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Comptes Rendus
Mathématique
Alexei Tsygvintsev
On the existence of generalised syzygies in the planar three-body
problem
Volume 361 (2023), p. 331-335
https://doi.org/10.5802/crmath.409
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 331-335
https://doi.org/10.5802/crmath.409
Dynamical systems / Systèmes dynamiques
On the existence of generalised syzygies in the
planar three-body problem
Sur l’existence des syzygies généralisées dans le problème
plan des trois corps
Alexei Tsygvintsev ∗, a
a ENS de Lyon, UMPA, 46 allée d’Italie, 69364 Lyon Cedex 07, FRANCE
E-mail: alexei.tsygvintsev@ens-lyon.fr (A. Tsygvintsev)
Abstract. We consider the Newtonian planar three-body problem. One defines a generalised syzygy as a
configuration where the three bodies or their velocities become collinear. Assuming that the motion is
bounded and collision-free, we provide a simple sufficient condition for the existence of such configurations.
Our proof is elementary and uses only basic tools from the Sturm–Liouville theory.
Résumé. Nous considérons le problème plan des trois corps. On définit une syzygie généralisée comme une
configuration où les trois corps ou leurs vitesses deviennent colinéaires. En supposant que le mouvement est
borné et sans collision, nous fournissons une condition suffisante pour l’existence de telles configurations.
Nos principaux outils sont élémentaires et basés sur la théorie de Sturm–Liouville.
2020 Mathematics Subject Classification. 92B05.
Manuscript received 22 June 2022, revised 27 July 2022, accepted 30 July 2022.
Version française abrégée
Considérons trois corps P1, P2, P3 dans le plan avec des masses m1 > 0, m2 > 0, m3 > 0 qui
s’attirent conformément à la loi de Newton. En supposant que le centre de gravité est fixe et placé
à l’origine on obtient les équations du mouvement sous la forme suivante
z̈1 = m2
z21
|z21|3
−m3
z13
|z13|3
, z̈2 = m3
z32
|z32|3
−m1
z21
|z21|3
, z̈3 = m1
z13
|z13|3
−m2
z32
|z32|3
, (1)
avec zk = xk + i yk ∈C, k = 1,2,3 et zkl = zk − zl .
Nous disons que les trois corps P1,P2,P3 forment une syzygie généralisée (éclipse) à l’instant
t0 ∈ I , si l’un des triplets (z1, z2, z3)(t0) (positions) ou (ż1, ż2, ż3)(t0) (vitesses) devient colinéaire.
En considérant deux corps quelconques, par exemple P1 et P2, on dit que pour t = 0 la
configuration de P1, P2, P3 est antisymétrique, si les aires orientées de deux parallélogrammes
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
332 Alexei Tsygvintsev
engendrés par les vecteurs (z1(0), z2(0)) et (ż1(0), ż2(0)) sont toutes les deux non nulles et de signes
opposés. Nous montrons ensuite, que toute solution définie sur R+ bornée dont la configuration
à l’instant t = 0 est antisymétrique, atteint nécessairement une syzygie généralisée en un temps
qui n’excède pas la valeur T = πα3/2/
p
2M , M = ∑i mi où α est la distance maximale entre les
corps. L’idée de la démonstration est de se ramener à l’étude de l’équation de Riccati Ċ +C 2 = A
où C = Ẋ X −1 et X est la matrice 2 × 2 formée par les coordonnées de P1 et P2. La matrice A
correspond alors à l’endomorphisme réduit de Wintner–Conley [1] dont la trace est toujours
négative. Les positions des singularités de C sont à leur tour contrôlées grâce à la théorie de
Sturm–Liouville. En remarquant qu’une syzygie généralisée correspond soit à l’annulation de
det(C ) soit à une explosion en temps fini de C , nous pouvons conclure.
1. Introduction
Let P1, P2, P3 be the three points in the plain with strictly positive masses m1,m2,m3 and
euclidean coordinates (xi , yi ) ∈ R2, i = 1,2,3. The Newtonian three-body problem [7] can be
written as follows
z̈1 = m2
z21
|z21|3
−m3
z13
|z13|3
, z̈2 = m3
z32
|z32|3
−m1
z21
|z21|3
, z̈3 = m1
z13
|z13|3
−m2
z32
|z32|3
, (2)
where zk = xk + i yk ∈ C, k = 1,2,3 and zkl = zk − zl . Assuming that the total linear momentum
(which is preserved) is zero, one can always admit that∑
k
mk żk = 0,
∑
k
mk zk = 0, (3)
by placing the center of mass of the bodies at the origin.
Let t 7→ zi (t ), i = 1,2,3 be any collision-free solution of equations (2) defined for t ∈ I = [0, a),
a > 0 and determined by a set of initial conditions (zi (0), żi (0)), i = 1,2,3.
Definition 1. We say that the three bodies P1,P2,P3 form a generalised syzygy (eclipse) at the
moment t0 ∈ I if one of the complex triplets (z1, z2, z3)(t0) (positions) or (ż1, ż2, ż3)(t0) (velocities) is
collinear.
Definition 2. Considering any two bodies, for example P1 and P2, one says that for t = 0 the
configuration of P1, P2, P3 is antisymmetric, if the oriented areas of the two parallelograms
spanned by the vectors (z1(0), z2(0)) and (ż1(0), ż2(0)) are both nonzero and have the opposites
signs. This condition is equivalent algebraically to
det
[
x1 y1
x2 y2
]
(0) ·det
[
ẋ1 ẏ1
ẋ2 ẏ2
]
(0) = (x1 y2 − y1x2) (0) · (ẋ1 ẏ2 − ẏ1ẋ2) (0) < 0. (4)
Remark 3. It is elementary to show, with the help of formulas (3) and (4), that the property to
be antisymmetric, does not depend on the choice of the pair of the bodies: once verified for two
bodies Pi , P j , i 6= j the condition (4) will be also satisfied for the two remaining choices of pairs.
We can formulate now our main result:
Theorem 4. Let t 7→ (z1(t ), z2(t ), z3(t )) be a solution of the three-body problem (2) defined for t ≥ 0
and satisfying the following conditions:
(a) It is collision-free and uniformly bounded, i.e. ∃ α> 0, |zi j (t )| ≤α, ∀ i 6= j , ∀ t ≥ 0
(b) The initial configuration at t = 0 is antisymmetric
Then, there exists 0 < t0 ≤ T such that the three bodies form a generalised syzygy at the moment
t = t0 where
T = πα
3/2
p
2M
, M =
∑
i
mi . (5)
Alexei Tsygvintsev 333
Montgomery in his work [5], using the different approach, has shown that, with the exception
of Lagrange’s solution, every solution to the zero angular momentum, Newtonian three-body
problem, suffers a syzygy - i.e. collinear positions at some instant. Our result is free from the
angular momentum hypothesis, but the generalised syzygies condition is weaker than the one
used in [5] since it now includes the possibility of collinear velocities.
2. The proof of the Main Result
We start by introducing the new variables wi = mi zi , i = 1,2,3 so that the relations (3) imply∑
i
wi = 0. (6)
Writing wk = Xk + i Yk , Xk = mk xk , Yk = mk yk , k = 1,2,3 and using the previous equality, one
derives from equations (2) the following 2×2 matrix equation
Ẍ = AX , X =
[
X1 Y1
X2 Y2
]
, A =
[−m2ρ3 −m13ρ2 m1 (ρ3 −ρ2)
m2
(
ρ3 −ρ1
) −m1ρ3 −m32ρ1
]
, (7)
where ρ1 = 1/|z32|3, ρ2 = 1/|z13|3, ρ3 = 1/|z21|3, mi j = mi +m j .
So, X is now a particular solution of a second-order linear differential equation if we consider
matrix A as a known function of time.
The matrix A is related to the Wintner–Conley endomorphism encoding the forces. Its higher
dimension version was investigated in [1, 3] in the study of n-body configurations and the same
equation (7) was used (up to reduction, transposition and some scaling factor).
Remark 5. It can be easily verified that both definitions of the generalised syzygy (Definition 1)
and antisymmetric configuration (Definition 2) are invariant when replacing xi , yi by Xi ,Yi and
ẋi , ẏi by Ẋi , Ẏi for i = 1,2,3.
The change of variable C = Ẋ X −1 transforms the linear equation (7) into the matrix Riccati
equation
Ċ +C 2 = A . (8)
Since the configuration of the bodies is antisymmetric for t = 0 (see Definition 1), by conti-
nuity, the matrix X (t ) is invertible and C is well defined with det(C (t )) < 0 for t ≥ 0 sufficiently
close to 0. The main idea of the proof is that the first generalised syzygy happens for t = t0 > 0
if det(C (t0)) = 0 or C has a blow-up (singularity) at t = t0. Indeed, according to the hypothesis
of Theorem 4, the function X is smooth for all t ≥ 0. So, C = Ẋ X −1 can have a singularity at
t = t0 > 0 only if det(X (t0)) = 0 i.e. the bodies are collinear. If det(C (t0)) = 0, then det(Ẋ (t0)) = 0
and the velocities of the bodies become collinear.
Taking the trace of the both sides of the equation (8), one derives the following scalar Riccati
equation
ω̇+ω2 =φ= Tr(A)+2det(C ), ω= Tr(C ) . (9)
Since the mutual distances |zi j | are bounded by α > 0, using the formula for A given in (7), it is
elementary to show that
Tr(A(t )) ≤−2 m1 +m2 +m3
α3
=−θ2, ∀ t ≥ 0. (10)
Now, if we assume that the motion of the bodies starts from an antisymmetric configuration at
t = 0 and that the first generalised syzygy happens at t = t0 > 0 (if not, we put t0 =+∞), one has
det(C (t )) < 0 and t 7→C (t ) is a smooth function for all t ∈ [0, t0). Therefore, according to (9), (10):
ω̇+ω2 =φ, φ(t ) ≤−θ2, ∀ t ∈ [0, t0) . (11)
334 Alexei Tsygvintsev
Linearising the above equation via the change ω= ẋ/x, one transforms it to the following form:
ẍ =φx, φ(t ) ≤−θ2, t ∈ [0, t0) . (12)
The well know fact from the Sturm–Liouville theory (see [2]) states that x admits a zero between
any two consecutive zeros of any nontrivial solution y of the linear equation ÿ = −θ2 y . But the
last one has the general solution given by y(t ) = B cos(θt +K ), B ,K ∈ R. Thus, the blow-up of
ω= ẋ/x happens necessarily for some 0 < t0 ≤ T with T given by the formula (5). Thus, we have
shown that the generalised syzygy happens in the interval (0,T ] and the proof of Theorem 4 is
finished. ä
3. Conclusion and numerical results
As can be seen from the proof of Theorem 4, the same result holds for solutions not necessary
bounded and collision-free on the whole R+ axis, but only during the sufficiently long period of
time preceding the first generalised syzygy. We also believe that our result can be generalised to
the case of d +1 bodies in d-dimensional space with d ≥ 3. In particular, as shown in [6], in this
situation, the any zero angular momentum bounded solution suffers infinitely many “coplanar”
instants were d +1 masses all lie on a single affine hyperplane. Finally, knowledge of the constant
angular momentum and energy, when applied to the equation (8), can bring undoubtedly more
precise information about the presence of syzygies (generalised or not).
Alexei Tsygvintsev 5
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure 1. Trajectories of the perturbed version of the figure eight solution taken from [4]:
m1 = m2 = m3 = 1, x1(0) = 0.97000436− 0.05, x2(0) = −0.97000436, y1(0) = −0.24308753,
y2(0) = 0.24308753, ẋ1(0) = 0.93240737/2−0.05, ẋ2(0) = 0.93240737/2, ẏ1(0) = 0.8647314/2,
ẏ2(0) = 0.8647314/2+0.04, 0 ≤ t ≤ 10
2 4 6 8 10
0.5
1.0
1.5
2.0
Figure 2. Mutual distances |zi j | of the bodies for 0 ≤ t ≤ 10
2 4 6 8 10
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
Figure 3. Determinants ∆1 (dashed line) and ∆2 for 0 ≤ t ≤ 10
References
[1] A. Albouy, A. Chenciner, “Le Problème des N corps et les distances mutuelles”, Invent. Math. 131 (1998), p. 151-184.
[2] R. Carroll, Transformation theory and application, North-Holland, 1985.
[3] A. Chenciner, “Non-avoided Crossings for n-Body Balanced Configurations in R3 Near a Central Configuration”,
Arnold Math. J 2 (2016), p. 213-10311.
[4] A. Chenciner, R. Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses”,
Annals of Mathematics 152 (2000), p. 881-901.
C. R. Mathématique — Draft, 27th July 2022
Figure 1. Trajectories of the perturbed version of the figure eight solution taken from [4]:
m1 = m2 = m3 = 1, x1(0) = 0.97000436− 0.05, x2(0) −0.97000436, y1(0) −0.24308753,
y2(0) 0.24308753, ẋ1(0) = 0.93240737/2−0.05, ẋ2(0) = 0.93240737/2, ẏ1(0) = 0.8647314/2,
ẏ2(0) = 0.8647314/2+0.04, 0 ≤ t ≤ 10
In order to illustrate numerically our Theorem 4, we choose the perturbed version of the figure
eight solution whose initial conditions were provided in [4]. It is bounded and collision-free in
time interval [0,10]. Figure 1 demonstrates the corresponding trajectories of the three bodies. We
plot the quantities ∆1 = x1 y2 − x2 y1 and ∆2 = ẋ1 ẏ2 − ẋ2 ẏ1 (determinants in (2)) as functions of
time t ∈ [0,10] (Figure 3). Then each intersection of the graphs of ∆1,2 with the horizontal axis
corresponds to a generalised syzygy. The formula (5) gives the value T = 4.18 were we estimate α
as α= 2.2 (see Figure 2).
Alexei Tsygvintsev 335
Alexei Tsygvintsev 5
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure 1. Trajectories of the perturbed version of the figure eight solution taken from [4]:
m1 = m2 = m3 = 1, x1(0) = 0.97000436− 0.05, x2(0) = −0.97000436, y1(0) = −0.24308753,
y2(0) = 0.24308753, ẋ1(0) = 0.93240737/2−0.05, ẋ2(0) = 0.93240737/2, ẏ1(0) = 0.8647314/2,
ẏ2(0) = 0.8647314/2+0.04, 0 ≤ t ≤ 10
2 4 6 8 10
0.5
1.0
1.5
2.0
Figure 2. Mutual distances |zi j | of the bodies for 0 ≤ t ≤ 10
2 4 6 8 10
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
Figure 3. Determinants ∆1 (dashed line) and ∆2 for 0 ≤ t ≤ 10
References
[1] A. Albouy, A. Chenciner, “Le Problème des N corps et les distances mutuelles”, Invent. Math. 131 (1998), p. 151-184.
[2] R. Carroll, Transformation theory and application, North-Holland, 1985.
[3] A. Chenciner, “Non-avoided Crossings for n-Body Balanced Configurations in R3 Near a Central Configuration”,
Arnold Math. J 2 (2016), p. 213-10311.
[4] A. Chenciner, R. Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses”,
Annals of Mathematics 152 (2000), p. 881-901.
C. R. Mathématique — Draft, 27th July 2022
Figure 2. Mutual distances |zi j | of the bodies for 0 ≤ t ≤ 10
Alexei Tsygvintsev 5
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure 1. Trajectories of the perturbed version of the figure eight solution taken from [4]:
m1 = m2 = m3 = 1, x1(0) = 0.97000436− 0.05, x2(0) = −0.97000436, y1(0) = −0.24308753,
y2(0) = 0.24308753, ẋ1(0) = 0.93240737/2−0.05, ẋ2(0) = 0.93240737/2, ẏ1(0) = 0.8647314/2,
ẏ2(0) = 0.8647314/2+0.04, 0 ≤ t ≤ 10
2 4 6 8 10
0.5
1.0
1.5
2.0
utual distances |zi j | of the bodies for 0 ≤ t ≤ 10
2 4 6 8 10
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
Figure 3. Determinants ∆1 (dashed line) and ∆2 for 0 ≤ t ≤ 10
References
[1] A. Albouy, A. Chenciner, “Le Problème des N corps et les distances mutuelles”, Invent. Math. 131 (1998), p. 151-184.
[2] R. Carroll, Transformation theory and application, North-Holland, 1985.
[3] A. Chenciner, “Non-avoided Crossings for n-Body Balanced Configurations in R3 Near a Central Configuration”,
Arnold Math. J 2 (2016), p. 213-10311.
[4] A. Chenciner, R. Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses”,
Annals of Mathematics 152 (2000), p. 881-901.
C. R. Mathématique — Draft, 27th July 2022
Figure 3. Determinants ∆1 (dashed line) and ∆2 for 0 ≤ t ≤ 10
As can be seen in Figure 3, one observes, for example, an antisymmetric configuration of the
bodies at t = 2 a d generalised syzygies occurring before the momentum t = 4. This is in perfect
agreement with the estimate given in Theorem 4.
Acknowledgments
I would like to express my special gratitude to Alain Chenciner and Richard Montgomery for their
useful suggestions and valuable remarks.
References
[1] A. Albouy, A. Chenciner, “Le Problème des N corps et les distances mutuelles”, Invent. Math. 131 (1998), no. 1, p. 151-
184.
[2] R. Carroll, Transformation theory and application, North-Holland, 1985.
[3] A. Chenciner, “Non-avoided Crossings for n-Body Balanced Configurations in R3 Near a Central Configuration”,
Arnold Math. J. 2 (2016), no. 2, p. 213-248.
[4] A. Chenciner, R. Montgomery, “A remarkable periodic solution of the three-body problem in the case of equal masses”,
Ann. Math. 152 (2000), no. 3, p. 881-901.
[5] R. Montgomery, “The zero angular momentum, three-body problem: All but one solution has syzygies”, Ergodic
Theory Dyn. Syst. 27 (2007), no. 6, p. 311-340.
[6] ——— , “Oscillating about coplanarity in the 4 body problem”, Invent. Math. 218 (2019), no. 1, p. 113-144.
[7] A. Wintner, The analytical foundations of Celestial Mechanics, Princeton Mathematical Series, vol. 5, Princeton
University Press, 1941.


On the existence of generalised syzygies in the planar three-body problem

Chenyan Wang

Huoxiong Wu

Comptes Rendus
Mathématique
Chenyan Wang and Huoxiong Wu
A note on singular oscillatory integrals with certain rational phases
Volume 361 (2023), p. 363-370
https://doi.org/10.5802/crmath.418
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 363-370
https://doi.org/10.5802/crmath.418
Harmonic analysis / Analyse harmonique
A note on singular oscillatory integrals with
certain rational phases
ChenyanWanga and HuoxiongWu∗, a
a School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
E-mails: chenyanwangxmu@163.com (C. Wang), huoxwu@xmu.edu.cn (H. Wu)
Abstract. LetΩ be homogeneous of degree zero with mean value zero, P and Q real polynomials on Rn with
Q(0) = 0 and Ω ∈ B0,0q (Sn−1) for some q > 1. This note extends and improves a classical result of Stein and
Wainger (Ann. Math. Stud. 112, pp. 307-355, (1986)) to the following general form∣∣∣∣p. v.∫
Rn
ei (P (x)+1/Q(x))Ω(x/|x|)|x|n d x
∣∣∣∣≤ B ,
where B depend only on ‖Ω‖
B0,0q (S
n−1), n and the degrees of P and Q, but not on their coefficients.
2020 Mathematics Subject Classification. 42B15, 42B20, 42A50, 42A45.
Funding. Supported by the NNSF of China (Nos. 12171399, 11871101).
Manuscript received 23 February 2022, accepted 1 September 2022.
1. Introduction
Let P be a polynomial on Rn of degree at most d with real coefficients and K be a homogeneous
function of degree −n on Rn , that is,
K (x) = Ω(x/|x|)|x|n ,
whereΩ is an integrable function on the unit sphere Sn−1 and satisfies
∫
Sn−1Ωdσ= 0.
In [8] Stein showed that ifΩ ∈ L∞(Sn−1), then∣∣∣p. v.∫
Rn
e i P (x)K (x)d x
∣∣∣≤Cd ,n,K , (1)
where Cd ,n,K is independent of the coefficients of P . The corresponding one-dimensional esti-
mation was obtained by Stein and Wainger in [9], see also [7] for the sharp bound. Subsequently,
Papadimitrakis and Parissis [6], Al-Qassem et al. [1] successively extended the estimate (1) to the
cases of thatΩ ∈ L logL(Sn−1) and H 1(Sn−1), the Hardy space defined on Sn−1. It is natural to ask
the following question.
Question. Can one extend the estimate (1) to phases which are general rational functions?
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
364 Chenyan Wang and Huoxiong Wu
In 2003, Folch-Gabayet and Wright [3] showed that for general rational phases, the estimate (1)
is not true. Meanwhile, they considered the rational phases of the form P (x)+1/Q(x), where P and
Q are real polynomials with Q(0) = 0, and forΩ ∈ L logL(Sn−1), obtained the following estimate:∣∣∣∣p. v.∫
Rn
e i (P (x)+1/Q(x))K (x)d x
∣∣∣∣≤ A, (2)
where A depends on ‖Ω‖L logL(Sn−1), n and the degrees of P and Q, but not otherwise on the
coefficients of P and Q. It is well known that
L∞
(
Sn−1
)
$
⋃
r >1
Lr
(
Sn−1
)
$ L logL
(
Sn−1
)
.
Therefore, the estimate (2) essentially improved and generalized the corresponding results in
[6, 8, 9].
On the other hand, to study the mapping properties of singular integrals with rough kernels
on Lp (Rn), Jiang and Lu introduced the following block spaces B 0,νq (S
n−1) for ν > −1 and q > 1
(see [5] for the details of block spaces).
Definition 1 ([5]). A q-block on Sn−1 is an Lq -function b (1 < q ≤∞) that satisfies
supp(b) ⊆Q, (i)
‖b‖Lq (Sn−1) ≤ |Q|1/q−1, (ii)
where Q = Sn−1 ∩ {y ∈Rn : |y −ζ| < ρ for some ζ ∈ Sn−1 and ρ ∈ (0,1]}.
Definition 2 ([5]). For ν>−1 and q > 1, the block spaces B 0,νq on Sn−1 are defined by
B 0,νq
(
Sn−1
)= {Ω ∈ L1 (Sn−1) :Ω(y ′) =∑
s
λs bs (y
′), M 0,νq ({λs }) <∞
}
.
where each λs is a complex number, each bs is a q-block supported in Qs ,
M 0,νq ({λs }) =
∑
s
|λs |
{
1+
(
log+
1
|Qs |
)1+ν}
,
and
‖Ω‖B 0,νq (Sn−1) = inf
{
M 0,νq ({λs }) : Ω(x
′) =∑
s
λs bs (x
′)
}
.
It is easy to check that
B 0,ν1q
(
Sn−1
)
$B 0,ν2q
(
Sn−1
)
, ∀ ν1 > ν2 >−1.
Moreover, it follows from [4, 10] that for any q > 1,⋃
r >1
Lr
(
Sn−1
)
$B 0,νq
(
Sn−1
)⊂ H 1 (Sn−1)+L (log+ L)1+ν (Sn−1) , ∀ ν>−1,
and B 0,νq (S
n−1)* L log+ L(Sn−1) for any ν ∈ (−1,0), in particular,⋃
r >1
Lr
(
Sn−1
)
$B 0,0q
(
Sn−1
)
, L log+ L
(
Sn−1
)
$ H 1
(
Sn−1
)
,
but the relationship between B 0,0q (S
n−1) and L log+ L(Sn−1) remains open. Therefore, it is interest-
ing to establish the estimate (2) under the assumption of thatΩ ∈ B 0,0q (Sn−1), and more generally
Ω ∈ H 1(Sn−1).
In this paper, we will establish the estimate (2) provided that Ω ∈ B 0,0q (Sn−1). As for the case
of that Ω ∈ H 1(Sn−1), which is more interesting, it is still open. Our results can be formulated as
follows.
Chenyan Wang and Huoxiong Wu 365
Theorem 3. Suppose that K (x) =Ω(x)/|x|n , where Ω ∈ B 0,0q (Sn−1) is homogeneous of degree zero
with mean value zero and q > 1, P and Q are real polynomials with Q(0) = 0. Then∣∣∣∣p. v.∫
Rn
e i (P (x)+1/Q(x))K (x)d x
∣∣∣∣≤ B , (3)
where B depends on ‖Ω‖B 0,0q (Sn−1), n and the degrees of P and Q, but not otherwise on the coefficients
of P and Q.
Remark 4. Employing the arguments in [3, Proposition 1.4], the following result shows that the
requirement Q(0) = 0 in Theorem 3 is most likely not necessary.
Theorem 5. With K and P as in Theorem 3, but now Q(x) = a + v · x, where a ∈R and v ∈Rn∣∣∣∣p.v.∫
Rn
e i (P (x)+1/Q(x))K (x)d x
∣∣∣∣≤ B , (4)
where B depends on ‖Ω‖B 0,0q (Sn−1), n and the degree of P but not otherwise on the coefficients of P.
As an immediate consequence of Theorems 3 and 5, we have the following result.
Corollary 6. With K and Q as in Theorem 3 or 5, but now P : Rn → Rm being a polynomial
mapping, convolution with the distribution
L(φ) = p.v.
∫
Rn
φ(P (x))e i /Q(x)K (x)d x, φ ∈S (Rm),
is bounded on L2(Rm).
Employing the arguments in proving [3, Theorem 1.1], the main ingredient of the proof of
Theorem 3 or 5 in the current paper is to establish the following integral estimate, which has an
independent interest.
Proposition 7. Let Ω ∈ B 0,0q (Sn−1) for some q > 1, P (x) =
∑
|α|=d cαxα be a homogeneous polyno-
mial of degree d on Rn . Write mP =∑|α|=d |cα|. Then∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω).Cd‖Ω‖B 0,0q (Sn−1). (5)
The rest of this paper is organized as follows. In Section 2 we will give some auxiliary lemmas.
The proofs of our main results will be given in Section 3. We remark that some ideas in our
arguments are taken from [3].
Finally, we make some conventions on notation. Throughout this paper, we denote by C a
positive constant which is independent of the main parameters, but it may vary from line to
line. Let A, B be complex-valued quantities. We use A . B or A = O(B) to denote the estimate
|A| ≤ C |B |. We use A ∼ B to denote the estimate A . B . A. For 1 ≤ γ ≤ ∞, γ′ is the conjugate
index of γ, and 1/γ+1/γ′ = 1.
2. Preliminaries
In this section, we recall and establish some auxiliary lemmas, which will be used in our argu-
ments later.
Lemma 8 ([3]). For any polynomial Q(r ) =∑dj=1 bk r k on R+, there is a finite collection {G j }Mj=1 of
disjoint intervals, called “gaps”, of R+ with M =O(1) such that
(1) The complement R+\
⋃M
j=1{G j } is the union of M −1 disjoint “dyadic” intervals, that is, the
ratio of the endpoints of such intervals is ∼ 1.
(2) For each G j , there is a k = k j ,1 ≤ k j ≤ d, such that for r ∈G j ,
|Q(r )| ∼
∣∣∣bk j ∣∣∣r k j and |Q ′ (r )| ∼ ∣∣∣bk j ∣∣∣r k j −1
366 Chenyan Wang and Huoxiong Wu
Also if L j and R j denote the left and right endpoints of G j respectively, then
(i) if R j <∞, then R j =Cd [|bl |/|bm |]1/m−l for some 1 ≤ l < m ≤ d and
(ii) if L j > 0, then L j =Cd [|br |/|bs |]1/s−r for some 1 ≤ r < s ≤ d.
Lemma 9 ([8]). Suppose φ is real-valued and smooth on (a,b), and that |φ(k)| ≥ λ > 0 for all
t ∈ (a,b). Then ∣∣∣∣∫ b
a
e iφ(t )d t
∣∣∣∣≤Ckλ−1/k
when either k ≥ 2, or k = 1 and φ′ (t ) is monotonic.
Lemma 10. Let γ > 1,Ω ∈ Lγ(Sn−1) and P (x) = ∑|α|=d cαxα be a homogeneous polynomial of
degree d on Rn . Write mP =∑|α|=d |cα|. Then∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω).Cdγ′‖Ω‖Lγ(Sn−1). (6)
Proof. We may assume mP = 1. By the Hölder inequality,∫
Sn−1
|Ω(ω)| · ∣∣log(|P (ω)|)∣∣dσ(ω) ≤ ‖Ω‖Lγ(Sn−1) (∫
Sn−1
∣∣log(|P (ω)|)∣∣γ′ dσ(ω))1/γ′ .
And(∫
Sn−1
∣∣log(|P (ω)|)∣∣γ′ dσ(ω))1/γ′ . (∫ 1
1/2
r n−1
∫
Sn−1
∣∣log(|P (ω)|)∣∣γ′ dσ(ω)dr )1/γ′
.
( ∑
k>0
(
kγ
′)γ′ ∫ 1
1/2
r n−1dr
∫{
ω∈Sn−1 :r d |P (ω)|62−kγ′
} dσ(ω)
)1/γ′
6 γ
′
( ∑
k>0
kγ
′ ∫{
1
2 6 |x|61:2−kγ
′ −16 |P (x)|62−kγ′
} d x
)1/γ′
6 γ
′
( ∑
k>0
kγ
′ ∣∣∣∣{12 6 |x|6 1 : |P (x)|6 2−kγ′
}∣∣∣∣
)1/γ′
,
which reduces matters to obtaining uniform sublevel set estimates for P under the normalisation
mP = 1. Using the fact that all norms are equivalent on the space of polynomials of degree at most
d , we can find a derivative ∂α, 0 ≤ |α| ≤ d , such that 1. |∂αP (x)| on |x|6 1. Ifα= 0, then the above
sublevel sets are empty for large k and so we may assumeα> 0. In this case, using the mean value
theorem, one can show that |{|x|6 1 : |P (x)|6 2−kγ′ }|. 2−kγ′/|α| (see e.g. [2]). Thus( ∑
k>1
kγ
′ ∣∣∣∣{12 6 |x|6 1 : |P (x)|6 2−kγ′
}∣∣∣∣
)1/γ′
6
( ∑
k>1
kγ
′
2−kγ
′
/|α|
)1/γ′
≤ ∑
k>1
k2−k/d .Cd .
This implies the desired conclusion and completes the proof of Lemma 10. 
3. Proofs of Main Results
In this section, we present the proofs of Proposition 7 and Theorem 3.
Chenyan Wang and Huoxiong Wu 367
Proof of Proposition 7. SinceΩ ∈ B 0,0q (Sn−1), we know by Definition 2 that there is a decomposi-
tion:Ω(x ′) =∑s λs bs (x ′), where each bs is a q-block, supported in Qs and∑
s
|λs |
(
1+ log+ 1|Qs |
)
<∞.
Therefore,∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω) ≤∑
s
|λs |
∫
Sn−1
|bs (ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω)
≤
 ∑
|Qs |≥e−q′
+ ∑
|Qs |<e −q′
 |λs |∫
Sn−1
|bs (ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω)
=: I + I I .
Recall that for each bs , supp(bs ) ⊂ Qs and ‖bs‖Lq (Sn−1) ≤ |Qs |1/q−1 = |Qs |−1/q
′
. If |Qs | ≥ e−q ′ , we
take γ= q and obtain
γ′ ‖bs‖Lγ(Sn−1) ≤ q ′ ‖bs‖Lq (Sn−1) ≤ q ′ |Qs |−1/q
′
. 1;
if |Qs | < e−q ′ , we take γ= log |Qs |/(1+ log |Qs |), then 1 < γ< q , γ′ = log(1/|Qs |) and
γ′ ‖bs‖Lγ(Sn−1) ≤ log
1
|Qs |
‖bs‖Lq (Sn−1) |Qs |1/γ−1/q ≤ log
1
|Qs |
|Qs |−1/γ
′
. log
1
|Qs |
.
These, combining with Lemma 10, lead to
I ≤Cd
∑
|Qs |≥e−q′
|λs |q ′ ‖bs‖Lq (Sn−1).Cd
∑
|Qs |≥e−q′
|λs | ,
and for γ= log |Qs |/(1+ log |Qs |),
I I ≤Cd
∑
|Qs |<e−q′
|λs |γ′ ‖bs‖Lγ(Sn−1).Cd
∑
|Qs |>e−q′
|λs | log 1|Qs |
.
Consequently, ∫
Sn−1
|Ω(ω)| ·
∣∣∣∣log( |P (ω)|mP
)∣∣∣∣dσ(ω).Cd ∑
s
|λs |
(
1+ log 1|Qs |
)
,
which completes the proof of Proposition 7. 
Proof of Theorem 3. The arguments are completely similar to those in proving [3, Theorem 1.1].
The only difference is replacing [3, Lemma 2.2] by Proposition 7 in the current setting. For
completeness, we present the details as follows.
We may assume P (0) = 0. Using polar coordinates write the integral in (3) as
I =
∫
Sn−1
Ω(ω)
∫ ∞
0
e i [Pω(r )+1/Qω(r )]
1
r
dr dσ(ω),
where Q(x) = Qω(r ) = ∑d ′j=1 q j (ω)r j , P (x) = Pω(r ) = ∑dk=1 p j (ω)r k and p j , qk are homogeneous
polynomials of degree j and k. Using Lemma 8, we may write I =∑ I j ,k +O(1), where
I j ,k =
∫
Sn−1
Ω(ω)
∫
G j ∩Fk
e i [Pω(r )+1/Qω(r )]
1
r
dr dσ(ω).
Here {G j } and {Fk } are the “gaps” of Qω(r ) and Pω(r ), respectively. Note that although the inner
integral of I j ,k depends on ω in a complicated way, we know the form of the endpoints of G j and
Fk as given by Lemma 8 and so it is at least measurable as a function of ω. It suffices to bound
each I j ,k separately.
We have |Qω(r )| ∼ |q jl (ω)|r jl and |Q ′ω(r )| ∼ |q jl (ω)|r jl−1 on G j , for some 1 ≤ jl ≤ d ′, and
|Pω(r )|∼ |pkm (ω)|r km and |P ′ω(r )|∼ |pkm (ω)|r km−1 on Fk , for some 1 ≤ km ≤ d .
368 Chenyan Wang and Huoxiong Wu
Therefore away from where r jl+km ∼ (|pkm (ω)| · |q jl (ω)|)−1 the size of the phase φω(r ) =
Pω(r ) + 1/Qω(r ) and its derivative is understood. In fact, on R j ,k = G j ∩ Fk ∩ [C (|pkm (ω)| ·
|q jl (ω)|)−1/(km+ jl ),∞) (for C large enough), we have∣∣φω(r )∣∣∼ ∣∣pkm (ω)∣∣r km and ∣∣φ′ω(r )∣∣∼ ∣∣pkm (ω)∣∣r km−1.
An application of van der Corput’s Lemma 9 shows∣∣∣∣∣
∫
{
r ∈R j ,k :r≥Θ
} e iφω(r ) 1r dr
∣∣∣∣∣=O(1),
whereΘ= |pkm (ω)|1/km . Since we are applying Lemma 9 with k = 1, we need to first split the inte-
gration of the above integral into O(1) intervals, whereφ′ω(r ) is monotone. In the complementary
interval, r <Θ, due to the size of φω(r ) on R j ,k , we see that∣∣∣∣∣
∫
{
r ∈R j ,k :r <Θ
}
[
e iφω(r )
1
r
−1
]
dr
∣∣∣∣∣=O(1).
Therefore for the part of I j ,k over R j ,k ,∫
Sn−1
Ω(ω)
∫
R j ,k
e iφω(r )
1
r
dr dσ(ω) =
∫
Sn−1
Ω(ω)
∫
{
r ∈R j ,k :r<Θ
} drr dσ(ω)+O(1).
Similarly for L j ,k =G j ∩Fk ∩(−∞, δ(|pkm (ω)| · |q jl (ω)|)−1/(km+ jl )] (for δ small enough), we have∫
Sn−1
Ω(ω)
∫
L j ,k
e iφω(r )
1
r
dr dσ(ω) =
∫
Sn−1
Ω(ω)
∫
{
r ∈L j ,k :r ≥Λ
} drr dσ(ω)+O(1),
whereΛ= |q jl (ω)|1/ jl . Therefore
I j ,k =
∫
Sn−1
Ω(ω)
∫
G j ∩Fk
e i [Pω(r )+1/Qω(r )]
1
r
dr dσ(ω)
=
∫
Sn−1
Ω(ω)
∫
{
r ∈R j ,k :r ≤Θ
} drr dσ(ω)
+
∫
Sn−1
Ω(ω)
∫
{
r ∈L j ,k :r ≥Λ
} drr dσ(ω)+O(1)
(7)
and these two last integrals can be shown to be O(1) by repeatedly applying Proposition 7.
In fact, by the structures of R j ,k and L j k , the integrals in (7):∫
Sn−1
Ω(ω)
∫
{
r ∈R j ,k :r ≤Θ
} drr dσ(ω), and
∫
Sn−1
Ω(ω)
∫
{
r ∈L j ,k :r ≥Λ
} drr dσ(ω)
can be written in the form ∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω),
where E(ω) is the intersection of O(1) intervals of the form [a(ω), ∞) or (−∞, a(ω)] with
a(ω) ∈
{(∣∣pk1 (ω)∣∣/ ∣∣pk2 (ω)∣∣)1/(k2−k1) ,(∣∣q j1 (ω)∣∣/ ∣∣q j2 (ω)∣∣)1/( j2− j1) ,∣∣pk (ω)∣∣−1/k , ∣∣q j (ω)∣∣−1/ j ,(∣∣pk (ω)∣∣ · ∣∣q j (ω)∣∣)−1/( j+k)} .
Without loss of the generality, we may say that E(ω) is the intersection of M half infinite inter-
vals such as [a(ω), ∞) or (−∞, a(ω)]. Let us write E(ω) = [a(ω), ∞)∩E ′(ω), where a(ω), say, is
(|pk1 (ω)|/|pk2 (ω)|)1/(k2−k1) and E ′(ω) is the intersection of M −1 half infinite intervals as in E(ω).
We can then use Proposition 7 to write∣∣∣∣∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣+O(1),
Chenyan Wang and Huoxiong Wu 369
where A1 = (mpk1 /mpk2 )1/(k2−k1), which is independent of ω. Indeed,∣∣∣∣∫
Sn−1
Ω(ω)
∫
[a(ω),∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣
+
∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1, a(ω)]∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣ ,
and for the second integral, by Lemma 9 and Proposition 7, we have∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1, a(ω)]∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣
≤
∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1, a(ω)]
dr
r
dσ(ω)
∣∣∣∣
≤ 1|k2 −k1|
[∫
Sn−1
|Ω(ω)|
∣∣∣∣∣log
( ∣∣pk1 (ω)∣∣
mpk1
)∣∣∣∣∣dσ(ω)+
∫
Sn−1
|Ω(ω)|
∣∣∣∣∣log
( ∣∣pk2 (ω)∣∣
mpk2
)∣∣∣∣∣dσ(ω)
]
.O(1).
For the other forms of a(ω), the argument is similar.
Similarly, we have∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩ [A2,∞)∩E ′′(ω)
dr
r
dσ(ω)
∣∣∣∣+O(1),
or ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩E ′(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
[A1,∞)∩ (−∞,A2]∩E ′′(ω)
dr
r
dσ(ω)
∣∣∣∣+O(1),
where A2 is independent of ω, and E ′′(ω) is the intersection of M −2 half infinite intervals as in
E(ω). Continuing this process, after M iterations, we obtain that∣∣∣∣∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
E
dr
r
dσ(ω)
∣∣∣∣+O(1),
where E is the intersection of M intervals of the form [A,∞) or (−∞, A], where A is independent
of ω. Therefore, by the mean value zero ofΩ on Sn−1, we have∫
Sn−1
Ω(ω)
∫
E
dr
r
dσ(ω) = 0.
This implies that∣∣∣∣∫
Sn−1
Ω(ω)
∫
E(ω)
dr
r
dσ(ω)
∣∣∣∣≤ ∣∣∣∣∫
Sn−1
Ω(ω)
∫
E
dr
r
dσ(ω)
∣∣∣∣+O(1) =O(1),
and completes the proof of Theorem 3. 
Proof of Theorem 5. By replacing [3, Lemma 2.2] by Proposition 7, the proof is similar to the
proof of [3, Proposition 1.4]. We omit the details here. 
References
[1] H. M. Al-Qassem, L. C. Cheng, A. Fukui, Y. Pan, “Weighted Estimates for Oscillatory Singular Integrals”, J. Funct.
Spaces Appl. 2013 (2013), article no. 543748 (3 pages).
[2] A. Carbery, M. Christ, J. Wright, “Multidimensional van der Corput and sublevel set estimates”, J. Am. Math. Soc. 12
(1999), no. 4, p. 981-1015.
[3] M. Folch-Gabayet, J. Wright, “An estimation for a family of oscillatory integrals”, Stud. Math. 154 (2003), no. 1, p. 89-
97.
[4] M. Keitoku, E. Sato, “Block spaces on the unit sphere in Rn ”, Proc. Am. Math. Soc. 119 (1993), no. 2, p. 453-455.
[5] S. Lu, M. H. Taibleson, G. Weiss, Spaces Generated by Blocks, Publishing House of Beijing Normal University, 1989.
[6] M. Papadimitrakis, I. R. Parissis, “Singular oscillatory integrals on Rn ”, Math. Z. 266 (2010), no. 1, p. 169-179.
370 Chenyan Wang and Huoxiong Wu
[7] I. R. Parissis, “A sharp bound for the Stein–Wainger oscillatory integral”, Proc. Am. Math. Soc. 136 (2008), no. 3, p. 963-
972.
[8] E. M. Stein, “Oscillation integrals in Fourier analysis”, in Beijing Lectures in Harmonic Analysis (Beijing, 1984), Annals
of Mathematics Studies, vol. 112, Princeton University Press, 1986, p. 307-355.
[9] E. M. Stein, S. Wainger, “The estimation of an integral arising in multiplier transformations”, Stud. Math. 35 (1970),
p. 101-104.
[10] X. Ye, X. Zhu, “A note on certain block spaces on the unit sphere”, Acta Math. Sin., Engl. Ser. 22 (2006), no. 6, p. 1843-
1846.


A note on singular oscillatory integrals with certain rational phases

Max Goering

Lukas Koch

A note on improved differentiability for the Banach-space valued Finsler $\gamma $-Laplacian

Gang Yu

Comptes Rendus
Mathématique
Gang Yu
A Note on Barker Sequences and the L1-norm of Littlewood Polynomials
Volume 361 (2023), p. 609-616
Published online: 2 March 2023
https://doi.org/10.5802/crmath.428
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 609-616
https://doi.org/10.5802/crmath.428
Number theory / Théorie des nombres
A Note on Barker Sequences and the L1-norm
of Littlewood Polynomials
Gang Yua
a Department of Mathematical Sciences, Kent State University, East Summit Street,
Kent, OH 45458, USA
E-mail: gyu@kent.edu
Abstract. In this note, we investigate the L1-norms of Barker polynomials and, more generally, Littlewood
polynomials over the unit circle, and give improvements to some existing results.
Mathematical subject classification (2010). 11B83, 11C08, 30C10.
Manuscript received 17 August 2022, revised 21 September 2022, accepted 22 September 2022.
1. Introduction
For a sequence of complex numbers a0 , a1 , . . . , an−1 , define its aperiodic autocorrelation sequence
{c j } by
c j =
n−1− j∑
l=0
al+ j al
for 0 ≤ j < n, and c− j := c j .
In 1953, Barker [2] asked to determine the ±1 sequences {ak } satisfying c j ∈ {0,−1} for all j ̸= 0.
This condition was later relaxed in the followup researches, and now we call a0, a1, . . . , an−1 ∈ {±1}
a generalized Barker sequence (or simply a Barker sequence) of length n if c j ∈ {0,±1} for all j ̸= 0.
Ignoring the trivial case n = 1, one can assume a0 = a1 = 1 in a Barker sequence because
negating every or every other term of a0, a1, . . . , an−1 ∈ {±1} does not change the value of |c j |.
Under this normalization, the Barker sequences of length ≤ 13 are given in the following table. In
fact, these are the only 8 Barker sequences that have been found so far.
n = 2 ++ n = 3 ++−
n = 4 +++− n = 4 ++−+
n = 5 +++−+ n = 7 +++−−+−
n = 11 +++−−−+−−+− n = 13 +++++−−++−+−+
(1)
In 1961, Turyn and Storer [16] proved that if the length n of a Barker sequence is odd, then
n ≤ 13. (See also [14] for an alternative proof.) Thus all odd length Barker sequences are included
in table (1). The even length case, on the other hand, is significantly harder. So far, with the efforts
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
610 Gang Yu
of many researchers, a lot of properties of Barker sequences of even length have been proved (cf.
Borwein and Mossinghoff [3] for more references). Moreover, some computational results have
been carried out since the 1990’s (c.f. [4, 6, 8–10, 13]). In particular, it is shown in [9] that there is
no Barker sequence of length 13 < n ≤ 4×1033.
Although the even case is unsettled, it is generally believed that there are no more Barker
sequences than those given in Table (1). This is the so-called strong Barker sequence conjecture. A
weaker form of this conjecture, the weak Barker sequence conjecture, asserts that there are finitely
many Barker sequences.
Barker sequences are closely related to the L1-norm of Littlewood polynomials. A degree n −1
Littlewood polynomial is a polynomial P (x) = s0 + s1x + s2x2 +·· ·+ sn−1xn−1 with s j = ±1 for all
j ≤ n −1. Here the Lp -norm of a polynomial refers to its Lp norm on the unit circle. In this note,
just for convenience, we abuse the terminology and call the trigonometric polynomial
f (α) = f
Sn
(α) = P (e2πiα) =
n−1∑
j=0
s j e( jα), where e(t ) = exp(2πi t ), (2)
the polynomial corresponding to the complex sequence Sn = {s0, s1, . . . , sn−1} and, in particular,
a Barker Polynomial if Sn is a Barker sequence. The Lp -norm of f is then accordingly defined by
∥ f ∥p :=
(∫ 1
0
| f (α)|p dα
) 1
p
.
There are numerous questions and conjectures concerning the Lp -norms of Littlewood poly-
nomials. In particular, a conjecture attributed to Newman is as follows.
Conjecture. There is a constant c < 1 such that ∥ f ∥1 ≤ c
p
n for every Littlewood polynomial f of
degree n −1.
This conjecture has a lot of implications. For example, it yields a conjecture of Erdös on the
maximum values of Littlewood polynomials, and the Turyn–Golay conjecture on the bounded-
ness of the merit factor of any binary sequence. It is also closely related to Littlewood’s flatness
conjecture (which was solved by Körner [7] for unimodular polynomials and, most recently, by
Balister et al. [1] for Littlewood polynomials). Although Newman’s conjecture is generally believed
to be true, it seems to be very difficult to prove at the moment.
The Barker sequence conjectures are related to some much weaker forms of Newman’s con-
jecture. According to Turyn [15], if f is a Barker polynomial of degree n −1, then ∥ f ∥1 >
p
n −1.
Thus even an upper bound as weak as ∥ f ∥1 ≲
p
n −1.1 for Littlewood polynomial of degree n −1
would at least settle the weak Barker sequence conjecture. It is unfortunate that the hitherto
best upper bound for ∥ f ∥1 is not sufficient for this purpose. For any degree n − 1 Littlewood
polynomial f , Newman [11] proved that ∥ f ∥21 < n − 0.03; Habsieger [5] improved the bound to
∥ f ∥21 < n − (3−2
p
2)+0.18n−1 and ∥ f ∥21 < n −0.175+o(1). Unaware of Habsieger’s result1, Bor-
wein and Mossinghoff [3] improved Newman’s bound to ∥ f ∥21 < n −0.09.
In this note, we shall first improve Turyn’s bound for L1-norm of Barker polynomials.
Theorem 1. Suppose n > 13 is an integer, and f (α) is a Barker polynomial of degree n −1. Then
n − 1
2
(
1+ 1
(
p
2+1)2
)
< ∥ f ∥21 ≤ n −
1
4
+ 1
64n
. (3)
1At the meeting of New Horizons in Additive Combinatorics(CRM, Montreal, October 2014), unaware of [5], the author
reported a bound that is essentially the same as the first result in [5]. He thanks Professor Habsieger for bringing [5] to his
attention.
Gang Yu 611
Note that 12
(
1+ 1
(
p
2+1)2
)= 0.5857. . . . Moreover, we have
∥ f ∥21 > n −λ(θ)+O(n−1) (4)
where
θ = sup
t>0
sin2 t
t
= 0.72461135. . .
and
λ(θ) = 1
2
(
1
(
p
1+θ+1)2
+ 1
(
p
1−θ+1)2
)
= 0.3084981. . . .
Our proof of Theorem 1 is quite simple and elementary. If f is a Barker polynomial of odd
degree n−1, then c j = 0 for all even j ̸= 0. This implies that | f (α)|2+| f (α+1/2)|2 = 2n for any real
α, and thus the sizes of | f (α)| and | f (α+1/2)| are mutually constrained. When bounding integrals
involving | f (α)|, we can gain some improvement by pairing up | f (α)| and | f (α+1/2)|. The same
idea applies to bounding the L1-norm of Littlewood polynomials of odd degrees.
Theorem 2. Suppose n is an even positive integer, and f is a Littlewood polynomial of degree n−1.
Then
∥ f ∥21 ≤ n −
1
4(1+γ) = n −0.21588. . . , (5)
where γ= 0.1580. . . is the only positive root of the polynomial 64x3 +112x2 +25x −7.
If a degree n − 1 Littlewood polynomial had L1-norm close enough to
p
n, then the sum of
|c j |’s (including c0) would tend to be close to 2n. In particular, when n ≡ 3 (mod 4), we would
have c j =−1 for most even −n < j < n. This observation enables us to better bound ∥ f ∥∞ in such
special case. Hence, while the approach used in Theorems 1 and 2 does not apply to Littlewood
polynomials of even degrees, we are able to improve the existing upper bound for the L1-norm in
the case n ≡ 3 (mod 4).
Theorem 3. Suppose n ≡ 3 (mod 4) is a positive integer. Then for any Littlewood polynomial f of
degree n −1, we have
∥ f ∥21 ≤ n −λ+o(1) (6)
where
λ= 1(√
2+2
√
4−2p2−2p2+1
)2 = 0.2151. . . .
We can make the bound in Theorem 3 uniform in n, but we shall not do so just to avoid
complicated calculations. Our primary purpose is to present our approach in this particular case.
2. Some Lemmas and Proof of Theorem 1
Before proving Theorem 1, we record two results about Barker sequences.
Lemma 4. Suppose a0, a1, . . . , an−1 is a {±1} sequence, and let {ck } denote the aperiodic autocorre-
lations. Then for every 1 ≤ j ≤ n −1,
c j + cn− j ≡ n (mod 4). (7)
If in addition n is even and {a j } is a Barker sequence, then n = 4m2 for some integer m.
Proof. This is from [15, Theorem 2.1]. One can also refer to [3, Theorem 2.1]. □
612 Gang Yu
Lemma 5. Suppose f is a Barker polynomial of degree n −1. Then∣∣| f (α)|2 −n∣∣
n
≤ θ+O(n−1), (8)
where
θ = sup
t>0
sin2 t
t
= 0.7246113537. . . .
Proof. This essentially belongs to Saffari [12], with an oversight corrected in [3, Theorem 3.1]. □
Now we prove Theorem 1. Suppose n is even, and f is a Barker polynomial of degree n−1. First
we note that c j = 0 for all non-zero even j . Thus, for any α ∈R,
| f (α)|2 +| f (α+1/2)|2 = 2c0 = 2n. (9)
From ∫ 1
0
(| f (α)|−pn)2dα= 2n −2pn∥ f ∥1,
we get
∥ f ∥1 =
p
n − 1
2
p
n
∫ 1
0
(| f (α)|−pn)2dα.
By the periodicity of f , we then have
∥ f ∥1 =
p
n − 1
4
p
n
∫ 1
0
[
(| f (α)|−pn)2 + (| f (α+1/2)|−pn)2]dα
=pn − 1
4
p
n
∫ 1
0
(| f (α)|2 −n)2 [(| f (α)|+pn)−2 + (| f (α+1/2)|+pn)−2]dα (10)
where the last step follows from (9). Writing x = x(α) = | f (α)|2n , we get
(| f (α)|+pn)−2 + (| f (α+1/2)|+pn)−2 = ((
p
x +1)−2 + (p2−x +1)−2)
n
= g (x(α))
n
, say.
From ∫ 1
0
(| f (α)|2 −n)2dα= n
and (10), then we have
p
n − max{g (x)}
4
p
n
≤ ∥ f ∥1 ≤
p
n − min{g (x)}
4
p
n
. (11)
On [0,2], g (x) is convex, symmetric about x = 1, and with only one critical number x = 1. Hence,
trivially, we have
max{g (x(α))} ≤ g (0), and min{g (x(α))} ≥ g (1).
Taking this into (11) and squaring all sides, we get (3).
From Lemma 5, we have 1−θ+O(n−1) ≤ x ≤ 1+θ+O(n−1). By applying this to (11), we have
the lower bound (4).
3. Proof of Theorem 2
Let
O := ∑
j odd
|c j |, and E :=
∑
j even
j ̸=0
|c j |.
Gang Yu 613
When n is even, c j has the same parity as j . Thus O ≥ n. First, we have
∥ f ∥21 = ∥ f ∥22 −
∫ 1
0
(| f (α)|−∥ f ∥1)2dα
= n −
∫ 1
0
(| f (α)|2 −∥ f ∥21)2
(| f (α)|+∥ f ∥1)2
dα
≤ n −
∫ 1
0 (| f (α)|2 −∥ f ∥21)2dα
(∥ f ∥∞+∥ f ∥1)2
≤ n −
∑
j ̸=0 c2j
(∥ f ∥∞+∥ f ∥1)2
. (12)
A trivial bound
∥ f ∥∞ ≤
√
n + ∑
j ̸=0
|c j | =
p
n +O +E
together with (12) gives
∥ f ∥21 ≤ n −
∑
j ̸=0 c2j
(
p
n +O +E +pn)2 ≤ n −
O +2E
(
p
n +O +E +pn)2 (13)
where the last inequality in (13) follows from the fact |c j |2 ≥ 2|c j | for even j . On the other hand,
from the periodicity of f (α) we have
∥ f ∥21 = ∥ f ∥22 −
1
2
(∫ 1
0
(| f (α)|−∥ f ∥1)2dα+
∫ 1
0
(∣∣∣∣ f (α+ 12 )
∣∣∣∣−∥ f ∥1)2dα)
≤ n − 1
4
∫ 1
0
(
| f (α)|−
∣∣∣ f (α+ 1
2
)∣∣∣)2dα
≤ n −
∫ 1
0
(| f (α)|2 − ∣∣ f (α+ 12 )∣∣2)2dα
4sup
{(| f (α)|+ ∣∣ f (α+ 12 )∣∣)2} . (14)
Note that ∫ 1
0
(
| f (α)|2 −
∣∣∣∣ f (α+ 12 )
∣∣∣∣2)2dα≥ 4O, (15)
and (
| f (α)|+
∣∣∣ f (α+ 1
2
)∣∣∣)2 ≤ 2(| f (α)|2 + ∣∣∣ f (α+ 1
2
)∣∣∣2)≤ 4(n +E). (16)
Thus from (14)–(16), and the fact that O ≥ n, we get another bound for ∥ f ∥21,
∥ f ∥21 ≤ n −
O
4(n +E) ≤ n −
n
4(n +E) . (17)
Next we prove (5) by using (13) and (17) in accordance with the size of E .
Case 1. E ≥ 2n. Note that the function
g (x) := x
(1+p1+x)2
is increasing on [0,∞), so if O +E ≥ 3n, then from (13) we have
∥ f ∥21 ≤ n −
O +E
(
p
n +O +E +pn)2 ≤ n −
3n
(
p
3n +n +pn)2 = n −
1
3
(18)
which is admissible.
614 Gang Yu
Case 2. E ≤ 2n. Let
g1(O,E) = n − O +2E
(
p
n +O +E +pn)2 .
When E ≤ 2n, we have
∂g1(O,E)
∂O
= E − (n +
p
n(n +O +E))p
n +O +E(pn +O +E +pn)3 < 0.
Thus from (13), we have
∥ f ∥21 ≤ n −
n +2E
(
p
2n +E +pn)2 . (19)
Let E = xn, and
g2(x) = n − 1+2x
(
p
2+x +1)2 , g3(x) = n −
1
4(1+x) .
Note that on [0,2], g2(x) is decreasing and g3(x) is increasing, and they intersect at x = γ =
0.15802. . . where γ is the only positive root of 64x3 +112x2 +25x −7. Hence from this and (17),
(19), we get
∥ f ∥21 ≤ n −
1
4(1+γ) = n −0.21588. . . . □
4. Proof of Theorem 3
Let
O := ∑
j odd
|c j |, and e :=
∑
j even
j ̸=0
|c j |− (n −1).
Since n is odd, we note that c j is even when j is odd, and odd when j is even. In particular, we
have e ≥ 0, 4 |O, 4 | e and, moreover,∑
j even
j ̸=0
|c j |2 =
∑
|c j |=1
|c j |+3
∑
|c j |=3
|c j |+5
∑
|c j |=5
|c j |+ · · · ≥ 4e +n −1. (20)
From this, and an argument similar to (12), we get
∥ f ∥21 ≤ n −
2O +4e +n −1
(∥ f ∥∞+
p
n)2
≤ n − 2O +2e +n
(∥ f ∥∞+
p
n)2
+O(n−1). (21)
We note that, trivially, ∥ f ∥∞ ≤
p
2n +O +e.
If O +e ≥ n5 , then
∥ f ∥21 ≤ n −
2O +2e +n
(
p
2n +O +e +pn)2 +O(n
−1) ≤ n − 7/5
(
p
11/5+1)2 +O(n
−1)
= n − 7
16+2p55 +O(n
−1) ≤ n −0.22+O(n−1), (22)
which is admissible.
We suppose O+e ≤ n5 henceforth. Note that there are at most e2 even non-zero j ’s with |c j | ̸= 1,
also there are at most O2 odd j ’s such that c j ̸= 0. Thus, from
c j + cn− j ≡ n ≡−1 (mod 4),
we conclude that there are at least n−1− O2 even non-zero j ’s satisfying c j ≡−1 (mod 4), among
which at least n −1− O+e2 even j ’s satisfy c j =−1.
For a positive integer K ≥ n−12 , let
hK (α) :=
∑
0≤| j |≤K
(
1− | j |
K
)
e( jα) = 1
K
∣∣∣∣ K∑
j=1
e( jα)
∣∣∣∣2.
Gang Yu 615
Then for any number δ> 0, we have
| f (α)|2 ≤ | f (α)|2 +δhK (2α) =
∑
−2K<n<2K
b j e( jα), (23)
where
b j =
{
c j , if j is odd
c j +δ
(
1− | j |2K
)
, if j is even.
Hence
| f (α)|2 ≤ ∑
−2K< j<2K
|b j | =O +
∑
−K< j<K
∣∣∣c2 j +δ(1− | j |
K
)∣∣∣. (24)
We suppose δ> 1 and
O +e
4
< (1−δ−1)K < n −1
2
. (25)
The purpose of introducing δhK (2α) in (23) is to reduce the contribution from |c2 j | for most
negative c2 j . We observe that the largest value of the last sum in (24) occurs when we have
c2 j =−1 for (1−δ−1)K < | j | ≤ n−12 and, other than the 2[(1−δ−1)K ]−O+e2 such j ∈ [−(1−δ−1)K , (1−
δ−1)K ], it holds that c2 j > 0 for all other j in the range. With this observation, we get from (24) that
| f (α)|2 ≤ 2n +O +e +δK −4
(
(1−δ−1)K − O +e
4
)
−4δ ∑
(1−δ−1)K< j≤ n−12
(1− j /K ). (26)
For convenience, we let O +e = ϵn, K = κn. Then (26) gives
| f (α)|2 ≤ g (ϵ;δ,κ)n +O(1), (27)
where
g (ϵ;δ,κ) = 2+2ϵ+δκ−δ(1−2(1−δ−1)κ)
(
1+δ−1 − 1
2κ
)
−4(1−δ−1)κ,
with (25) and the other conditions on the variable ϵ and parameters δ, κ translated into
0 ≤ ϵ≤ 1
5
, κ> 1
2
, δ> 1, and ϵ
4
≤ (1−δ−1)κ< 1
2
. (28)
Now by (21), we have
∥ f ∥21 ≤ n −
1+2ϵ
(
√
g (ϵ;δ,κ)+1)2
+O(n−1). (29)
It turns out that the optimal choices for δ and κ are
κ= κ0 = 2−1/2, and δ= δ0 =
√
1+2−1/2.
It is easy to check that all conditions in (28) are satisfied under these choices and 0 ≤ ϵ ≤ 15 .
Moreover, the function 1+2ϵ
(
p
g (ϵ;δ0,κ0)+1)2
is strictly increasing on [0,1/5]. Hence from (29) we have
∥ f ∥21 ≤ n −
1
(
√
g (0;δ0,κ0)+1)2
+O(n−1) = n − 1(√
2+2
√
4−2p2−2p2+1
)2 +O(n−1)
and thus the theorem is proved. □
Acknowledgements
The author is indebted to the anonymous referee for many helpful comments and suggestions
that greatly improved the quality of the manuscript.
616 Gang Yu
References
[1] P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, M. Tiba, “Flat Littlewood polynomials exist”, Ann. Math. 192
(2020), no. 3, p. 977-1004.
[2] R. H. Barker, “Group synchronizing of binary digital systems”, in Communication Theory, Butterworths Sci. Pub.,
1953, p. 273-287.
[3] P. Borwein, M. J. Mossinghoff, “Barker sequences and flat polynomials”, in Number theory and polynomials, London
Mathematical Society Lecture Note Series, vol. 352, Cambridge University Press, 2008, p. 71-88.
[4] ——— , “Wieferich pairs and Barker sequences II”, LMS J. Comput. Math. 17 (2014), p. 24-32.
[5] L. Habsieger, “Sur un problème de Newman”, C. R. Acad. Sci. Paris 324 (1997), no. 7, p. 765-769.
[6] J. Jedwab, S. Lloyd, “A note on the nonexistence of Barker sequences”, Des. Codes Cryptography 2 (1992), no. 1, p. 93-
97.
[7] T. W. Körner, “On a polynomial of Byrnes”, Bull. Lond. Math. Soc. 12 (1980), no. 3, p. 219-224.
[8] K. H. Leung, B. Schmidt, “The field descent method”, Des. Codes Cryptography 36 (2005), no. 2, p. 171-188.
[9] ——— , “The anti-field-descent method”, J. Comb. Theory, Ser. A 139 (2016), p. 87-131.
[10] M. J. Mossinghoff, “Wieferich pairs and Barker sequences”, Des. Codes Cryptography 53 (2009), no. 3, p. 149-163.
[11] D. J. Newman, “Norms of polynomials”, Am. Math. Mon. 67 (1960), p. 778-779.
[12] B. Saffari, “Barker sequences and Littlewood’s “two-sided conjectures” on polynomials with ±1 coefficients”, in
Séminaire d’analyse harmonique. Année 1989/90., Université de Paris-Sud, Mathématiques, 1989, p. 139-151.
[13] B. Schmidt, “Cyclotomic integers and finite geometry”, J. Am. Math. Soc. 12 (1999), no. 4, p. 929-952.
[14] K.-U. Schmidt, J. Willms, “Barker sequences of odd length”, Des. Codes Cryptography 80 (2016), no. 2, p. 409-414.
[15] R. Turyn, “On Barker codes of even length”, IEEE Trans. Inf. Theory 51 (1963), no. 9, p. 1256.
[16] R. Turyn, J. Storer, “On the binary sequences”, Proc. Am. Math. Soc. 12 (1961), p. 396-399.


A Note on Barker Sequences and the $L_1$-norm of Littlewood Polynomials

Serge Hippolyte Arnaud Kanga

Ouagnina Hili

Sophie Dabo-Niang

Comptes Rendus
Mathématique
Serge Hippolyte Arnaud Kanga, Ouagnina Hili and Sophie Dabo-Niang
On nonparametric conditional quantile estimation for non-stationary spatial
processes
Volume 361 (2023), p. 847-852
Published online: 18 July 2023
https://doi.org/10.5802/crmath.400
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 847-852
https://doi.org/10.5802/crmath.400
Statistics / Statistiques
On nonparametric conditional quantile
estimation for non-stationary spatial
processes
Sur l’estimation non paramétrique du quantile
conditionnel des processus spatiaux non stationnaires
Serge Hippolyte Arnaud Kangaa, Ouagnina Hili∗, a and Sophie Dabo-Niangb
a UMRI Mathématiques et Nouvelles Technologies de l’Information, Institut National
Polytechnique Félix Houphouët Boigny, BP 1093 Yamoussoukro, Côte d’Ivoire
b Laboratoire Paul Painlevé UMR CNRS 8524, INRIA-MODAL Université de Lille,
France
E-mails: arnaudkanga@gmail.com (S. H. A. Kanga), o_hili@yahoo.fr (O. Hili),
sophie.dabo@univ-lille.fr (S. Dabo-Niang)
Abstract. A kernel conditional quantile estimate of a real-valued non-stationary spatial process is proposed
for a prediction goal at a non-observed location of the underlying process. The originality is based on
the ability to take into account some local spatial dependency. Large sample properties based on almost
complete and Lq -consistencies of the estimator are established.
Résumé. Dans cette note, nous présentons un estimateur à noyau du quantile conditionnel d’un processus
spatial non-stationnaire, pour un but de prédiction du processus considéré en un site non-observé. L’ori-
ginalité vient du fait que l’estimateur permet de prendre en compte une éventuelle dépendance locale des
données. Une étude asymptotique basée sur les convergences presque complète et en moyenne d’ordre q de
l’estimateur est proposée.
Mathematical subject classification (2010). 62H11, 62G20, 62M30.
Manuscript received 8 May 2020, accepted 26 June 2022.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
848 Serge Hippolyte Arnaud Kanga, Ouagnina Hili and Sophie Dabo-Niang
1. Introduction
Spatial statistical modeling is undergoing significant developments because of its recurrent use
in many fields such as epidemiology, econometrics, environmental and earth sciences, forestry,
agronomy, images analyses, among others. A main goal in spatial analyses, is prediction of some
features on un-sampled locations by taking into account some spatial dependence. There is a
wide literature on parametric spatial prediction compare to the nonparametric setting.
Some results in this direction are those of Biau and Cadre [1] on kernel prediction of a strictly
stationary real-valued random field indexed in (N∗)N . This work has been extended to several
directions. Dabo-Niang and Yao [5] considered kernel regression estimation and prediction of
continuously indexed and strictly stationary random fields.
Dabo-Niang et al. [3] proposed a new kernel spatial predictor of non-stationary processes
based on a conditional mean regression model. This last predictor depends on two kernels in
order to control both the distance between observations and that between spatial locations. The
regression model used by these authors may not be relevant in some situations. This could be
for instance the case of presence of extreme or outliers values. In fact, mean regression model is
much more sensitive to extreme or outliers values than the quantile one. Conditional quantile
spatial regression model may then be an alternative to mean regression one and is our interest in
this work.
In the spatial real-valued setting, we refer to Koenker and Mizera [11] for parametric quantile
estimation while Hallin et al. [8], proposed a non-parametric local linear conditional quantile es-
timator for strictly stationary spatial data. The last authors established consistency of their esti-
mate by Bahadur representation and an asymptotic normality result. Dabo-Niang and Thiam [4]
studied a kernel conditional quantile estimator for strictly stationary spatial processes and estab-
lished its asymptotic normality. Abdi et al. [12] established consistency in L2r (r ∈N∗) of a condi-
tional quantile estimator for a strictly stationary spatial random field. Some of these results have
been extended in the case of functional data.
All non-parametric conditional quantile estimators considered in the previous papers dealt
with strictly stationary spatial processes. For a recent review on non-parametric estimation in a
spatial context, refer to El Machkouri et al. [6], among others. The current work goes beyond and
extends the spatial regression model of Dabo-Niang et al. [3] on non-stationary processes to the
quantile regression context.
Namely, this note proposes a nonparametric conditional quantile estimator of locally sta-
tionary processes for a prediction purpose. Compare to Abdi et al. [12] and Dabo-Niang and
Thiam [4], the proposed estimator depends on a multiplicative kernel taking into account the
spatial positions of observations. This allows to take into account in the predictor, the number
of closest neighbors locations of the non-observed location to predict the process of interest.
In addition, the considered model does not assume strict stationary of the spatial process. This
idea was used for density function and prediction estimations in Dabo-Niang et al. [2] and Dabo-
Niang et al. [3], respectively.
The note is organized as follow. Section 2 introduces the proposed quantile predictor and gives
some large sample properties, namely almost complete and Lq -norm (q ∈N∗) consistencies with
rates when the considered sample is α-mixing. Some comments and main lines of the proofs are
given in the last section.
2. New kernel quantile estimator
We consider a spatial process (Zi = (Xi,Yi) ∈ Rd × R, i ∈ ZN ) defined over some probability
space (Ω,F ,P). The main purpose of this work is to estimate the quantile of (Yi, i ∈ ZN ), given
Serge Hippolyte Arnaud Kanga, Ouagnina Hili and Sophie Dabo-Niang 849
(Xi, i ∈ ZN ) at an unobserved location i0 ∈ ZN , in order to predict Yi0 . We assume that the
process is observable over the rectangular domain In = {i = (i1, . . . , iN ),1 ≤ ik ≤ nk , k = 1, . . . , N }
(a basic assumption in the non-parametric literature) and observed on a spatial set On ⊂ In,
n = (n1, . . . , nN ) of finite cardinality tending to ∞ as n →∞ (defined later), with i0 ∉On.
We do not suppose strict stationarity, rather we consider the following hypothesis. Let (Xi0 ,Yi0 )
has the same distribution as (X ,Y ) and the variables (Xi,Yi)i∈On be locally identically distributed
(see for instance Klemelä [10] who considered density estimation for locally identically time-
series data). Assume that, there is a sufficient number of (Xi,Yi) with a distribution closed to
that of (X ,Y ). One may imagine that when i is closed to i0, and if there is enough sites i closed
to i0, then the sequence (Xi,Yi)i∈On may be used to predict Yi0 under the condition Xi0 = xi0 ,
denoted by x in the following with abuse of notation. Then, a spatial predictor of Yi0 may account
which locations of On have an influence on the site i0 of interest. Let (X ,Y ) and (Xi,Yi) have
unknown densities with respect to Lebesgue measure, let fX Y and f be the densities of (X ,Y )
and X respectively.
Let n̂ = n1 × . . . ×nN be the size of In. We assume for simplicity that n1 = n2 = . . . = nN = n.
However the following results can be extended to a more general framework. We write n →∞ if
n →∞.
For the location i0, let kn = kn,i0 =
∑
i 1[∥i−i0∥≤dn] denote the number of neighbors i for which
the distance between i and i0 is less than or equal to dn > 0 such that dn → ∞ as n → ∞. This
last assumes that the number of neighbors locations increases (eventually) as the sample size
increases.
The used method is model free. It uses nonparametric kernel techniques to estimate condi-
tional quantiles, it is useful for prediction and may be an alternative to classical regression, see
Huang and Nguyen [9] for more details.
Let the conditional distribution function of Y given X , denoted by
F x (y) =P(Y ≤ y ∣∣ X = x) .
For 0 <α< 1,
tα = tα(x) = inf
{
y : F x (y) ≥α} ,
is called the α-th conditional quantile of Y given X = x.
This quantile is the solution of
F x (tα(x)) =α.
To insure the existence and uniqueness of tα(x), we assume that F x is strictly increasing. The
conditional distribution F x is estimated by the following proposed estimator
F̂ x (y) =

∑
i∈On K1
(
x−Xi
bn
)
K2
(
ρ−1n
∥∥∥ i0−in ∥∥∥)K3( y−Yihn )∑
i∈On K1
(
x−Xi
bn
)
K2
(
ρ−1n
∥∥∥ i0−in ∥∥∥) , if
∑
i∈On
K1
(
x−Xi
bn
)
K2
(
ρ−1n
∥∥∥ i0−in ∥∥∥) ̸= 0
0 otherwise,
(1)
where K1 defined on Rd and K2 defined on R are kernels functions, while K3 defined on R is a
distribution function. Moreover, bn, ρn and hn are sequences of positive numbers tending to 0
such as n̂ρNn b
d
n hn →∞, when n goes to ∞.
Let in what follow K2,ρn (∥i0 − i∥) = K2(ρ−1n ∥ i0−in ∥) and note that
∥ i0−in ∥ denotes the distance between normalized locations i0n = i01n , . . . , i0Nn and in = i1n , . . . , iNn .
As in Dabo-Niang et al. [13], ρ−1n ∥ i0−in ∥ ≤ 1 stands for ∥i0 − i∥ ≤ nρn.
The conditional quantile kernel estimate t̂α(x) is linked to the conditional distribution esti-
mate in the following way
F̂ x
(
t̂α(x)
)=α.
850 Serge Hippolyte Arnaud Kanga, Ouagnina Hili and Sophie Dabo-Niang
The proposed predictor of Yi0 is then Ŷ i0 = t̂α(x). Its large sample properties are studied in the
following with the help of some assumptions.
2.1. Assumptions
Let C or C ′ denote any nonnegative constant whose value is unimportant and may vary from line
to line. Let ε be an arbitrary small nonnegative number and un = ∏Ni=1(logni )(loglogni )1+ε. It is
obvious that
∑
1/(n̂un) <∞ (see Dabo-Niang and Thiam [4]), where the summation is over the
indice n such that ni ≥ 2, for all 1 ≤ i ≤ N . Let us assume the following conditions.
The random field (Zi, i ∈ZN ) is α-mixing: there is a function ϕ such that ϕ(t ) ↓ 0 as t →∞;
α
(
B(E),B
(
E′
))= sup
B ∈B(E),C ∈B(E′ )
|P(B ∩C )−P(B)P(C )|
≤ψ(Card(E),Card (E′))ϕ(dist (E,E′)) . (2)
where Card (E) (resp. Card (E′)) is the cardinality of the spatial set E (resp. E′), dist(E, E′) the eu-
clidian distance between E and E′ and ψ :N2 →R+ is a symmetric positive function nondecreas-
ing in each variable such that ϕ(0) = 1. We recall that (Zi, i ∈ ZN ) is said to be strongly mixing if
ψ≡ 1. In addition, we assume that
ϕ(t ) ≤C t−µ, µ> 4N . (3)
(H1) The densities fX Y and fX are continuous on Rd+1 and Rd respectively and fX > 0.
(H2) F x is of class C 1, f x (tα(x)) > 0, where f x is the conditional density of Y given X = x.
If f x (tα(x)) = 0, then one can use: F x (.) is of class C j , F x(k) (tα(x)) = 0 for 1 ≤ k ≤ j and
0 <C < |F x(k) (y)| <C ′ <∞. See Ferraty et al. [7] for more details.
(H3) For all (x1, x2) ∈Nx ×Nx , for all (y1, y2) ∈R2,∣∣F x (y1)−F x (y2)∣∣≤C (∥x1 −x2∥b1 + ∣∣y1 − y2∣∣b2) , b1 > 0, b2 > 0,
with Nx , a neighborhood of x.
(H4) (a) The densities fi and fXiYi of Xi and (Xi,Yi) respectively, verify
sup
x∈Rd ,∥i−i0∥≤h
∣∣ fi(x)− fX (x)∣∣= oP(1),
sup
x∈Rd ,∥i−i0∥≤h
∣∣ fXiYi (x, y)− fX Y (x, y)∣∣= oP(1), where h → 0.
(b) Local dependence condition. The joint probability density fXi Xi0 of (Xi, Xi0 ) exists, is
bounded such that | fXi Xi0 (z, t )− fi(z) fi0 (t )| ≤ C , for all z, t ∈ Rd , for all i ̸= i0, where
C is a positive constant.
(H5) There exist two constants C1i and C2i with 0 <C1i <C2i <∞, i = 1,2, such that
C111[0,1](s
′s) < K1(s) <C121[0,1](s′s) for s ∈Rd ,
where s′ is the transpose of s.
C211[0,1](t ) < K2(t ) <C221[0,1](t ) for t ∈R.
(H6) K3 is of class C 1 and symmetric and its derivative K
(1)
3 is a bounded kernel of compact
support. Moreover, we assume that the restriction of K3 on the set {t ∈ R,K3(t ) ∈ (0,1)} is
a strictly increasing function.
(H7) ψ(m,n) ≤C min(n,m) and n̂bdµ1n ρNµ1n hµ1n (log n̂)µ2 uµ3n →∞withµ> N (q +2), q ≥ 2 and
µ1 = µ
µ−N (q +2) , µ2 =
µ−2N
µ−N (q +2) , µ3 =
2N
N (q +2)−µ .
Serge Hippolyte Arnaud Kanga, Ouagnina Hili and Sophie Dabo-Niang 851
(H8) ψ(m,n) ≤ C (m + n + 1)ν and n̂bdµ
′
1
n h
µ′1
n ρ
Nµ′1
n (log n̂)
µ′2 u
µ′3
n → ∞ with µ > N (q + 1 + 2ν),
ν> 1, q ≥ 2 and
µ′1 =
d(N +µ)
µ−N (q +1+2ν) , µ
′
2 =
µ−N
µ−N (q +1+2ν) ,µ
′
3 =
2N
N (q +1+2ν−µ) .
Remark 1. (H5) is imposed for sake of simplicity and is satisfied by several kernels with compact
support. (H6) is classical in non-parametric quantile inference and is satisfied by kernels with
compact support. Gaussian kernel K (1)3 is also possible, it suffices to replace the compact support
assumption by
∫
Rd |t |b2 K (1)3 (t )d t <∞. (H6) ensures the existence and uniquness of the quantile
estimate t̂α(x). This hypothesis is discussed in Abdi et al. [12].
(H7)-(H8) concern the bandwidths and are based on the mixing condition to achieve the follow-
ing consistency results.
2.2. Main results
The main following result gives the almost complete convergence (a.c) of the conditional quantile
estimate t̂α(x) considered as a predictor of Yi0 .
Theorem 2. Under hypotheses (H1)-(H6),(H7) or (H8) and (2)-(3), we have∣∣t̂α(x)− tα(x)∣∣→ 0 a.c. (4)
A weak consistency is given in the following result.
Theorem 3. Under hypotheses (H1)-(H6) and (2)-(3), we have∥∥t̂α(x)− tα(x)∥∥q = E(∣∣t̂α(x)− tα(x)∣∣q )1/q =O (bb1n )+O (hb2n )+O ((n̂bdnρNn )−1/2) . (5)
Conclusion
The Note presents a kernel quantile estimate as a predictor of a real-valued spatial locally
stationary process at a given location. Large sample properties of the predictor are given towards
an almost complete and mean of order q consistencies under mixing conditions. These results are
the baseline of a work in progress, concerning asymptotic normality and finite sample properties
of the proposed predictor.
3. Main lines of the proofs
Sketch of the proof of Theorem 2. Under (H2), F̂ x (.) is continuous and strictly increasing, then
F̂ x
−1
(.) is also continuous and strictly increasing. The result is obtain, by using same arguments
as in the proof of Abdi et al. [12, Theorem 1] and applying Lemma 4. □
Lemma 4. Under conditions of Theorem 2,
F̂ x (tα(x))−F x (tα(x)) → 0, a.c. (6)
Sketch of the proof of Theorem 3. By the same decomposition as in the proof of Abdi et al. [12,
Theorem 2], and applying Lemmas 5, 6 and 7, we get the result. □
Lemma 5. Under assumptions of Theorem 3, we have
f̂ x (tα(x))− f x (tα(x)) → 0 a.c.
852 Serge Hippolyte Arnaud Kanga, Ouagnina Hili and Sophie Dabo-Niang
Lemma 6. Under assumptions H2 −H5,
γn1 =O
(
bb1n +hb2n
)
,
with
γn1 =
∥∥∥∥∥ ∑i∈In Wi
(
E (Hi (tα(x)) |Xi)−F x (tα(x))
)∥∥∥∥∥
q
,
Wi = K1iK2i∑
i∈In K1iK2i
, K1i = K1
(
x −Xi
bn
)
, K2i = K2,ρn (∥i0 − i∥), and Hi(.) = K3
(
.−Yi
hn
)
.
Lemma 7. If the conditions of Theorem 3 are satisfied, then we get γn2 = O((n̂ρNn bdn )−1/2), and
γn3 =O((n̂ρNn bdn )−1/2), where
γn2 =
∥∥∥∥∥ ∑i∈In Wi (Hi (tα(x))−E (Hi (tα(x)) |Xi))
∥∥∥∥∥
q
,
and
γn3 =
(
P
( ∑
i∈In
Wi = 0
))1/q
.
References
[1] G. Biau, B. Cadre, “Nonparametric Spatial Prediction”, Stat. Inference Stoch. Process. 7 (2004), no. 3, p. 327-349.
[2] S. Dabo-Niang, L. Hamdad, C. Ternynck, A.-F. Yao, “A Kernel Spatial Density Estimation Allowing for the Analysis
of Spatial Clustering. Application to Monsoon Asia Drought Atlas data”, Stochastic environmental research and risk
assessment 28 (2014), no. 8, p. 2075-2099.
[3] S. Dabo-Niang, C. Ternynck, A.-F. Yao, “Nonparametric Prediction of Spatial Multivariate data”, J. Nonparametric
Stat. 28 (2016), no. 2, p. 428-458.
[4] S. Dabo-Niang, B. Thiam, “Robust Quantile Estimation and Prediction for Spatial Processes”, Stat. Probab. Lett. 80
(2010), no. 17-18, p. 1447-1458.
[5] S. Dabo-Niang, A.-F. Yao, “Kernel Regression Estimation for Continuous Spatial Processes”, Math. Methods Stat. 16
(2007), no. 4, p. 298-317.
[6] M. El Machkouri, X. Fan, L. Reding, “On the Nadaraya–Watson kernel regression estimator for irregularly spaced
spatial data”, J. Stat. Plann. Inference 205 (2020), p. 92-114.
[7] F. Ferraty, A. Rabhi, P. Vieu, “Conditional Quantiles for Dependent Functional Data with Application to the Climatic
El Niño Phenomenon”, Sankhyā 67 (2005), no. 2, p. 378-398.
[8] M. Hallin, Z. Lu, K. Yu, “Local Linear Spatial Quantile Regression”, Bernoulli 15 (2009), no. 3, p. 659-686.
[9] M. Huang, C. Nguyen, “A Nonparametric Approch for Quantile Regression”, J. Stat. Distrib. Appl. 5 (2018), article
no. 3.
[10] J. Klemelä, “Density Estimation with Locally Identically Distributed Data and with Locally Stationary Data”, J. Time
Ser. Anal. 29 (2008), no. 1, p. 125-141.
[11] R. Koenker, I. Mizera, “Penalized Triograms: Total Variation Regularization for Bivariate Smoothing”, J. R. Stat. Soc.,
Ser. B, Stat. Methodol. 66 (2004), no. 1, p. 145-164.
[12] S. A. Ould Abdi, S. Dabo-Niang, A. Diop, A. Ould Abdi, “Consistency of a Nonparametric Conditional Quantile
Estimator for Random Fields”, Math. Methods Stat. 19 (2010), no. 1, p. 1-21.
[13] C. Ternynck, “Contributions à la Modélisation de Données Spatiales et Fonctionnelles: Applications”, PhD Thesis,
Université Charles de Gaulle-Lille III, Lille, France, 2014.


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