fock spaces[END_REF][START_REF] Borichev | Riesz Bases of reproducing kernels in Fock type spaces[END_REF][START_REF] Seip | Density theorems for sampling interpolation in the Bargmann-Fock space[END_REF][START_REF] Seip | Density theorems for sampling and interpolation in the Bargmann-Fock space[END_REF] and the references therein. Also, for the case when the corresponding Riesz measure of the weight ϕ is doubling, Marco, Massaneda and Ortega-Cerdà in [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF] proved that F 2 ϕ possesses no Riesz bases of normalized reproducing kernels. Finally, it follows from [START_REF] Isaev | On unconditional bases of reproducing kernels in Fock type spaces[END_REF] that there is no criterion for the existence of Riesz bases in a Fock-type space F 2 ϕ consisting of reproducing kernels in terms of the growth of the weight function ϕ only.

The main question we deal with in this paper was posed in [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF]. One looks for a characterization of Riesz bases of normalized reproducing kernels for F 2 ϕ β , where 0 < β < 1. Indeed, we obtain a complete description of such bases. Our results are analogous to the well known Avdonin theorem [START_REF] Avdonin | On the question of Riesz bases of exponential functions in L 2[END_REF] established for Riesz bases of complex exponentials in the Paley-Wiener spaces (see [START_REF] Young | An introduction to nonharmonic Fourier series[END_REF]Theorem 14,p. 178] and [2, p. 102] for more general statement) and to those obtained for F 2 ϕ 1 (see [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF]Theorem 1.1]). We also give a characterization of complete interpolating sequences in F ∞ ϕ β for 0 < β < 1. The case when β = 1 was studied in [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF]Theorem 1.2].

1.1. Description of Riesz bases of F 2 ϕ . Let ϕ be an increasing function defined on [0, ∞) tending to infinity. We extend ϕ into the whole complex plane C by ϕ(z) = ϕ(|z|) and we define the corresponding Fock type space

F 2 ϕ := f ∈ Hol(C) : f 2 ϕ,2 := C |f (z)| 2 e -2ϕ(z) dm(z) < ∞ ,
here m stands for the area Lebesgue measure. Since point evaluations are bounded linear functionals on F 2 ϕ , by the representation Riesz Theorem the space F 2 ϕ endowed with the inner product

f, g ϕ := C f (z)g(z)e -2ϕ(z) dm(z), f, g ∈ F 2 ϕ ,
is a reproducing kernel Hilbert space (RKHS). Let k z be its reproducing kernel at z ∈ C, that is

f (z) = f, k z ϕ , f ∈ F 2 ϕ . We denote by k z = k z / k z ϕ,2 the normalized reproducing kernel of F 2 ϕ .
Let Γ be a sequence of complex numbers. A system of normalized reproducing kernels

K Γ = {k γ } γ∈Γ is said to be a Riesz basis for F 2 ϕ if K Γ is complete and if for some constant C ≥ 1 we have 1 C γ∈Γ |a γ | 2 ≤ γ∈Γ a γ k γ 2 ϕ,2 ≤ C γ∈Γ |a γ | 2 ,
for each finite sequence {a γ }. This means that Γ is a complete interpolating sequence for F 2 ϕ . In an equivalent way, the operator

T Γ : F 2 ϕ -→ ℓ 2 (Γ) f -→ ( f, k γ ) γ∈Γ is bounded and invertible from F 2 ϕ to ℓ 2 (Γ). We next set d(z, w) = |z -w| 1 + min(|z|, |w|) , z, w ∈ C. A set Σ ⊂ C is said to be d-separated if there is d Σ > 0 such that inf {d(σ, σ * ) : σ, σ * ∈ Σ, σ = σ * } ≥ d Σ .
We denote by D(z, r) the Euclidean disk of radius r centered at z. For δ < 1 and 0 = z ∈ C the ball corresponding to the distance d is given by

D d (z, δ) := {w ∈ C : d(z, w) < δ}.
(1) Γ is d-separated, (2) δ n /(1 + n) 1 β -1 n≥0
is a bounded sequence,

(3) there exists N 1 such that

lim sup n 1 (1 + n + N) 1 β -(1 + n) 1 β n+N k=n+1 δ k < 1 2(1 + β) 1 β
.

(1.1)

1.2. Complete interpolating sequences in uniform Fock spaces F ∞ ϕ . In this subsection we consider the uniform Fock type space, associated with ϕ, given as follows

F ∞ ϕ := f ∈ Hol(C) : f ϕ,∞ := sup z∈C |f (z)|e -ϕ(z) < ∞ . F ∞ ϕ endowed with the norm . ϕ,∞ is clearly a Banach space. Let Γ = {γ n } be a sequence in C. Γ is said to be sampling for F ∞ ϕ whenever there exists C ≥ 1 such that f ϕ,∞ ≤ C f ϕ,∞,Γ , for every f ∈ F ∞ ϕ , where f ϕ,∞,Γ := sup γ∈Γ |f (γ)|e -ϕ(γ) . Γ is called an interpolating set for F ∞ ϕ if for every sequence v = (v γ ) γ∈Γ such that v ϕ,∞,Γ < ∞ there exists f ∈ F ∞ ϕ that satisfies f (γ) = v γ , for every γ ∈ Γ.
Finally, Γ is said to be a complete interpolating sequence for F ∞ ϕ if Γ is simultaneously a sampling and an interpolating set for F ∞ ϕ .

The following result describes complete interpolating sequences for F ∞ ϕ , the case β = 1 was obtained in [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF].

Theorem 1.2. Let ϕ(r) = log + r β+1 where 0 < β < 1. Let Γ be a sequence of complex numbers and let

γ * ∈ C \ Γ. Then Γ ∪ {γ * } is a complete interpolating set for F ∞ ϕ if and only if K Γ is a Riesz basis for F 2 ϕ .
1.3. Remarks. Several remarks are in order before we turn to the proofs of our theorems:

1. Complete interpolating sequences for F p ϕ , p = 2, ∞ are characterized by comparing them to the sequence

Λ = λ n := exp 1 + n 1 + β 1 β e iθn , θ n ∈ R : n ≥ 0 , (1.2)
which is a complete interpolating sequence for F 2 ϕ and Λ ∪ {λ * } is a complete interpolating sequence for F ∞ ϕ where λ * ∈ C \ Λ (for p = 2 see [3, Theorem 1.2] and Lemma 2.6 and for the case p = ∞ see Lemma 2.7 ).

2.

The diagonal asymptotic estimates of the reproducing kernel play an important role in the study of Riesz bases. Let ρ(z) := (∆ϕ(z)) -1/2 = |z| (log |z|) 1-β 2 , as stated in Corollary 2.3 the kernel admits the following estimates

e 2ϕ(z) ρ(z) 2 k z 2 ϕ,2 e 2ϕ(z) |z|ρ(z) , |z| > 1.
Furthermore the upper estimate is attained on a subset of z (see Lemma 2.2 for the precise diagonal asymptotic estimates).

3. The case β = 1 was treated in [5, Theorem 1.1] under conditions (1), (2) and under the hypothesis that there exists N 1 such that

sup n 1 N n+N k=n+1 δ k < 1 4 . (1.3)
The family K Λ = {k λ } λ∈Λ (associated with β = 1) is a Riesz basis for F 2 ϕ 1 . Condition (1.1) may appear different to (1.3) in the case β = 1; however, these two conditions are equivalent for the problem studied here (see Remark 5.1).

Condition (1.1) can be written as

∆ N := lim sup n 1 log |λ n+N | -log |λ n | n+N k=n+1 δ k < 1 2 . (1.4) For N = 1, condition (1.1) is equivalent to sup n≥1 log |γ n | -log |λ n | n 1 β -1 = sup n≥1 |δ n | n 1 β -1 < 1 2β(1 + β) 1 β , (1.5)
(see Remark 5.1). Conditions (1.4) and (1.5) look similar to those used in some related results. We mention here the well known 1/4 Kadets-Ingham's theorem [START_REF] Kadets | The exact value of the Paley-Wiener constant[END_REF] on Riesz bases of exponentials in the Paley-Wiener spaces P W 2 α , and the results by Marzo and Seip [START_REF] Marzo | The Kadets 1/4 Theorem for polynomials[END_REF] for spaces of polynomials. For fixed N, (1.4) looks similar to Avdonin's condition [START_REF] Avdonin | On the question of Riesz bases of exponential functions in L 2[END_REF]. It is interesting to note that conditions (1)-(3) describe completely Riesz bases of reproducing kernels in F 2 ϕ ; on the contrary, in the Paley-Wiener spaces, these conditions are just sufficient.

5.

We notice again that our spaces possess Riesz bases of normalized reproducing kernels at real points, and hence they can be viewed as de Branges spaces [START_REF] Baranov | Spectral synthesis in de Branges spaces[END_REF][START_REF] Baranov | Fock type spaces with Riesz Bases of reproducing kernels and de Branges spaces[END_REF]. Using the boundedness and invertibility results on the discrete Hilbert transform on lacunary sequences, Belov, Mengestie and Seip gave another characterization of Riesz bases, where our summability condition (1.1) corresponds to a Muckenhoupt-type condition; for further details we refer to [START_REF] Mengestie | Discrete Hilbert transforms on sparse sequences[END_REF]Theorem 1.1]. Our approach consists of using Bari's theorem on Riesz bases in Hilbert spaces, as in the proof of [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF]Theorem 1.1].

The plan of our paper is as following. In the next section we state diagonal asymptotic estimates on the reproducing kernel. We then show that Λ and Λ ∪ {λ * } are complete interpolating sequences for F 2 ϕ and F ∞ ϕ , respectively. Furthermore, we deal with the separation condition. Section 3 is devoted to proving Theorem 1.1. The proof of Theorem 1.2 is presented in Section 4. We end our paper by some remarks in the last section.

Throughout the paper, we use the following notations:

• A B means that there is an absolute constant C such that A ≤ CB.

• A ≍ B if both A B and B A hold.

Acknowledgement. The authors are grateful to A. Borichev, O. El-Fallah, Yu. Lyubarskii and J. Ortega-Cerdà for helpful discussions. We also would like to thank the referees for their detailed comments and suggestions.

Key lemmas and preliminary results

In this section we prove some preliminary results and some key lemmas. First, we establish estimates on the reproducing kernel at the diagonal also get necessary estimates on certain infinite products and also preliminary results concerning the separation condition. Throughout the rest of this paper we denote ϕ(r) = ϕ β (r) = log + r β+1 , where 0 < β < 1.

2.1.

Estimates on the norm of the reproducing kernel. We begin by the following lemma which was proved in [3, Lemma 3.1] and which will be used repeatedly throughout the paper.

Lemma 2.1. For every nonnegative integer n we have

z n 2 ϕ,2 ≍ 1 + n 1 + β 1-β 2β exp 2β 1 + n 1 + β 1+β β
.

Next we produce an asymptotic estimate on the reproducing kernel k z of F 2 ϕ on the diagonal. Note that such estimate was obtained for the points λ ∈ Λ by Baranov, Belov and Borichev in [START_REF] Baranov | Fock type spaces with Riesz Bases of reproducing kernels and de Branges spaces[END_REF]Lemma 3.2]. They proved the estimate directly by using Hardy's convexity theorem. Here we use Laplace's method to deal with the rest of complex numbers z ∈ C. We denote by [x] the integer part of a real x.

Lemma 2.2. Let |z| = e s , n z = [(1 + β)s β ]. Set g s (t) = st -β t 1+β 1+β β . Let ϕ(z) = max(g s (n z ), g s (n z + 1)). We have k z 2 ϕ,2 ≍ e 2 ϕ(z) |z|ρ(z) + e 2ϕ(z) ρ 2 (z) , |z| > 1,
where ρ(z

) := (∆ϕ(z)) -1/2 = |z| (log |z|) 1-β 2 . Proof. Let |z| = e s . Note that g ′ s (t) = 0 for t = (1 + β)s β and ϕ(z) ≤ ϕ(z). Since k z 2 ϕ,2 = n≥0 |z| 2n / z n 2 ϕ,2 , by Lemma 2.1 we have e 2s k e s 2 ϕ,2 ≍ n≥1 e 2gs(n) (1 + n) 1-β 2β
nz 0 e 2gs(t)

(1 + t)

1-β 2β

dt + e 2gs(nz)

(1 + n z ) 1-β 2β + e 2gs(nz+1) (2 + n z ) 1-β 2β + ∞ nz+1 e 2gs(t) (1 + t) 1-β 2β dt e 2 ϕ(z) s 1-β 2 + ∞ 0 e 2s 1+β x-β x 1+β 1+β β s 1-β 2 (1 + x) 1-β 2β s β dx := e 2 ϕ(z) s 1-β 2 + s β s 1-β 2 ∞ 0 e -2s 1+β ψ(x) f (x)dx, where ψ(x) = β x 1+β 1+β β -x and f (x) = 1/(1 + x) 1-β 2β . We have ψ ′ (x) = x 1+β 1 β -1. Let x 0 = 1 + β. Then ψ ′ (x 0 ) = 0 and ψ(x 0 ) = -1. Since ψ ′′ (x) = x 1 β -1 β(1+β) 1 β > 0, x ≥ 1, we get by Laplace's method [9] ∞ 0 e -Sψ(x) f (x)dx ∼ 2π Sψ ′′ (x 0 ) f (x 0 )e -Sψ(x 0 ) , S → +∞.
For S = 2s 1+β → +∞, we get

k z 2 ϕ,2 = k e s 2 ϕ,2 e 2 ϕ(z) e 2s s 1-β 2 + e 2s 1+β e 2s s 1-β = e 2 ϕ(z) |z| 2 (log |z|) 1-β 2 + e 2ϕ(z) |z| 2 (log |z|) 1-β .
On the other hand, using Laplace's method again, we obtain

I := 1+β 0 e 2s 1+β x-β x 1+β 1+β β dx ∼ 2π 2s 1+β ψ ′′ (x 0 ) e 2s 1+β = πβ(1 + β) s 1+β e 2s 1+β .
Also, we have

J := 1+β 1+β-2/s β e 2s 1+β x-β x 1+β 1+β β dx = 2/s β 0 exp 2s 1+β (1 + β -u) -β 1 - u 1 + β 1+β β du = 2/s β 0 exp 2s 1+β 1 - 1 2β(β + 1) u 2 du ≤ e 2s 1+β ∞ 0 e -s 1+β u 2 β(β+1) du = 1 2 β(β + 1)π s 1+β e 2s 1+β ∼ 1 2 I.
So, we get

e 2s k e s 2 ϕ,2 nz-1 0 e 2gs(t)
(1 + t)

1-β 2β

dt + e 2gs(nz)

(1 + n z ) 1-β 2β + e 2gs(nz+1) (2 + n z ) 1-β 2β e 2 ϕ(z) s 1-β 2 + s β s 1-β 2 1+β-2/s β 0 e 2s 1+β x-β x 1+β 1+β β (1 + x) 1-β 2β dx = e 2 ϕ(z) s 1-β 2 + s β s 1-β 2 I -J ≍ e 2 ϕ(z) s 1-β 2 + s β s 1-β 2 e 2s 1+β √ s 1+β .
Therefore,

k z 2 ϕ,2 e 2 ϕ(z) |z| 2 (log |z|) 1-β 2 + e 2ϕ(z) |z| 2 (log |z|) 1-β , |z| > 1.
This completes the proof.

We have the following corollary, with identity (2.2) obtained already in [3, Lemma 3.2]

Corollary 2.3. We have

e 2ϕ(z) ρ(z) 2 k z 2 ϕ,2 e 2ϕ(z) |z|ρ(z) , |z| > 1. (2.1)
Furthermore, for every λ n ∈ Λ, where Λ is the sequence given in (1.2), we have

k λn 2 ϕ,2 ≍ e 2ϕ(λn) |λ n |ρ(λ n ) ≍ |e n (λ n )| 2 , (2.2
)

where e n (z) = z n / z n ϕ,2 . Also, if σ n = exp n+1/2 1+β 1 β , n ≥ 2, then k σn 2 ϕ,2 ≍ e 2ϕ(σn) ρ(σ n ) 2 . ( 2
:= [(1 + β)s β ], g s (t) = st -β t 1+β β+1 β . By Lemma 2.2 we have k z 2 ϕ,2 ≍ 1 |z| 2 (log |z|) 1-β 2
e 2 max{gs(nz),gs(nz+1)} + e 2ϕ(z)-2 1-β 4 log log |z| .

We write n z = (1 + β)s β + δ z , for some δ z ∈ (-1, 0]. Simple calculations show that for j ∈ {0, 1} we have

g s (n z + j) -ϕ(z) + 1 -β 4 log s = s (1 + β)s β + δ z + j -β (1 + β)s β + δ z + j 1 + β β+1 β -s β+1 + 1 -β 4 log s = 1 -β 4 log s - 1 + o(1) 2β(1 + β) (δ z + j) 2 s 1-β -→ |z|→∞ -∞, if and only if β(1 -β 2 ) 2 + o(1) log s s 1-β = o (δ z + j) 2 , |z| → ∞. Now, if z = σ n , then δ 2 z = (δ z + 1) 2 = 1 4
and hence the latter estimate holds. Thus (2.3) follows immediately.

Remark. For regular radial weights satisfying the estimate (log r) 2 = O(ϕ(r)), when r → ∞, the reproducing kernel of F 2 ϕ possesses the property k z (z) ≍ ∆ϕ(z)e 2ϕ(z) (see [START_REF] Baranov | Fock type spaces with Riesz Bases of reproducing kernels and de Branges spaces[END_REF][START_REF] Borichev | Sampling and interpolation in large Bergman and Fock spaces[END_REF][START_REF] Borichev | Riesz Bases of reproducing kernels in Fock type spaces[END_REF]). This estimate remains valid for the kernel of F 2 ϕ when the associated Riesz measure of the weight ϕ is doubling [START_REF] Marco | Interpolating and sampling sequences for entire functions[END_REF]Lemma 21]. Estimate (2.2) proves that the case β < 1 is quite different from the case when β = 1. Indeed, for β = 1 the reproducing kernel admits the estimate k z 2 ϕ 1 ,2 ≍ ∆ϕ 1 (z)e 2ϕ 1 (z) , for every z ∈ C. However, in (2.2) we have for every

λ ∈ Λ that k λ 2 ϕ β ,2 ≍ e 2ϕ β (λ) / |λ| 2 log(λ) 1-β 2
, and hence

∆ϕ β (λ)e 2ϕ β (λ) = o k λ 2 ϕ β ,2
, whenever 0 < β < 1. This could explain the difference in the results obtained about Riesz bases of normalized reproducing kernels in the situation β = 1 and 0 < β < 1.

Complete interpolating sequences in

F p ϕ , p = 2, ∞.
With a given sequence Γ of complex numbers, we associate the infinite product

G Γ (z) = γ∈Γ 1 - z γ , z ∈ C, whenever it converges.
In what follows, we denote by dist(z, Γ) the Euclidean distance between z and the sequence Γ.

The following lemma provides estimates on the infinite product G Γ associated with a sequence Γ defined by

Γ := γ n := exp 1 + n 1 + β 1 β e δn e iθn , θ n ∈ R : n 0 . (2.4)
Recall that dist(z, Γ)

∆ N := lim sup n→∞ 1 log |λ n+N | -log |λ n | n+N k=n+1 δ k .
(1 + |z|) 3 2 +∆ N +ε ≤ |G Γ (z)| e -ϕ(z) ≤ C dist(z, Γ) (1 + |z|) 3 2 -∆ N -ε , z ∈ C. Proof. Let z ∈ C with |z| = e t = exp s 1+β 1 β . Let also m be the integer such that |γ m-1 | ≤ |z| < |γ m | and suppose that dist(z, Γ) = |z -γ m-1 |. Then log |G Γ (z)| = 0 n<m-1 log 1 - z γ n + log 1 - z γ m-1 + O(1) = m-1 n=0 log z γ n + log dist(z, Γ) -t + O(1) = mt - m n=1 k 1 + β 1 β - m-1 n=1 δ k + log dist(z, Γ) -t + O(1), t → ∞.
Furthermore, for every α = -1 we have

m k=1 k α = m α+1 α + 1 + m α 2 (1 + o(1)) . ( 2
δ k ∆ N + ε 2 p-1 j=0 (u jN +N -u jN ) + O(m 1 β -1 ) ∆ N + ε 2 t + o(t), t → ∞.
Consequently,

log |G Γ (z)| mt -β m 1 + β 1 β +1 - 3 2 ∓ ∆ N ∓ ε 2 m 1 + β 1 β + log dist(z, Γ) + o(t) t β+1 - 3 2 ∓ ∆ N ∓ ε 2 t + log dist(z, Γ) + o(t).
Finally, there exist two positive numbers A and B depending on ε such that

A dist(z, Λ) (1 + |z|) 3 2 +∆ N +ε ≤ |G Γ (z)| e -ϕ(z) ≤ B dist(z, Λ) (1 + |z|) 3 2 -∆ N -ε , z ∈ C.
We need the following standard ingredient, which we single out as a lemma.

Lemma 2.5. Let Γ be a sequence defined as in (2.4). We have

C dist(z, Γ) 2 (1 + |z|) 3+α dm(z) < ∞ ⇐⇒ α > 1.
Proof. Indeed, for every

1 2 (|γ n-1 | + |γ n |) r 1 2 (|γ n | + |γ n+1 |), we have dist(re iθ , Γ) ≍ |re iθ -γ n |. So, we get C dist(z, Γ) 2 (1 + |z|) 3+α dm(z) ≍ n 0 1 2 (|γn|+|γ n+1 |) 1 2 (|γ n-1 |+|γn|) 2π 0 |re iθ -γ n | 2 (1 + r) 3+α rdθdr ≍ n 0 1 |γ n | α-1 .
Therefore, the last sum is finite if and only if α > 1.

Next we extend a result of Borichev and Lyubarskii [8, Theorem 2.8]. They proved that {k λn } n≥0 is a Riesz basis for F 2 ϕ where λ 0 = 0, | λ n | = exp [(w n+1 -w n-1 )/4] and w n := log z n 2 = c(n + 1) 1+ 1 β + O(log n). Their proof was based on Bari's theorem (see [15, section A.5.7.1]). Using the estimate on the moments z n ϕ,2 given in Lemma 2.1, we produce a family of sequences of similar kind. These sequences could be viewed as a reference family of complete interpolating sequences for F 2 ϕ and will play a crucial tool in the proof of the main theorems. A proof of this lemma was given by Baranov-Belov-Borichev in [3, Theorem 1.2] using the results of Belov-Mengestie-Seip [START_REF] Mengestie | Discrete Hilbert transforms on sparse sequences[END_REF]. However, for the sake of completeness, we give here another proof that uses Bari's theorem as in [START_REF] Borichev | Riesz Bases of reproducing kernels in Fock type spaces[END_REF].

Lemma 2.6. Let Λ be a sequence defined by (1.2). Then K Λ = {k λ } λ∈Λ is a Riesz basis for F 2 ϕ . Proof. Let n 0, e n (z) = z n / z n ϕ,2 , and h n := e -inθn e n . We have

h n -k λn 2 ϕ,2 = e -inθn e n - e n (λ n ) k λn ϕ,2 e n - k =n e k (λ n ) k λn ϕ,2 e k 2 ϕ,2 = 1 - |e n (|λ n |)| k λn ϕ,2 2
J 1 + k =n |e k (λ n )| 2 k λn 2 ϕ,2 J 2 .
Furthermore,

J 1 = 1 - |e n (|λ n |)| k λn ϕ,2 2 ≤ 1 - |e n (λ n )| 2 k λn 2 ϕ,2 = J 2 .
Consequently,

h n -k λn 2 ϕ,2 ≍ k =n |e k (λ n )| 2 k λn 2 ϕ,2 k =n e k (λ n ) e n (λ n ) 2 .
On the other hand, for k = n we have

|e n (λ n )| 2 |e k (λ n )| 2 ≍ 1 + k 1 + n 1-β 2β e 2c(n,k) , where c(n, k) := (n -k) 1 + n 1 + β 1 β + β 1 + k 1 + β 1+β β - 1 + n 1 + β 1+β β .
Therefore, by simple computations we get

c(n, k) (1 + β) -1+β β |k -n| 2 (1 + n) 1 β -1 . Thus, ∞ n=0 h n -k λn 2 ϕ,2 ≍ ∞ n=0 k =n |e k (λ n )| 2 k λn 2 ϕ,2 ∞ n=0 k =n 1 + n 1 + k 1-β 2β e -2(1+β) -1+β β |k-n| 2 (1+n) 1 β -1 < ∞.
Note that Λ = Γ with δ k = 0. So Lemmas 2.4 and 2.5 imply that Λ is a uniqueness set for F 2 ϕ , that is the unique function in F 2 ϕ that vanishes on Λ is the zero function. Also, for any λ ∈ Λ the sequence Λ \ {λ} is a zero sequence of F 2 ϕ and, hence, the system K Λ is complete and minimal in F 2 ϕ and n h nk λn 2 ϕ,2 < ∞. Bari's theorem [15, section A.5.7.1] ensures that K Λ is a Riesz basis for F 2 ϕ .

For F ∞ ϕ we have the following result

Lemma 2.7. Let λ * ∈ C \ Λ. Then Λ ∪ {λ * } is a complete interpolating sequence for F ∞ ϕ .
Proof. First let us prove that Λ = Λ ∪ {λ * } is a uniqueness set for F ∞ ϕ . Indeed, suppose that λ * = 0 and take f ∈ F ∞ ϕ vanishing on Λ ∪ {λ * }, so that f (z) = (1 -z/λ * )h(z)G Λ (z) where h is an entire function. By Lemma 2.4 we get

|h(z)| dist(z, Λ) (1 + |z|) 1/2+ε |f (z)|e -ϕ(z) 1, z ∈ C.
It follows that h is the zero function. It remains to show that Λ is an interpolating set for F ∞ ϕ . For this purpose let v = (v λ ) λ∈ Λ be a sequence of complex numbers such that v ϕ,∞, Λ < ∞. Put F (z) = (1 -z/λ * )G Λ (z) and consider the function

F v (z) = λ∈ Λ v λ F (z) F ′ (λ)(z -λ) , z ∈ C.
We have

|F v (z)| ≤ λ∈ Λ |v λ | F (z) F ′ (λ)(z -λ) v ϕ,∞, Λ λ∈ Λ e ϕ(λ) |z| |λ| G Λ (z) G ′ Λ (λ)(z -λ) . Let z ∈ C and p ∈ N such that |λ p-1 | ≤ |z| < |λ p |. Suppose that 0 = |λ * | < |λ 1 | and write Λ = (λ n ) n≥-1 , where λ -1 = λ * . Let |z| = e t and u n = log |λ n |. We have λ∈ Λ e ϕ(λ) |z| |λ| G Λ (z) G ′ Λ (λ)(z -λ) ≍ n≤p-1
. . .

I 1 + n≥p e ϕ(λn) dist(z, Λ) |z -λ n | |z| p |λ n | n p-1 k=0 1 |λ k | n-1 k=0 |λ k | I 2
. By (2.5),

I 1 = n≤p-1 e ϕ(λn) dist(z, Λ) |z -λ n | |z| p |λ n | n p-1 k=0 1 |λ k | n-1 k=0 |λ k | ≤ |z| p n≤p-1 exp u β+1 n -(n + 1)u n + n k=0 u k - p-1 k=0 u k = e pt-βu β+1 p-1 n≤p-1 exp u n 2 + d β u 1-β n - u p-1 2 + d β u 1-β p-1 ≤ e pt-βu β+1 p-1 n≥0 e -c|u p-1 -un| exp ϕ(z) - 1 2β(β + 1) + o(1) (p -s) 2 t 1-β ≤ e ϕ(z)
and

I 2 ≍ |z| p+1 n≥p exp u β+1 n -(1 + n)u n + n-1 k=p u k ≤ e (p+1)t-βu β+1 p n≥0 exp - 1 2 |u n -u p | + d β u 1-β n -u 1-β p e (p+1)t-βu β+1 p ≤ e ϕ(z) .
Thus, the interpolating function F v belongs to F ∞ ϕ . This completes the proof.

2.3. d-separated and log-separated sequences. A sequence {µ n } of real numbers is said to be separated whenever there exists a constant δ > 0 such that

inf n =m |µ n -µ m | ≥ δ.
Let Γ be a sequence of complex numbers. It is not difficult to see that if log Γ := {log |γ| : γ ∈ Γ} is separated then Γ is d-separated, and also if Γ is d-separated then log Γ is a finite union of separated sequences. Hence Γ is a finite union of d-separated sequences if and only if log Γ is a finite union of separated real sequences.

The following lemma was established by using a Bernstein type inequality in [START_REF] Baranov | Sampling, interpolation and Riesz Bases in small Fock spaces[END_REF] in the case β = 1. The proof when 0 < β < 1 is different. Lemma 2.8. Let Γ be a sequence defined by (2.4).

Then γn∈Γ | f, k γn | 2 ≤ C(Γ) f 2 ϕ,2 , (2.6)
for every f ∈ F 2 ϕ if and only if {log |γ n | : γ n ∈ Γ} is a finite union of separated sequences.

Proof. Let f be a function in F 2 ϕ . According to Lemma 2.6, the system K Λ is a Riesz basis for F 2 ϕ and, hence,

f (z) = λn∈Λ f, k λn g λn (z), z ∈ C,
where G Λ := {g λn : λ n ∈ Λ} is the unique biorthogonal system of K Λ given by

g λn (z) = k λn G ′ Λ (λ n ) G Λ (z) z -λ n , z ∈ C, (2.7)
and G Λ is the infinite product associated with Λ. We then get

m≥0 f, k γm 2 = m≥0 n≥0 f, k λn g λn , k γm 2 = m≥0 n≥0 f, k λn g λn , k γm 2 .
Since the operator f → ( f, k λn ) is an isomorphism between F 2 ϕ and ℓ 2 , relation (2.6) is equivalent to the fact that the matrix C = (C n,m ) n,m defines a bounded operator on ℓ 2 , where

|C n,m | = | g λn , k γm | = k λn ϕ,2 k γm ϕ,2 G Λ (γ m ) G ′ Λ (λ n )(λ n -γ m )
.

Set p m = [(1 + β) (log |γ m |) β ] -1. We have |G Λ (γ m )| ≍ dist(γ m , Λ) γ m pm k=0 γ m λ k and |G ′ Λ (λ n )| ≍ 1 |λ n | n k=0 λ n λ k .
Hence,

|C n,m | ≍ k λn ϕ,2 k γm ϕ,2 dist(γ m , Λ) |λ n -γ m | |γ m | pm |λ n | n pm k=0 1 |λ k | n k=0 |λ k |.
(2.8)

By (2.
2), we have

k λn ϕ,2 ≍ |e n (λ n )| and k γm ϕ,2 ≥ |e pm (γ m )| thus we obtain |C n,m | z pm ϕ,2 z n ϕ,2 dist(γ m , Λ) |λ n -γ m | pm k=0 1 |λ k | n k=0 |λ k | ≍ 1 + p m 1 + n 1-β 4β dist(γ m , Λ) |λ n -γ m | e -α(n,m) , where α(n, m) = β 1 + n 1 + β 1+β β -β 1 + p m 1 + β 1+β β + pm k=0 1 + k 1 + β 1 β - n k=0 1 + k 1 + β 1 β = 1 2 1 + p m 1 + β 1 β - 1 + n 1 + β 1 β + 1 + o(1) 12β(1 + β) 1 + p m 1 + β 1-β β - 1 + n 1 + β 1-β β . Now, if |γ m | ≥ |λ n |, then we have dist(γ m , Λ) ≤ |λ n -γ m |, and hence, |C n,m | e -c β (1+pm) 1 β -(1+n) 1 β
.

(2.9)

If |γ m | < |λ n |, then dist(γ m , Λ) ≤ |λ pm | (or |γ m |) and |λ n -γ m | ≍ |λ n |.
Again we have (2.9). Thus (2.9) holds in both cases.

On the other hand, by Corollary 2.3 and by (2.8), we get

|C n,m | log |γ m | log |λ n | 1-β 4 e ϕ(λn)-ϕ(γm) dist(γ m , Λ) |λ n -γ m | |γ m | pm+1 |λ n | n+1 pm k=0 1 |λ k | n k=0 |λ k | ≍ 1 + p m 1 + n 1-β 4β dist(γ m , Λ) |λ n -γ m | e -α(n,m) ,
where

α(n, m) = 1 + p m 1 + β 1+β β - 1 + n 1 + β 1+β β + (1 + n) 1 + n 1 + β 1 β -(1 + p m ) 1 + p m 1 + β 1+β β + pm k=0 1 + k 1 + β 1 β - n k=0 1 + k 1 + β 1 β = 1 2 1 + p m 1 + β 1 β - 1 + n 1 + β 1 β + 1 + o(1) 12β(1 + β) 1 + p m 1 + β 1-β β - 1 + n 1 + β 1-β β .
In the same way we get

|C n,m | e -c β (1+pm) 1 β -(1+n) 1 β
.

Finally,

e -c β (1+pm) 1 β -(1+n) 1 β |C n,m | e -c β (1+pm) 1 β -(1+n) 1 β . If log |γ| : γ ∈ Γ is a finite union of separated sequences, then C = (C n,m ) is bounded on ℓ 2 .
In the opposite direction, since K Λ is a Riesz basis, let g λ be the function given in (2.7), g λ 2 ϕ,2 ≍ 1 and as before we have

1 γ∈Γ |g λ (γ)| 2 k γ (γ) γ∈Γ∩D d (λ,δ) |g λ (γ)| 2 k γ (γ) γ∈Γ∩D d (λ,δ) e -c β | log |γ|-log |λ|| ≍ Card(Γ ∩ D d (λ, δ)). Since dist(γ, Λ) ≍ |γ -λ|, we get sup λ∈Λ Card(Γ ∩ D d (λ, δ)) < ∞.
Therefore Γ is a finite union of d-separated sequences.

Proof of Theorem 1.1

Proof. "⇐="

First of all, since the sequence Γ is d-separated, by Lemma 2.8 the operator T Γ is bounded from F 2 ϕ to ℓ 2 . On the other hand, the sequence Γ is a uniqueness set for F 2 ϕ , and for any γ ∈ Γ, the sequence Γ \ {γ} is a zero set for F 2 ϕ . Indeed, let f ∈ F 2 ϕ be such that f | Γ = 0. By Hadamard's factorization theorem [START_REF] Levin | Lectures on entire functions[END_REF] we can write f = hG Γ , for an entire function h. Since f ∈ F 2 ϕ , by Lemma 2.3 we have

|f (z)| f ϕ,2 k z ϕ,2 e ϕ(z)
|z|(log |z|)

1-β 4 , |z| > 1. Lemma 2.4 implies that |h(z)| dist(z, Γ) (1 + |z|) 3/2+c N |f (z)|e -ϕ(z) 1 |z|(log |z|) 1-β 4 , |z| > 1,
where c N = ∆ N + ε, for a small enough ε so that c N < 1/2. Then h must be a polynomial. Combining this fact with Lemma 2.5, we conclude that h must be identically zero, and hence Γ is a uniqueness set for F 2 ϕ , therefore T Γ is injective. Futhermore T Γ g n = e n+1 , n ≥ 0, where e n+1 is the (n + 1)-th element of the canonical basis of ℓ 2 , and

g n (z) = k γn ϕ,2 G ′ Γ (γ n ) G Γ (z) z -γ n , z ∈ C.
Therefore, the range of T Γ is dense in ℓ 2 . In order to prove that T Γ is onto, we associate with each a ∈ ℓ 2 the function H a as follows :

H a (z) = γ∈Γ a γ k γ ϕ,2 G ′ Γ (γ) G Γ (z) z -γ , z ∈ C.
The estimate of the reproducing kernel given in Corollary 2.3 and the estimate on G Γ proved in Lemma 2.4 ensure that the above series converges uniformly on every compact set of C. It remains now to show that H a ∈ F 2 ϕ . Indeed, since K Λ is a Riesz basis for F 2 ϕ (see Lemma 2.6), we have

H a 2 ϕ,2 ≍ m 0 | H a , k λm | 2 = m 0 n 0 a n G Γ (λ m ) G ′ Γ (γ n )(λ m -γ n ) k γn ϕ,2 k λm ϕ,2 2
.

Hence {H a } a∈ℓ 2 ⊂ F 2 ϕ if and only if the matrix A = (A n,m ) n,m 0 defines a bounded operator on ℓ 2 , where

|A n,m | = G Γ (λ m ) G ′ Γ (γ n )(λ m -γ n ) k γn ϕ,2 k λm ϕ,2 .
Recall that for every z ∈ C there exists p ∈ N such that |γ p-1 | ≤ |z| < |γ p |. We have

|G Γ (z)| ≍ dist(z, Γ) |z| p-1 k=0 |z/γ k | and |G ′ Γ (γ n )| ≍ 1 |γ n | n-1 k=0 γ n γ k . Since δ n = O (1 + n) 1 β -1
, there exists M > 0 such that for every m ≥ 0 there exists |i| ≤ M such that the index p corresponding to λ m is m + i. Consequently,

|A n,m | ≍ k γn ϕ,2 k λm ϕ,2 dist(λ m , Γ) |λ m -γ n | |γ n | λ m m+i-1 k=0 λ m γ k n-1 k=0 γ k γ n .
By Lemma 2.2 we have

k z 2 ϕ,2 ≍ |e nz-1 (z)| 2 + |e nz (z)| 2 + e 2ϕ(z) ρ(z) 2 ,
where

n z = [(1 + β) log β |z|]. Again, since δ n = O((1 + n) 1 β -1 ), for every n ≥ 0 there exists |j| ≤ M satisfying k γn 2 ϕ,2 ≍ |e n+j (γ n )| 2 + e 2ϕ(γn) |γ n | 2 (log |γ n |) 1-β .
Also for every m ≥ 0 we have

k λm ϕ,2 ≍ |e m (λ m )| ≍ e ϕ(λn) |λ m | (log |λ m |) 1-β 4
.

By these estimates we can write |A n,m | ≍ I n,m + J n,m , where

I n,m ≍ |e n+j (γ n )| |e m (λ m )| |λ m | |λ m -γ n | |γ n | |λ m | m+i-1 k=0 λ m γ k n-1 k=0 γ k γ n =: |λ m | |λ m -γ n | e Θ(n,m)
and ,m) .

J n,m ≍ (log |λ m |) 1-β 4 (log |γ n |) 1-β 2 e ϕ(γn)-ϕ(λm) |λ m | |λ m -γ n | m+i-1 k=0 λ m γ k n-1 k=0 γ k γ n =: |λ m | |λ m -γ n | e θ(n
To estimate the coefficients I n,m and J n,m , we first write

K n,m = m+i-1 k=0 λ m γ k n-1 k=0 γ k γ n .
We next put u k := log |λ k | for every k ≥ 0. We have

log K n,m = (m + i)u m -n(u n + δ n ) - m+i-1 k=0 u k + n-1 k=0 u k - m+i-1 k=0 δ k + n-1 k=0 δ k = u m+i -u n 2 + βu β+1 n -n(u n + δ n ) -βu β+1 m+i -(m + i)u m + d β u 1-β n -u 1-β m+i - m+i-1 k=0 δ k + n-1 k=0 δ k .
Hence,

Θ(n, m) := log K n,m + log |e n+j (γ n )| -log e m (λ m ) + (u n + δ n ) -u m = u n -u m+i 2 + 1 2β(β + 1) + o(1) j 2 u 1-β n -i 2 u 1-β m+i + d β u 1-β n -u 1-β m+i - m+i-1 k=0 δ k + n k=0 δ k + 1 -β 4 log 1 + m 1 + n . Now if n ≤ m + i, then I n,m ≍ |λ m | |λ m -γ n | e Θ(n,m) ≍ e Θ(n,m)
and

Θ(n, m) = - 1 2 + o(1) (u m+i -u n ) - m+i-1 k=n+1 δ k . If n ≥ m + i, then I n,m ≍ |λ m | |λ m -γ n |
e Θ(n,m) ≍ e Θ(n,m)+um-un-δn and

Θ(n, m) + u m -u n -δ n = - 1 2 + o(1) (u n -u m+i ) + n-1 k=m+i δ k .
Hence,

I n,m ≍ exp - 1 2 + o(1) |u n -u m+i | ± n-1 k=m+i δ k .
Similarly, we have

θ(n, m) := log K n,m + ϕ(γ n ) -ϕ(λ m ) + 1 -β 4 log 1 + m (1 + n) 2 = u n -u m+i 2 + 1 2β(β + 1) + o(1) j 2 u 1-β n -i 2 u 1-β m+i + 1 -β 4 log 1 + m (1 + n) 2 + d β u 1-β n -u 1-β m+i - m+i-1 k=0 δ k + n k=0 δ k .
If n ≤ m + i, we have

J n,m ≍ |λ m | |λ m -γ n | e θ(n,m) ≍ e θ(n,m) , and
θ(n, m) = - 1 2 + o(1) (u m+i -u n ) - m+i-1 k=n+1 δ k . If n ≥ m + i, then J n,m ≍ |λ m | |λ m -γ n | e θ(n,m) ≍ e θ(n,m)+um-un-δn and θ(n, m) + u m -u n -δ n = - 1 2 + o(1) (u n -u m+i ) + n-1 k=m+i δ k .
Hence,

J n,m ≍ exp - 1 2 + o(1) |u n -u m+i | ± n-1 k=m+i δ k . Consequently, |A n,m | ≍ I n,m + J n,m ≍ exp - 1 2 + o(1) |u n -u m+i | ± n-1 k=m+i δ k .
Recall that

∆ N := lim sup n 1 u n+N -u n n+M k=n+1 δ k < 1 2 .
This implies that for a very small ε (chosen in such a way that ∆ N + ε < 1/2) and for every n ≥ 0 we have

n+M k=n+1 δ k ≤ ∆ N + ε 2 (u n+N -u n ) .
Thus,

|A n,m | exp - 1 2 -∆ N + ε 2 + o(1) |u n -u m+i | . Since ∆ N < 1 2 , the matrix A = (A n,m
) defines a bounded operator on ℓ 2 and, hence, K Γ is a Riesz basis for F 2 ϕ .

"=⇒" Proof of (1). Let ϕ(r

) = ϕ β (r) = log + r 1+β , 0 < β ≤ 1. If K Γ is a Riesz basis for F 2 ϕ β
then Γ is an interpolating sequence for F 2 ϕ β and hence for F 2 ϕ 1 . By [5, Corollary 2.3] Γ is d-separated.

Proof of [START_REF] Avdonin | Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems[END_REF]. Let Γ = {γ n } be a sequence of complex numbers such that K Γ is a Riesz basis for F 2 ϕ . Then for every γ ∈ Γ there exists a unique function f γ ∈ F 2 ϕ that satisfies the interpolation problem :

f γ , k γ = 1 and f γ , k γ = 0, γ ′ ∈ Γ \ {γ}.
Consequently, Γ \ {γ} is a subset of the zero set of the function f γ . Since K Γ is complete, Γ is a uniqueness set, then Γ \ {γ} is exactly the zero set of f γ . By Hadamard's factorization theorem we have

f γ (z) = c G Γ (z) G ′ Γ (γ)(z-γ) , for some constant c ∈ C. Since f γ , k γ = 1, we get c = k γ ϕ,2 . Therefore, f γ (z) = k γ ϕ,2 G Γ (z) G ′ Γ (γ)(z -γ) , z ∈ C. Furthermore f γ ϕ,2 ≍ 1. Hence, |f γ (λ m )| ≤ f γ ϕ,2 k λm ≍ |e m (λ m )| ,
for every λ m ∈ Λ.

Suppose now that the sequence (δ n /(1+n) 1 β -1 ) is unbounded. Without loss of generality we assume the existence of a subsequence δ n k /(1+n k ) 1 β -1 which tends to +∞ (the case of convergence to -∞ is similar). Then, for every k there exists m k such that |n k -m k | → ∞ and γ n k is close to λ m k . We obtain

f γn k (λ m k ) k λm k ϕ,2 ≍ |e m k (λ m k )| . (3.1)
On the other hand, we have

k γn k ϕ,2 ≥ |e m k (γ n k )| and |G Γ (λ m k )| ≍ |λ m k -γ n k | |λ m k | n k -1 j=0 λ m k γ j , |G ′ Γ (γ n k )| ≍ 1 |γ n k | n k -1 j=0 γ n k γ j .
For the sake of brevity, we denote n := n k and m := m k . Identity (3.1) becomes

|e m (λ m )| |f γn (λ m )| |γ n | |λ m | |e m (γ n )| n-1 j=0 λ m γ j γ j γ n .
Therefore,

e m (λ m ) e m (γ n ) λ m γ n n-1 ⇐⇒ λ m γ n n-m-1 1. (3.2)
Note that we can suppose

|λ m | ≤ |γ n | and, hence, m = n + δ ′ n + δ n,m , where (δ ′ n ) is a se- quence tending to infinity δ ′ n = Const(β)δ n /(1 + n) 1 β -1 and (δ n,m
) is a bounded negative sequence such that δ n,m ≤ -1 (otherwise we replace m by m -m ′ for a suitable integer m ′ ). Thus,

log λ m γ n = u m -(u n + δ n ) = b β (1 + β)(u n + δ n ) β + δ n,m 1 β -(u n + δ n ) = 1 β(β + 1) + o(1) δ n,m (u n + δ n ) 1-β . Since n -m -1 = n + δ ′ n -m -δ ′ n -1 = -δ n,m -δ n -1, (3.2) becomes 1 -(δ n + δ n,m + 1) 1 β(β + 1) + o(1) δ n,m (u n + δ n ) 1-β ≥ c β δ n u 1-β n ,
which is impossible.

Proof of (3). Recall first that the matrix of the coefficients

|A n,m | = G Γ (λ m ) G ′ Γ (γ n )(λ m -γ n ) k γn ϕ,2 k λm ϕ,2
defines a bounded operator on ℓ 2 . From the proof of the first part we have

|A n,m | exp - |u m+i -u n | 2 + c β u 1-β n -u 1-β m+i ∓ m+i-1 k=n+1 δ k .
Assume that for every N ≥ 1, we have

∆ N := lim sup n 1 u n+N -u n n+N k=n+1 δ k = 1 2 + ε N ,
for a nonnegative sequence (ε N ). For every N ≥ 1 there exists an integer n N (sufficiently large) such that

n N +N k=n N +1 δ k ≥ u n N +N -u n N 2 + ε N u n N +N -u n N 2 .
• Assume that there exists a subsequence (N l ) such that

n l +N l k=n l +1 δ k > 0 (n l is the integer n N l ). We get |A n l +N l ,n l | exp - |u n l -u n l +N l | 2 + c β u 1-β n l +N l -u 1-β n l + n l +N l k=n l +1 δ k exp c β u 1-β n l +N l -u 1-β n l .
This implies that (A n l +N l ,n l ) is unbounded (0 < β < 1) and, hence, the matrix (A n,m ) cannot represent a bounded operator on ℓ 2 .

• There exists N 0 ≥ 1 such that for every N ≥ N 0 we have

n l +N l k=n l +1
δ k < 0. For every n we have

|A n,n+N | exp - u n+N -u n 2 -c β u 1-β n+N -u 1-β n - n+N k=n+1 δ k .
If (ε N ) contains a subsequence (ε N ) N ∈J which is bounded below by some ε > 0, where J is an infinite subset of N. Then for N ∈ J, we have

|A n N ,n N +N | exp -c β u 1-β n N +N -u 1-β n N + ε u n N +N -u n N 2 .
This ensures that (A n N ,n N +N ) tends to infinity. Therefore, the matrix A cannot define a bounded operator on ℓ 2 . Suppose now that (ε N ) converges to zero. Simple computations " =⇒ " Suppose now that Γ ∪ {γ * } is a complete interpolating set for F ∞ ϕ . To prove that K Γ is a Riesz basis for F 2 ϕ , it suffices to verify that Γ satisfies conditions (1)-( 3) of Theorem 1.1. First Γ is d-separated because every interpolating sequence for F ∞ ϕ β is also an interpolating sequence for F ∞ ϕ 1 . Furthermore, remark that

|B n,m | ≍ e ϕ(γn)-ϕ(λm) |λ m | |γ n | G Γ (λ m ) G ′ Γ (γ n )(λ m -γ n ) (log γ n ) 1-β 2 (log λ n ) 1-β 4
|A n,m |.

Arguing as in the proof of Theorem 1.1, if (δ n ) contains a subsequence (δ n k ) such that δ n k /(1 + n k ) 1 β -1 is unbounded, then (A n,m ) is unbounded and, consequently, (B n,m ) is unbounded too. Thus (2) holds. Suppose now that ∆ N = 1 2 + ε N , for a nonnegative sequence (ε N ). Again as in the proof of Theorem 1.1, the sequence |A n k +N k ,n k |+|A n k ,n k +N k | is unbounded and, hence, (B n,m ) is unbounded too. This proves (3) and completes the proof.

Final remarks

The following remark shows that the superior limit in condition (3) in Theorem 1.1 can be replaced by a supremum. This shows that in the case β = 1, Theorem 1.1 and [5, Theorem 1.1] are equivalent. Consequently, K Γ is also a Riesz basis for F 2 ϕ .

Lemma 2 . 4 . 1 β

 241 Let Γ = {γ n } n be a sequence defined in(2.4). We write γ n = λ n e δn e iθn , where δ n , θ n ∈ R, and suppose that|γ n | ≤ |γ n+1 | and |δ n | = O (1 + n) -1 .The infinite product G Γ converges on every compact set of C and for every small ε > 0 there exists a positive constant C such that 1 C

. 5 )

 5 Now for every N ∈ N there exist two positive integers p and r such that m = pN + r and 0 r < N. Set u k := log |λ k |. We have

where F Γ = ( 1 - 1 e

 11 z/γ * )G Γ . Let us verify that L v ∈ F ∞ ϕ . According to Lemma 2.7, Λ ∪ {λ * } is a complete interpolating set for F ∞ ϕ and consequentlyL v ϕ,∞ ≍ sup m≥--ϕ(λm) n≥0 v n F Γ (λ m ) F ′ Γ (γ n )(λ m -γ n ) = sup m≥-1 n≥0v n e -ϕ(γn) B n,m , where|B n,m | = e ϕ(γn)-ϕ(λm) F Γ (λ m ) F ′ Γ (γ n )(λ m -γ n ) ≍ e ϕ(γn)-ϕ(λm) |λ m | |γ n | G Γ (λ m ) G ′ Γ (γ n )(λ m -γ n ) ≍ |A n,m | e o(1)|um-un| , and u n = log |λ n |. The estimates on the matrix A = (A n,m ) imply that L v belongs to F ∞ ϕ .

Remark 5 . 1 .

 51 Let (δ n ) be a sequence of real numbers such that δ n /(1 + n)

1 β - 1 1 2β 1 N 1 N 1 β

 111111 ∈ ℓ ∞ . Let N be a positive integer and δ = Γ = {γ n } be a d-separated sequence of complex numbers ordered in such a way that |γ n | ≤ |γ n+1 |. Set γ n = λ n e δn e iθn . Suppose that∆ N := lim sup n Let ε ∈ (0, δ -∆ N ). There exists m ≥ 0 such that sup n≥m ≤ ∆ N + ε < δ.Thus, K Γ is a Riesz basis for F 2 ϕ , whereΓ := e (n+1 1+β ): 0 ≤ n ≤ m -1 ∪ {γ n ∈ Γ : n ≥ m} .

  Hence, D d (z, δ) is comparable to D(z, c δ |z|) with a suitable constant c δ depending on δ. Thus Σ is d-separated if and only if there exists c > 0 such that the Euclidean disks

D(σ, c|σ|), σ ∈ Σ, are disjoint. Our first main result in this paper is the following Theorem 1.1. Let ϕ(r) = log + r β+1 where 0 < β < 1, and let Γ = {γ n : n ≥ 0} be a sequence of complex numbers such that |γ n | |γ n+1 |. We write γ n = exp 1+n 1+β 1 β e δn e iθn , for every n ≥ 0, where (δ n ) n and (θ n ) are real sequences. Then K Γ = {k γ } γ∈Γ is a Riesz basis for F 2 ϕ if and only if the following three conditions hold:

The research of the first author is partially supported by by the project ANR-18-CE40-0035 and by the Joint French-Russian Research Project PRC CNRS/RFBR,2017-2019

The second author is supported by CNRST Grant 84UM52016.

Choose

Thus (A n N ,n N +N ) N is unbounded and, hence, the matrix A = (A n,m ) cannot define a bounded operator on ℓ 2 . This completes the proof.

4. Proof of Theorem 1.2

" ⇐= " Suppose that K Γ is a Riesz basis for F 2 ϕ . Then Γ satisfies conditions (1), ( 2) and (3) of Theorem 1.1.

First, the sequence Γ = Γ ∪ {γ * } is a uniqueness set for F ∞ ϕ . Indeed, if f is a function from F ∞ ϕ that vanishes on Γ, then f = (1 -z/γ * )G Γ h, for an entire function h. Our estimates of G Γ imply that

Since ∆ N < 1/2, the function h must be identically zero.

Let us prove that Γ = {γ n } n≥-1 is an interpolating sequence for F ∞ ϕ , where γ -1 = γ * . For this, let v = (v n ) be a sequence such that v ϕ,∞, Γ < ∞ and consider the entire function


Andrzej Duda
email: andrzej.duda@imag.fr

Martin Heusse
email: heussemartin.heusse@imag.fr

Spatial Issues in Modeling LoRaWAN Capacity

Keywords: Networks → Network performance modeling, Very longrange networks, Sensor networks LoRaWAN, Packet Delivery Ratio, capacity, modeling, inhomogeneous density

whether they are published or not. The documents may come

INTRODUCTION

LoRa [START_REF] Lora | A Technical Overview of LoRa and LoRaWAN[END_REF] is a recent example of a Low Power Wide Area Network (LPWAN) that can provide wireless connectivity to a large number of IoT devices over long distances. It defines a physical layer based on the Chirp Spread Spectrum (CSS) modulation [START_REF] Berni | On the Utility of Chirp Modulation for Digital Signaling[END_REF] and a simple channel access method similar to ALOHA called LoRaWAN [START_REF] Sornin | LoRaWAN 1.1 Specification[END_REF]. The LoRa CSS modulation results in good sensitivity enabling transmissions over long distances: a range of several kilometers outdoors and hundreds of meters indoors.

A LoRa end device can vary several transmission parameters: channel bandwidth (BW), transmission power (TP), coding rate (CR), and spreading factor (SF). The achievable data rates depend on some of the parameters: a higher bit rate results from lower SF (at the cost of a shorter range), higher BW, and CR of 4/5. The bit rates range from 293 b/s to 11 kb/s, the low bit rate resulting in long transmission times: 2.466 s for sending 59 B at SF12.

In addition to the physical layer parameters, the LoRa performance in terms of the Packet Delivery Ratio (PDR) and scalability to a large number of devices strongly depend on the LoRaWAN access method similar to unslotted ALOHA: a device may wake up at any instant and start transmitting a packet without testing for on-going transmissions.

The previous analytical studies investigated LoRa performance for an increasing number of nodes around a gateway with an important assumption: the density of nodes within the gateway range is uniform [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF][START_REF] Li | 2D Time-Frequency Interference Modelling Using Stochastic Geometry for Performance Evaluation in Low-Power Wide-Area Networks[END_REF][START_REF] Waret | LoRa Throughput Analysis with Imperfect Spreading Factor Orthogonality[END_REF][START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF][START_REF] Heusse | How Many Sensor Nodes Fit in a LoRAWAN Cell? submitted for publication[END_REF][START_REF] Caillouet | Optimal SF Allocation in LoRaWAN Considering Physical Capture and Imperfect Orthogonality[END_REF]. However, such an assumption is not realistic for cells covering large areas of several square kilometers for two reasons.

First, measurement studies in cellular networks showed that spatial traffic distribution is highly non-uniform across different cells [START_REF] Gotzner | Spatial Traffic Distribution in Cellular Networks[END_REF][START_REF] Lee | Spatial Modeling of the Traffic Density in Cellular Networks[END_REF][START_REF] Mirahsan | Measuring the Spatial Heterogeneity of Outdoor Users in Wireless Cellular Networks Based on Open Urban Maps[END_REF][START_REF] Wang | Spatio-Temporal Analysis and Prediction of Cellular Traffic in Metropolis[END_REF][START_REF] Taufique | Planning Wireless Cellular Networks of Future: Outlook, Challenges and Opportunities[END_REF] with complex patterns that include hot spots with a high density and other less dense places. For instance, Lee et al. [START_REF] Lee | Spatial Modeling of the Traffic Density in Cellular Networks[END_REF] demonstrated based on traffic measurements that the spatial distribution of the traffic density across different cells can be approximated by the log-normal or Weibull distributions depending on time and space. As cells in cellular networks can be smaller than LPWANs (target range of several kilometers), we expect that spatial traffic distribution in LoRa networks will also be highly non-uniform. We can also consider that the distribution of devices follows in fact the same pattern as we can usually observe for population and building densities in cities: apart from the saturated downtown, the density decreases with the distance from the centerit is much higher downtown than in suburbs. The deployment of gateways by LPWAN operators will probably also follow the same strategy as in cellular networks-place networks close to potential users and create hot spots near high density areas.

Second, there is a discouraging effect when placing a device far from the gateway in a LoRaWAN cell: the device needs to use large SF (e.g., SF11 or SF12) to get its transmissions through, which means long transmission times, so increased contention (more collisions) and higher energy consumption. For instance, a device using SF11 will roughly consume 10 times more energy for a transmission than when using SF7. Moreover, if we consider an equidistant distribution of spreading factors, which is the most popular spatial model adopted in the analyses with concentric annuli spaced at 1 km intervals, there are more devices with larger SF such as SF11 and SF12 than devices in the area of SF7. There are also no devices outside the zone of SF12, a kind of a disruptive irregularity difficult to observe in real world deployments.

We claim that uniform node density is highly unlikely for large cells so we propose to analyze LoRa performance under a more realistic assumption: the density of nodes decreases with the inverse square of the distance to the gateway. The inverse-square law is common in physics stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. For instance, radio wave transmission in free space follows an inverse square law for power density. We consider that such a spatial model corresponds better to real-world LoRaWAN deployments than the models based on the constant density.

In this paper, we use the model by Georgiou and Raza [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF] to find the Packet Delivery Ratio (PDR) for inhomogeneous spatial distributions of devices around a gateway and obtain the number of devices that benefit from a given level of PDR. Unlike Georgiou and Raza, we adopt a realistic assumption about generated traffic: nodes send packets at the smallest possible time interval determined by the largest duty cycle possible at SF12. Under this assumption, devices generate packets in a way defined by a sensing application unlike the model by Georgiou and Raza that assumes 1% duty cycle of all devices regardless of SF, which means that a device that changes from SF8 to SF7 starts sending packets twice more often. We also modify the model expression for traffic intensity so it correctly reflects the unslotted ALOHA behavior (instead if slotted ALOHA).

In the rest of the paper, we describe the basics of LoRa networks (Section 2) and present the PDR model (Section 3). In Section 4, we analyze LoRaWAN capacity in terms of PDR for different spatial configurations. Finally, we discuss related work (Section 5) and draw some conclusions (Section 6).

LORAWAN BASICS

We briefly recall the basic characteristics of LoRaWAN.

Devices can control the physical layer of LoRaWAN through the following parameters [START_REF]SX1272/73 -860 MHz to 1020 MHz Low Power Long Range Transceiver[END_REF]:

• Bandwidth (BW): it is the range of transmission frequencies.

We can configure the bandwidth between 7.8 kHz and 500 kHz. A larger bandwidth allows for a higher data rate, but results in lower sensitivity. traffic model where all nodes have the same traffic tensity, in which case it is relevant to express the cell apacity in terms of number of nodes. This assumption is e most realistic, since traffic generation is determined y the application, for example periodic sensing or etering, regardless of the distance to the gateway. n SF allocation to improve and even out the PDR roughout the cell. We optimize the SF boundaries to alance the opposite effects of attenuation and collisions. wo last factors are antagonistic because switching to a F results in more robust transmissions but with longer n, which increases contention. We will see that it is wise ol the number of nodes using higher SFs because they

II. PDR IN A LORAWAN CELL FOR HOMOGENEOUS

TRAFFIC

In a LoRaWAN cell, a frame may be lost for two reasons (and maybe both): i) the SNR is below the reception threshold or ii) a collision occurs and the signal is not strong enough relatively to the interference.

We restrict our analysis to the basic LoRa CSS modulations with BW of 125 kHz and SF in 12, 11,. . . 7, which corresponds to data rates DRj, with j = 0, 1, . . . 5, and SF = 12 j.

A. Channel model

We use the Okumura-Hata model for path loss attenuation (also used by Bankov [START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF] and Magrin [START_REF] Heusse | How Many Sensor Nodes Fit in a LoRAWAN Cell? submitted for publication[END_REF]), using the suburban environment variant with an antenna height of 15 m. This empirical model is slightly less favorable than adopting an arbitrary path loss exponent as in most of the previous work we cite. We have chosen the Okumura-Hata model because it is more realistic, but the results are qualitatively similar. We consider a GW-side antenna gain of 6 dB which compensates for a receiver noise factor of 6 dB. LoRaWAN defines an access method similar to ALOHA: a device wakes up at any instant and sends a packet right away. It then wakes up after a delay to receive a downlink frame, if the transmitted frame was of the Confirmed Data type. Collisions occur when two devices transmit packets at the same time. However, unlike in the pure ALOHA scheme, a receiver can correctly receive a frame in presence of interfering signals, because the LoRa physical layer is robust enough to resist significant interference. The capture effect has important impact on LoRa performance. Taking into account the capture effect, Haxhibeqiri et al. [START_REF] Haxhibeqiri | LoRa Scalability: A Simulation Model Based on Interference Measurements[END_REF] showed that when the number devices increases to 1000 per gateway, the packet loss rate only increases to 32%, which is low compared to 90% in pure ALOHA for the same load.

In the following sections, we assume LoRa modulations with 125 kHz bandwidth and spreading factors SF{12 -j} that result in data rates DRj, with j = 0, 1, . . . 5 (see Table 1).

MODEL FOR SUCCESSFUL PACKET DELIVERY

We consider a LoRaWAN cell in which devices choose SF (so the data rate) based on the distance to the gateway, which corresponds to the annuli view presented in Figure 1. l j denotes the distance to the farthest device that uses SF{12 -j} so its data rate is DRj. l 0 is the maximum transmission range. We assume that devices are located at random in the annulus at l j according to a Poisson Point Process (PPP) with intensity ρ j . Inside the annulus, the spatial density is constant, so the number of nodes using SF{12 -j} and data rate DRj is proportional to ρ j and the surface of the annulus between l j+1 and l j . Table 2 presents the values of l j for the equidistant SF configuration with maximum range l 0 of 6 km and a homogeneous density (Mahmood et al. considered such boundaries [START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF] and Georgiou and Raza provided numerical examples for the double maximum range: l 0 of 12 km [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF]). The number of devices in a disk or annulus is proportional to surface S j so it increases with the distance. Note that energy consumption of a node in annulus j is proportional to airtime τ j , which attains large values for high SF.

We give below the details of the model [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF] to compute PDR based on the notation in Table 3.

Provided that there is no collision, a frame transmission succeeds as long as the SNR at the receiver for this transmission is above q j , the minimum SNR for the corresponding spreading factor [START_REF] Goursaud | Dedicated networks for IoT : PHY / MAC state of the art and challenges[END_REF]. We consider a Rayleigh channel, so that the received signal power is affected by a multiplicative random variable with an exponential distribution of unit mean (and standard deviation). Recent measurements confirm the validity of the hypothesis for LoRa transmissions [START_REF] Attia | Experimental Characterization of LoRaWAN Link Quality[END_REF]. Thus, the signal power depends on the distance and the Rayleigh fading gain, whereas the noise power is the constant thermal noise for a 125 kHz-wide band: N = -123 dBm. We consider the maximum transmission power of P = 14 dBm.

Thus, the probability of successful transmission at distance l j with data rate DRj is [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF]:

H (l j ) = exp - Nq j Pд(l j ) , (1)
where д(l j ) is the average channel gain at distance l j . We use the Okumura-Hata model for path loss attenuation.

We use an approximate expression for the success probability in presence of concurrent traffic [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF] reflecting the behavior of unslotted ALOHA with capture:

Q 1 (l j , v j ) = 2 exp(-2v j )l η j (η + 2)S j π 2v j l η+2 j + l η j (2(η + 2)S j -2πv j l 2 j ) , ( 2
)
where η is the path loss exponent (we assume η = 4 in the numerical examples below, which gives relative attenuation values that closely match those of the propagation model). Note that we double the traffic intensity in the expression by Georgiou and Raza to reflect correctly the behavior of unslotted ALOHA.

Finally, PDR of nodes in annulus j is the following:

PDR(l j , v j ) = H (l j ) × Q 1 (l j , v j ) (3)
To use these expressions, we need to define traffic intensity v j . We consider that nodes generate traffic according to a Poisson process of intensity λ t . For SF = 12 -j and n j , the number of contending nodes in annulus j, traffic occupancy is

v j = n j τ j λ t ( 4
)
in Erlang and the number of nodes in annulus j is:

n j = S j ρ j n j S j ρ j (5)
We set λ t to the traffic intensity of nodes operating at DR0 and SF12 at their maximal duty cycle and using 59 B packets, the maximum size at this rate. The duty cycle depends on the frequency band: LoRa devices have to limit their occupation of each frequency band to 1% of time with 3 to 5 frequency channels in each band in Europe. The airtime of maximum size packets at DR0 corresponds to 2.47 s so they can be sent every 747 s to achieve 0.33% duty cycle per frequency channel. Thus,

λ t = 1 747 s . ( 6
)

PROBABILITY OF SUCCESSFUL PACKET DELIVERY FOR DIFFERENT SPATIAL CONFIGURATIONS

In this section, we analyze PDR for different spatial configurations.

SF Allocation Strategies

There are several ways of allocating SF to nodes, which means finding the annuli boundaries l j (we restrict our analysis to nonoverlapping allocations in which all nodes in a given annulus use the same value of SF) [START_REF] Li | 2D Time-Frequency Interference Modelling Using Stochastic Geometry for Performance Evaluation in Low-Power Wide-Area Networks[END_REF][START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF][START_REF] Lim | Spreading Factor Allocation for Massive Connectivity in LoRa Systems[END_REF]:

(1) Equidistant SF allocation with l j+1 -l j = l 0 /6 (also called equal-interval-based [START_REF] Lim | Spreading Factor Allocation for Massive Connectivity in LoRa Systems[END_REF]). (2) Equal-area-based SF allocation with l j = l 0 √ j/6 [START_REF] Lim | Spreading Factor Allocation for Massive Connectivity in LoRa Systems[END_REF].

(3) SNR-based SF allocation with l j = {d : H (d ) ≥ θ } (also called path loss based [START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF]). ( 4) PDR-based SF allocation with l j = {d : SF allocation and showed that it performs the best with the equalarea-based SF allocation ranked second [START_REF] Lim | Spreading Factor Allocation for Massive Connectivity in LoRa Systems[END_REF]. However, they did not take into account the capture effect important to evaluate the probability of success reception under the LoRaWAN access method. The analyses assumed a constant node density within the range. Note that the complexity of computing SF allocations increases with the order of allocations presented above: equidistant and equalarea-based allocations only depend on the distance, the SNR-based allocation requires solving Eq. 1 numerically, and PDR-based allocations lead to a nonlinear optimization problem.

H (d ) × Q 1 (d ) ≥ θ }.
Q 1 (Aloha) H × Q 1 (PDR) v j (
The SNR-based allocation resulting from solving Eq. 1 numerically for each j gives the values of l j presented in Table 4 for three values of threshold θ : 90%, 95%, and 99%. We can observe that increasing the threshold results in smaller cells.

The SNR-based and PDR-based allocations can be implemented in a similar way to the Adaptive Data Rate (ADR) algorithm defined in LoRaWAN. In ADR, the gateway estimates the average SNR level for the last 20 packets of a node. It then chooses SF and TP suitable for the given level of SNR, while keeping a 5 dB margin, and sends the parameters in the LinkADRReq frame to the node. In a similar way, the gateway can estimate PDR and choose the right parameters for the node. Nevertheless, such allocations require sending downlink messages, whereas the gateways have very limited transmission capacity.

Inhomogeneous Node Density

We have already discussed the reasons for which assuming constant node density ρ for all annuli is not realistic. Moreover, the allocation of SF in LoRaWAN strongly impacts energy consumption so that far devices that need to use high values of SF will consume much

S j /π × ρ (l j )
1.40 0.44 0.44 0.44 0.37 0.37 more energy than devices with low SF, which will discourage the placement of nodes far from the gateway. There is also another adverse effect of using high values of SF: increased transmission times lead to more contention and collisions, thus affecting the probability of successful packet reception.

To take into account these considerations, we adopt a more realistic model for the spatial distribution of nodes based on the inverse-square law for node density:

ρ j ρ j-1 = l 2 j-1 l 2 j ( 7
)
Table 4 also presents node density ρ j for each annulus based on this relation and the value proportional to the number of devices. To see the effect of the decreasing node density, we can observe that such a distribution favors the annuli close to the gateway with a higher number of devices using SF7 and results in less devices with SF 11 and SF12.

Figure 2 compares the spatial models-Figures 2a and2b visualize the distribution of nodes for homogeneous and inverse squared density (n = 1200 nodes generated with a Monte Carlo method, randomly placed in equidistant annuli). We can observe that the number of nodes in the SF12 annulus is much lower for inverse squared density: 108 vs. 367 (see Figures 2c and2d).

LoRaWAN capacity for different spatial configurations

In this section, we present figures with PDR computed according to the model in Section 3. The total number of nodes (1200, 1700, and 2100) for the figures was chosen so that they give the maximal number of nodes that benefit from PDR > 80% for the respective SNR thresholds (we discuss this aspect at the end of this Section). 2c shows PDR and its components: channel attenuation H and Q 1 , the success probability under ALOHA with capture. We can observe that Q 1 goes down at each frontier because of increased traffic v j that comes from an increased number of nodes. As the annuli surface increases with the distance, the homogeneous node density results in the number of nodes in annuli growing with the distance and attaining 367 for SF12 with only 33 nodes using SF7. The estimated number of nodes that benefit from PDR > 80% is 300. The spatial model assumes that there are no nodes outside the last annuli. Figures 2b and2d present the distribution of nodes and PDR when the density of nodes is inversely proportional to the square of distance. PDR goes down with the distance much slowly than in Figure 2c because there are less nodes in higher SF annuli: 352 nodes using SF7 and 108 for SF12. The estimated number of nodes that benefit from PDR > 80% is 809. We can observe that the number of nodes in each annulus decreases with the distance.

This basic example shows the importance of the spatial model for evaluating LoRaWAN capacity: just changing the spatial distribution of nodes raises the number of nodes with good PDR from 300 to 809. So, the choice of the spatial model may result in misleading results on LoRaWAN capacity and scalability. For H (l j ) ≥ 90% threshold, the range of the cell is relatively large with 5.3 km. 787 nodes benefit from PDR > 80% out of 1200.

SNR-based SF frontiers.

Figure 4 and Figure 5 present the same data for H (l j ) ≥ 95% and H (l j ) ≥ 99% thresholds and the total number of nodes n = 1700 and n = 2100, respectively. The increased value of the threshold results in smaller cells (2.82 km for H (l j ) ≥ 99%) because of the dependance of funtion H on the distance. The number of nodes that benefit from PDR > 80% is 1115 and 1377, respectively.

We can observe that the assumption of the inhomogeneous density results in an interesting effect: smaller cells can provide good PDR for an increased number of nodes. It evokes "cell breathing" in cellular networks in which heavily loaded cells decrease in size. Note that the nodes benefiting from PDR > 80%, use low SF values: SF7, SF8, and SF9, which also means that their energy consumption stays low.

l j [m] success probability % H (attenuation) Q 1 (Aloha) H × Q 1 (PDR) v j (
l j [m] success probability % H (attenuation) Q 1 (Aloha) H × Q 1 (PDR) v j (traffic) 0
Figure 6 presents the results for a homogeneous density to compare with Figure 5: there are 776 nodes with PDR > 80%.

l j [m] success probability % H (attenuation) Q 1 (Aloha) H × Q 1 (PDR) v j (
The optimization starts with the maximal range l 0 that we set to the thresholds for two extreme SNR-based allocations: 5.30 km for 90% and 2.82 km for 99% (we skip the intermediate threshold of 95%), and looks for more uniform distribution of PDR values across SF annuli with the maxmin objective. We still assume the density of nodes inversely proportional to the square of distance.

Figure 7 presents PDR in the first case of a large cell: l 0 = 5.30 km for 90% for the same total number of nodes as in Figure 3. Compared to Figure 3, the minimum value of PDR is higher (PDR > 60%), however, there are less nodes that benefit from good PDR > 80% (460 vs. 787).

Similarly, Figure 8 shows PDR for the small cell: 2.82 km for 99%. When comparing Figure 5, we can see a similar effectthe minimal value of PDR is high (almost reaching 80%), but still the number of nodes that benefit from good PDR > 80% is low, which shows that the call has attained its capacity. If we lower the total number of nodes in the network to 1500, the maxmin PDR allocation gives very good results: Figure 9 shows that all nodes achieve PDR > 80%. Note that the allocation exhibits strong asymmetry-it favors lower SF over high SF: there are 739 nodes in the SF7 annuli compared to only 8 nodes with SF12. As H is as high as 99%, the most important factor for PDR is Q 1 , which depends on the traffic load almost constant for all annuli with low SF (see Figure 9) in this allocation. Such an allocation has also an advantage of low overall energy consumption as only a few nodes use SF11 and SF12, expensive in terms of energy.

Scalability.

Figure 10 shows the number of nodes that benefit from good PDR > 80% in function of the total number of nodes in the network for the small cell of the 2.82 km range. We fix the allocation of annuli frontiers l j to the optimal allocation presented in Figure 9 and we vary the total number of nodes. At the beginning, the number of nodes with PDR > 80% increases and achieves the maximum for n = 1500. Then, the number of nodes with PDR > 80% decreases because PDR begins to drop below 80%, which means that the network has attained its capacity.

A question remains: how does the SNR allocation perform compared to the PDR based one? Figure 11 shows the corresponding data for the SNR allocation. It achieves a lower maximum number of nodes with PDR > 80% (1377 nodes for n = 2100, see also Figure 5), but it can handle a slightly larger total number of nodes.

RELATED WORK

We briefly review previous work on modeling LoRa capacity and inhomogeneous spatial models.

LoRa capacity models

Georgiou and Raza [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF] provided a stochastic geometry framework for modeling the performance of a single gateway LoRa network. They showed that the coverage probability drops exponentially as the number of contending devices grows. Their model assumes that the airtime is filled up-nodes use the shortest interval between packet transmissions allowed at given SF, which means that switching to lower SF results in generating twice as much traffic. Moreover, the model of the access method corresponds to slotted ALOHA. In this paper, we have used the model with the modifications concerning the intensity of packet generation and the expression for the success probability reflecting the behavior of unslotted ALOHA.

Mahmood et al. [START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF] proposed an analytical model of a single-cell LoRa system that takes into account interference among transmissions over the same SF (co-SF) as well as different SFs (inter-SF). They derived the signal-to-interference ratio (SIR) distributions for several interference conditions. Due to imperfect orthogonality, inter-SF interference exposes the network for additional 15% coverage loss for a small number of concurrently transmitting enddevices.

Li et al. [START_REF] Li | 2D Time-Frequency Interference Modelling Using Stochastic Geometry for Performance Evaluation in Low-Power Wide-Area Networks[END_REF] analyzed interference in the time-frequency domain using a stochastic geometry model assuming transmissions as patterns on a two-dimensional plane to quantify the capture effect. They use the model to analyze LoRaWAN by characterizing the outage probability and throughput.

Based on a simple model for collisions and capture effect, Caillouet et al. [START_REF] Caillouet | Optimal SF Allocation in LoRaWAN Considering Physical Capture and Imperfect Orthogonality[END_REF] introduced a theoretical framework for maximizing the LoRaWAN capacity in terms of the number of end nodes.

All the presented models assumed a homogeneous node density around a gateway.

Inhomogeneous spatial models

Gotzner and Rathgeber [START_REF] Gotzner | Spatial Traffic Distribution in Cellular Networks[END_REF] challenged the homogeneous assumption in spectrum frequency analysis and proposed to model the spatial inhomogeneity of real cellular traffic with log-normal distributions.

Lee et al. [START_REF] Lee | Spatial Modeling of the Traffic Density in Cellular Networks[END_REF] observed that modeling and simulation of a cellular network typically assume the target area divided into regular hexagonal cells and a uniform distribution of mobile devices scattered in each cell. In reality, the spatial traffic distribution is highly non-uniform across different cells, which requires adequate spatial traffic models. They reported on traffic measurements collected from commercial cellular networks and demonstrated that the spatial distribution of the traffic density (the traffic load per unit area) can be approximated by the log-normal or Weibull distribution depending on time and space.

Mirahsan et al. [START_REF] Mirahsan | Measuring the Spatial Heterogeneity of Outdoor Users in Wireless Cellular Networks Based on Open Urban Maps[END_REF] used maps of Paris, France to study the spatial traffic heterogeneity of outdoor users in dense areas of the city center. They found that the statistical distribution of spatial metrics is close to Weibull. Their results show that the building topology in a city imposes a significant degree of heterogeneity on the spatial distribution of the wireless traffic.

Taufique et al. [START_REF] Taufique | Planning Wireless Cellular Networks of Future: Outlook, Challenges and Opportunities[END_REF] investigated the problem of planning future cellular networks. They noticed that the cell size increasingly adapts to the spatial traffic variation. Instead of having the same cell size throughout, areas with low traffic density can have larger cells compared to areas with high traffic density, resulting in energy and cost savings. As planning future cellular networks faces heterogeneous and ultra dense networks, the issue is to find the optimal base station placement jointly for macrocells and small cells in a non uniform user density scenario. They showed an example of such a deployment for a Gaussian spatial user distribution.

Wang et al. [START_REF] Wang | Spatio-Temporal Analysis and Prediction of Cellular Traffic in Metropolis[END_REF] characterized temporal and spatial dynamics in cellular traffic through a big cellular usage dataset covering 1.5 million users and 5,929 cell towers in a major city of China. Their results reveal highly non-uniform spatial distribution of the traffic density.

CONCLUSIONS

In this paper, we have shown that adopting inhomogeneous spatial node distribution leads to much different results on LoRaWAN capacity than that reported previously. The existing measurement studies of the traffic density in cellular networks showed high diversity of the node density in urban settings. We expect that LoRaWAN networks will follow the same deployment pattern with the placement of gateways close to high density areas.

We have used the model by Georgiou and Raza [START_REF] Georgiou | Low Power Wide Area Network Analysis: Can LoRa Scale?[END_REF] to analyze the capacity of a LoRaWAN cell for various types of SF allocations: equidistant, SNR-based, and PDR-based. We can draw several conclusions from the numerical results presented in this paper:

• For a required PDR level and a target communication range, we can find an allocation of annuli l j that results in the maximal number of nodes that benefit from the PDR level. • There is a natural trend towards configurations composed of smaller cells that concentrate nodes close to the gateway.

In this way, nodes benefit from low SF, which also means lower energy consumption. • To provide the required PDR level to more nodes, we need to consider multiple gateways that will increase the overall capacity while maintaining moderate energy consumption. In future work, we plan to explore a model in which the density of nodes is a continuous distribution in function of the distance from the gateway, which may better reflect realistic deployment scenarios.

We also want to develop models for capacity prediction in case of multiple gateways.

Figure 1 :

 1 Figure 1: Annuli of SF allocation around the gateway

Figure 1 :

 1 Figure 1: Annuli of SF configurations around a gateway

  Mahmood et al. showed that equidistant and SNR-based SF allocations performed the best, nevertheless, they did not consider the PDR-based allocation[START_REF] Mahmood | Scalability analysis of a lora network under imperfect orthogonality[END_REF]. Lim and Han analyzed the PDR-based

Figure 2 :

 2 Figure 2: Comparison of spatial models, n = 1200.

Figure 3 presents

 3 PDR in SNR-based SF allocation for the inverse squared density and n = 1200 nodes.

Figure 5 :

 5 Figure 5: Inverse squared density and SNR SF annuli for H (l j ) ≥ 99%, n = 2100. 1377 nodes with PDR > 80%.

Figure 6 :

 6 Figure 6: Homogeneous density and SNR SF annuli for H (l j ) ≥ 99%, n = 2100. 776 nodes with PDR > 80%.

Figure 8 :

 8 Figure 8: Inverse squared density and PDR SF annuli for H (l j ) ≥ 99%, n = 2100. No nodes with PDR > 80%.

Figure 9 :

 9 Figure 9: Inverse squared density and PDR SF annuli for H (l j ) ≥ 99%, n = 1500. All 1500 nodes with PDR > 80%.

Figure 10 :

 10 Figure 10: Number of nodes with PDR > 80% in function of the total number of nodes in the network. Optimal allocation of annuli frontiers l j as in Figure 9.

Figure 11 :

 11 Figure 11: Number of nodes with PDR > 80% in function of the total number of nodes in the network. SNR allocation of annuli frontiers l j .

Table 1 presents

 1 SFj, data rate DRj, Signal-to-Noise Ratio (SNR) limit, airtime τ j , and s max , the maximum payload size. τ j denotes

	practice)

• Spreading Factor (SF) characterizes the number of bits carried by a chirp: SF bits are mapped to one of N = 2 SF possible frequency shifts in a chirp. SF varies between 6

(7 in

Table 1 :

 1 LoRa parameters for BW of 125 kHz.

	SFj SNR limit	Airtime	Bit rate s max
		q j	τ j	DRj
	7	-7.5 dB 102.7 ms DR5: 5469 b/s 230 B
	8	-10 dB 184.8 ms DR4: 3125 b/s 230 B
	9	-12.5 dB 328.7 ms DR3: 1758 b/s 123 B
	10	-15 dB 616.5 ms DR2: 977 b/s	59 B
	11	-17.5 dB 1315 ms DR1: 537 b/s	59 B
	12	-20 dB 2466 ms DR0: 293 b/s	59 B

Table I :

 I Notations

	Spatial density of nodes Traffic generation intensity Frame transmission duration at Data Rate DRj Distance of farthest node using DRj Traffic occupancy (in Erlang) at DRj Average channel gain at distance d SNR threshold for DRj Transmission power, in-band noise power Success probability, due to attenuation, fading, thermal noise Success probability, due to collisions	⇢ t ⌧ j l j v j g (d) q j P , N H Q

Table 2 :

 2 Equidistant SF boundaries [km], S j /π [km 2 ]: value proportional to the number of devices (constant node density ρ in all annuli).

		SF7 SF8 SF9 SF10 SF11 SF12
		l 5	l 4	l 3	l 2	l 1	l 0
	l j [km] S j /π [km 2 ]	1 1	2 3	3 5	4 7	5 9	6 11

Table 3 :

 3 Notation

	Traffic generation intensity	λ t
	Frame transmission duration at data rate DRj	τ j
	Distance of farthest node using DRj	l j
	Surface of annulus j	S j
	Spatial density of nodes in annulus j	ρ j
	Total number of nodes	n
	Average number of nodes in annulus j	n j
	Traffic occupancy (in Erlang) at DRj	v j
	Average channel gain at distance d	д(d )
	SNR threshold for DRj	q j
	Transmission power, in-band noise power	P, N
	Success probability, due to attenuation, fading	H
	Success probability, due to collisions	Q 1

Table 4 :

 4 SNR-based SF boundaries [km], H (l j ) is the success probability due to attenuation, fading, thermal noise. S j /π [km 2 ]: value proportional to the number of devices. Node density in an annulus based on the inverse-square law so S j /π × ρ (l j ) is proportional to the number of devices.

		SF7 SF8 SF9 SF10 SF11 SF12
		l 5	l 4	l 3	l 2	l 1	l 0
	l j : H (l j ) ≥ 90% 2.23 2.68 3.23 3.89 4.54 5.30 S j /π [km 2 ] 4.96 2.23 3.24 4.69 5.49 7.47
	ρ j	1 0.69 0.48 0.33 0.24 0.18
	S j /π × ρ (l j )	4.96 1.54 1.54 1.54 1.32 1.32
	l j : H (l j ) ≥ 95% 1.84 2.21 2.66 3.20 3.74 4.37 S j /π [km 2 ] 3.38 1.50 2.19 3.17 3.74 5.1
	ρ j	1 0.69 0.48 0.33 0.24 0.18
	S j /π × ρ (l j )	3.38 1.03 1.05 1.05	0.9	0.9
	l j : H (l j ) ≥ 99% 1.18 1.43 1.72 2.07 2.41 2.82 S j /π [km 2 ] 1.40 0.63 0.91 1.33 1.55 2.11
	ρ j	1 0.69 0.48 0.33 0.24 0.18

  Inverse squared density and SNR SF annuli for H (l j ) ≥ 90%, n = 1200. 787 nodes with PDR > 80%.homogeneous density and n = 1200 nodes. The existing studies use this distribution and equidistance frontiers of SF allocation to analyze the LoRa capacity. Figure

		100
	success probability %	20 40 60 80	H (attenuation) Q 1 (Aloha) H × Q 1 (PDR) v j (traffic)
		0
		0		2228	l j [m]	3229	3889	4539	5299
	#nodes in annuli:	487	151	151	151	130	130
	Figure 3:

4.3.1 Equidistant SF frontiers. Figures

2a and 2c

present the spatial distribution of nodes and PDR in equidistant SF annuli for a

  Inverse squared density and SNR SF annuli for H (l j ) ≥ 95%, n = 1700. 1115 nodes with PDR > 80%.

		traffic)
	0		1836	2661	3204	3741	4367
	#nodes in annuli:	690	214	214	214	184	184
	100 Figure 4: 0 20 40 60 80 success probability %	H (attenuation) Q 1 (Aloha) H × Q 1 (PDR) v j (traffic)
	0		l j [m] 1184 1426 1717	2067	2413	2817
	#nodes in annuli:	853	265	265	265	227	227

  Inverse squared density and PDR SF annuli for H (l j ) ≥ 90%, n = 1200. 460 nodes with PDR > 80%.

		traffic)
	0		2646 3193	4082		4759 5232
	#nodes in annuli:	555	174	215	147	96
	100 Figure 7: 0 20 40 60 80 success probability %	H (attenuation) Q 1 (Aloha) H × Q 1 (PDR) v j (traffic)
	0		l j [m] 1465	2155		2598
	#nodes in annuli:	1049	564	327	154

ACKNOWLEDGMENTS

This work has been partially supported by the French Ministry of Research project PERSYVAL-Lab under contract ANR-11-LABX-0025-01.