Numérisation de Documents Anciens Mathématiques

Ash, J. Marshall

Catoiu, Stefan

Fejzić, Hajrudin

Comptes Rendus
Mathématique
J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić
A new proof of the GGR conjecture
Volume 361 (2023), p. 349-353
https://doi.org/10.5802/crmath.413
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. 349-353
https://doi.org/10.5802/crmath.413
Functional analysis / Analyse fonctionnelle
A new proof of the GGR conjecture
J. Marshall Asha, Stefan Catoiu∗, a and Hajrudin Fejzićb
a Department of Mathematics, DePaul University, Chicago, IL 60614
b Department of Mathematics, California State University, San Bernardino, CA 92407
E-mails: mash@depaul.edu (J. M. Ash), scatoiu@depaul.edu (S. Catoiu),
hfejzic@csusb.edu (H. Fejzić)
Abstract. For each positive integer n, function f , and point x, the 1998 conjecture by Ginchev, Guerragio,
and Rocca states that the existence of the nth Peano derivative f(n)(x) is equivalent to the existence of all
n(n +1)/2 generalized Riemann derivatives,
Dk,− j f (x) = lim
h→0
1
hn
k∑
i=0
(−1)i
(
k
i
)
f (x + (k − i − j )h),
for j ,k with 0 ≤ j < k ≤ n. A version of it for n ≥ 2 replaces all − j with j and eliminates all j = k − 1. Both
the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on
highly non-trivial combinatorial algorithms. This article provides a second, inductive, algebraic proof to each
of these theorems, based on a reduction to (Laurent) polynomials.
Mathematical subject classification (2010). 26A24, 13F20, 15A03, 26A27.
Funding. This paper is in final form and no version of it will be submitted for publication elsewhere.
Manuscript received 10 June 2021, revised 30 April 2022, accepted 24 August 2022.
Arguably, the most important application of higher order derivatives is Taylor’s theorem,
asserting that an n times differentiable function f at a point c is approximated near c by a
polynomial p with error f (c +h)− p(c +h) = o(hn) as h → 0. It is also well known that Taylor’s
theorem provides only a sufficient condition for this approximation to happen, and all functions
f with this property are said to be n times Peano differentiable at c.
In 1998, Ginchev, Guerragio, and Rocca (GGR) conjectured the following result:
Conjecture (GGR Conjecture). When n ≥ 2, the following two conditions,
(i) f is n −1 times Peano differentiable at c and
(ii) limh→0 1hn
∑n
j=0(−1)n− j
(n
j
)
f (c + ( j −k)h) exists for all k with 0 ≤ k ≤ n −1,
are sufficient to make f an n times differentiable function at c.
They proved the theorem by hand for n = 2,3,4 in [7], and with the use of a computer they
proved it for n = 5,6,7,8 in [8], leaving the rest as a conjecture. The GGR conjecture was recently
proved in [2] and is now a theorem. The result proved in [2] is slightly stronger for n odd, by
eliminating the second condition for k = 0. A variant of the conjecture, where the bounds for k
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
350 J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić
are replaced by −(n −2) ≤ k ≤ 0, is proved in [4]. Article [6] sheds some light towards extending
the GGR conjecture for n = 1.
The original statement of the GGR conjecture is actually an equivalent version of the above, by
the principle of mathematical induction: If Dn,k denote the above limits, Ginchev, Guerragio, and
Rocca conjectured that the
(n+1
2
)
limits Dm,k for 0 ≤ k < m ≤ n will be enough for f to be n times
Peano differentiable at c. The limit Dn,0 = Dn,0 f (c) is called the nth Riemann derivative of f at c,
and Dn,n/2 f (c) is the nth symmetric Riemann derivative of f at c. Both of these derivatives were
invented by Riemann in the mid 1800s, see [11]. The Peano derivatives were invented by Peano
in [10] in 1892, and then developed greatly by de la Vallée Poussin in [12]. For this reason, they are
often referred to as de la Vallée Poussin derivatives.
The purpose of this note is to provide a new, simple proof to both the GGR conjecture and its
variant.
∗ ∗ ∗
Let Rn(h) be the difference defined recursively by R1(h) = f (c + h) − f (c), and Rn(h) =
Rn−1(2h)−2n−1Rn−1(h) for n ≥ 2. Closed form formulas for generalizations of these differences
are deduced in [3]; they involve the Gaussian or q-binomial coefficients, so they are q-analogues
of the Riemann differences. Other q-analogues of Riemann differences are found in [1] and [5].
We will use the following 1936 result of Marcinkiewicz and Zygmund in [9].
Theorem 1. Suppose f is n −1 times Peano differentiable at c. If limh→0 Rn(h)/hn exists, then f
is n times Peano differentiable at c.
If we denote ∆k (h) as the difference
∑n
j=0(−1)n− j
(n
j
)
f (c + ( j −k)h) in the GGR limit condition,
then this condition can be concisely written as limh→0∆k (h)/hn exists. It is also obvious that if all
GGR limit conditions are met then the following linear combination
lim
h→0
∑
k ck∆k (sk h)
hn
exists, where ck , sk are arbitrary real constants.
Hence, if we can show that R(h) is a linear combination
∑
k ck∆k (sk h), then the GGR conjecture
follows from Theorem 1.
That this is indeed the case will follow from the analogous result for polynomials, via the linear
isomorphism, ∆(h) := ∑c j f (c + b j h) 7→ d(t ) := ∑c j t b j , from the R-space of all differences of
f at c and h with integer nodes (the b j ), to the R-space R[t , t−1] of all Laurent polynomials in
indeterminate t with real coefficients. In this way, (1) if
∆k (h) =
n∑
j=0
(−1)n− j
(
n
j
)
f (c + ( j −k)h),
then
dk (t ) =
n∑
j=0
(−1)n− j
(
n
j
)
t j−k = t−k (t −1)n ;
(2) the polynomial corresponding to ∆k (sh) is dk (t
s ); and (3) if rn(t ) is the polynomial that
corresponds to Rn(h) under this linear isomorphism, then its recursive definition is r1(t ) = t −1,
and rn(t ) = rn−1(t 2)−2n−1rn−1(t ) for n ≥ 2.
Based on these properties of the above linear isomorphism, we will be done by showing that
the following result is true.
Theorem 2. There are constants ck and sk such that rn(t ) =
∑
ck dk (t
sk ).
Before proceeding with the proof of Theorem 2, we need to make a clarification. Our solution
to the theorem has the numbers sk non-negative integers instead of real numbers, so we can
think of sk as s, with s ≥ 0. Then the ck are viewed as ck,s , for a more precisely indexed sum
J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić 351
∑
ck dk (t
sk ) =∑∞s=0 ∑k ck,s dk (t s ) =∑∞s=0 ∑k ck,s t−sk (t s−1)n , where the ranges for k will be clarified
later, since these are different for different cases in the proof of the theorem. Summarizing, in
order to prove Theorem 2, it suffices to show that
rn(t ) ∈Vn := span
{
t−sk (t s −1)n
∣∣∣k = (0),1, . . . , n −1, s = 1,2, . . .} ,
where (0) means that the value 0 is taken only for n even.
∗ ∗ ∗
The proof of Theorem 2 is much shorter for the variant of the GGR conjecture than it is for the
conjecture itself. For this reason, we deal with the variant first.
Proof of Theorem 2 (Variant Case)
In this case, by replacing the index of summation k with −k, the range −(n −2) ≤ k ≤ 0 becomes
0 ≤ k ≤ n −2. In this way, Vn becomes
Vn = span
{
t sk
(
t s −1)n ∣∣∣k = 0, . . . , n −2; s = 1,2, . . .} .
The following lemma provides a new set of generators for the space Vn .
Lemma 3. Vn = span{(t s −1)n+k | k = 0,1, . . . , n −2, s = 1,2, . . .}.
Proof. It suffices to show the following equality of subspaces:
span
{
t k (t −1)n
∣∣∣k = 0,1, . . . , n −2}= span {(t −1)n+k ∣∣∣k = 0,1, . . . , n −2} .
Indeed, this is the result of multiplying by (t −1)n both sides of the obvious equation
span
{
1, t , t 2, . . . , t n−2
}= span{1, t −1,(t −1)2, . . . , (t −1)n−2} . 
We are now ready to proceed with the proof of Theorem 2 in its variant case.
Proof of Theorem 2 (Variant Case). Induct on n. When n = 2, r2 = (t − 1)2 is clearly in V2.
Suppose rn ∈ Vn , for some n, n ≥ 2, and prove the same property for n + 1. By Lemma 3, rn is
a linear combination of polynomials of the form (t s − 1)n+k , where s is a positive integer and
k = 0,1, . . . ,n − 2. By the recursion, rn+1(t ) = rn(t 2) − 2nrn(t ) will be a linear combination of
polynomials (
t 2s −1)n+k −2n (t s −1)n+k , for various k and s.
By Lemma 3, these polynomials belong to Vn+1 in all cases, except for k = 0, when(
t 2s −1)n −2n (t s −1)n = (t s −1)n ((t s +1)n −2n)= (t s −1)n+1 p (t s) ,
where p is a polynomial in t of degree n − 1, so that (t s − 1)n+1p(t s ) belongs to the subspace
span{(t s −1)n+1, (t s −1)n+2, . . . , (t s −1)2n} of Vn+1. 
Proof of Theorem 2 (GGR Case)
The GGR case in the proof of Theorem 2 is similar to the variant case. The proof of the inductive
step is now split further into two subcases, n odd and n even. In both cases, following the refined
result of the GGR Theorem from [2],
Vn = span
{
t−sk
(
t s −1)n ∣∣∣k = (0),1, . . . , n −1; s = 1,2, . . .} ,
where (0) means that 0 is taken for n even, and not taken for n odd. More explicitly, this is
Vn =
{
span
{
(t −1)n , t−1(t −1)n , . . . , t−(n−1)(t −1)n , . . .} , n even,
span
{
t−1(t −1)n , t−2(t −1)n , . . . , t−(n−1)(t −1)n , . . .} , n odd,
352 J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić
where the last dots in both cases mean that the generating set also includes the previously listed
generators evaluated at t s , for all s at least 2. Let Wn be the subspace of Vn spanned by all
generators with s = 1. Then Wn has the expression
Wn =
{
span
{
(t −1)n , t−1(t −1)n , . . . , t−(n−1)(t −1)n} , n even,
span
{
t−1(t −1)n , t−2(t −1)n , . . . , t−(n−1)(t −1)n} , n odd.
The following lemma provides new sets of generators for Wn in both parity cases.
Lemma 4. With the above notation,
Wn =
{
span
{
t−n/2(t −1)n , t−1(t −1)n+1, . . . , t−(n−1)(t −1)n+1} , n even,
span
{
t−(n−1)/2(t −1)n , t−2(t −1)n+1, . . . , t−(n−1)(t −1)n+1} , n odd.
Proof. When n is even, the result follows from t−n/2(t − 1)n being one of the generators in the
definition of Wn , and t−k (t −1)n+1 = t−(k−1)(t −1)n − t−k (t −1)n , for each k = 1, . . . , n−1. The case
when n is odd has a similar proof. 
We are now ready to prove Theorem 2 in the GGR case.
Proof of Theorem 2 (Case GGR). Induct on n. When n = 1, r1 = t−1 ∈V1. We assume that rn ∈Vn
and prove that rn+1 ∈Vn+1 in two possible cases:
Case 1. When n is even,
Vn+1 = span
{
t−1(t −1)n+1, t−2(t −1)n+1, . . . , t−n(t −1)n+1, . . .} .
The inductive hypothesis and Lemma 4 imply that rn(x) is a linear combination of
t−n/2(t −1)n and t−k (t −1)n+1, for k = 1, . . . , n −1,
and their evaluations at t s , for s at least 2. Then rn+1(t ) = rn(t 2)−2nrn(t ) is a linear combination
of two kinds of polynomials and their evaluations at t s , for s ≥ 2. The first kind of polynomial has
the form t−n(t 2 −1)n −2n t−n/2(t −1)n
= t−n(t −1)n ((t +1)n −2n t n/2)= t−n(t −1)n+1p(t ),
where p(t ) is a polynomial degree n −1, hence the whole expression lives inside of
(t −1)n+1 span {t−1, t−2, . . . , t−n} ,
a subspace of Vn+1. The polynomials of the second kind are polynomials of the form
t−2k
(
t 2 −1)n+1 −2n t−k (t −1)n+1, for k = 1, . . . , n −1.
Their second term is a scalar multiple of a generator of Vn+1, while their first term is an (s = 2)-
dilation of the same generator, so all polynomials of the second kind also belong to Vn+1. We
conclude that rn+1 ∈Vn+1, as needed.
Case 2. When n is odd,
Vn+1 = span
{
(t −1)n+1, t−1(t −1)n+1, . . . , t−n(t −1)n+1, . . .} .
The inductive hypothesis and Lemma 4 imply that rn(x) is a linear combination of
t−(n−1)/2(t −1)n and t−k (t −1)n+1, for k = 2, . . . , n −1,
and their evaluations at t s , for s at least 2. Then rn+1(t ) = rn(t 2)−2nrn(t ) is a linear combination
of two kinds of polynomials and their evaluations at t s , for s ≥ 2. The first kind of polynomial is
of the form t−(n−1)(t 2 −1)n −2n t−(n−1)/2(t −1)n
= t−(n−1)(t −1)n ((t +1)n −2n t (n−1)/2)= t−(n−1)(t −1)n+1p(t ),
where p(t ) is a polynomial degree n −1, hence the above expression lives inside of
(t −1)n+1 span {1, t−1, t−2, . . . , t−(n−1)} ,
J. Marshall Ash, Stefan Catoiu and Hajrudin Fejzić 353
a subspace of Vn+1. The polynomials of the second kind are polynomials of the form
t−2k
(
t 2 −1)n+1 −2n t−k (t −1)n+1, for k = 2, . . . , n −1.
Their second term is a scalar multiple of a generator of Vn+1, while their first term is an (s = 2)-
dilation of the same generator, so all polynomials of the second kind also belong to Vn+1. We
conclude that rn+1 ∈Vn+1 in this case as well. 
References
[1] J. M. Ash, S. Catoiu, “Quantum symmetric Lp derivatives”, Trans. Am. Math. Soc. 360 (2008), no. 2, p. 959-987.
[2] J. M. Ash, S. Catoiu, “Characterizing Peano and symmetric derivatives and the GGR conjecture’s solution”, Int. Math.
Res. Not., IMRN (2022), no. 10, p. 7893-7921.
[3] J. M. Ash, S. Catoiu, H. Fejzić, “Gaussian Riemann derivatives”, https://arxiv.org/abs/2211.09209, to appear in Israel
J. Math., 23 pp., online first, 2022.
[4] ——— , “Two pointwise characterizations of the Peano derivative”, https://arxiv.org/abs/2209.04088, preprint, 2022.
[5] J. M. Ash, S. Catoiu, R. Ríos-Collantes-de Terán, “On the nth quantum derivative”, J. Lond. Math. Soc. 66 (2002), no. 1,
p. 114-130.
[6] S. Catoiu, “A differentiability criterion for continuous functions”, Monatsh. Math. 197 (2022), no. 2, p. 285-291.
[7] I. Ginchev, A. Guerraggio, M. Rocca, “Equivalence of Peano and Riemann derivatives”, in Generalized convexity and
optimization for economic and financial decisions (Verona, 1998), Pitagora Editrice, Bologna, 1998, p. 169-178.
[8] I. Ginchev, M. Rocca, “On Peano and Riemann derivatives”, Rend. Circ. Mat. Palermo 49 (2000), no. 3, p. 463-480.
[9] J. Marcinkiewicz, A. Zygmund, “On the differentiability of functions and summability of trigonometric series”,
Fundam. Math. 26 (1936), p. 1-43.
[10] G. Peano, “Sulla formula di Taylor”, Atti Acad. Sci. Torino 27 (1892), p. 40-46.
[11] B. Riemann, Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe, Ges. Werke, 2. Aufl., Leipzig,
Dieterichschen Buchhandlung, 1892 (1867), 227-271 pages.
[12] C.-J. de la Vallée Poussin, “Sur l’approximation des fonctions d’une variable réelle et de leurs dérivées par les
pôlynomes et les suites limitées de Fourier”, Belg. Bull. Sciences 1908 (1908), p. 193-254.


A new proof of the GGR conjecture

Kempf, Volker

Comptes Rendus
Mathématique
Volker Kempf
Brezzi–Douglas–Marini interpolation on anisotropic simplices and
prisms
Volume 361 (2023), p. 437-443
https://doi.org/10.5802/crmath.424
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. 437-443
https://doi.org/10.5802/crmath.424
Numerical analysis / Analyse numérique
Brezzi–Douglas–Marini interpolation on
anisotropic simplices and prisms
Interpolation Brezzi–Douglas–Marini sur les simplexes
et prismes anisotropes
Volker Kempf ∗, a
a Universität der Bundeswehr München, Germany
E-mail: volker.kempf@unibw.de (V. Kempf)
Abstract. The Brezzi–Douglas–Marini interpolation error on anisotropic elements has been analyzed in two
recent publications, the first focusing on simplices with estimates in L2, the other considering parallelotopes
with estimates in terms of Lp -norms. This contribution provides generalized estimates for anisotropic
simplices for the Lp case, 1 ≤ p ≤∞, and shows new estimates for anisotropic prisms with triangular base.
Résumé. L’erreur d’interpolation de Brezzi–Douglas–Marini sur les éléments anisotropes a été analysée dans
deux publications récentes, la première se concentrant sur les simplices avec des estimations dans L2, l’autre
considérant les parallelotopes avec des estimations en termes de normes Lp . Notre contribution fournit des
estimations généralisées pour les simplexes anisotropes pour le cas Lp , 1 ≤ p ≤ ∞, et montre de nouvelles
estimations pour les prismes anisotropes à base triangulaire.
2020 Mathematics Subject Classification. 65D05, 65N30.
Manuscript received 12 May 2022, revised 9 September 2022, accepted 15 September 2022.
1. Introduction
The Brezzi–Douglas–Marini (BDM) finite element [8] was introduced to approximate H div by
polynomials. This proves useful for problems of incompressible fluid flow, where recent ap-
proaches employ H div-conforming discretizations to approximate the velocity solution. The cor-
responding interpolation operator can also be used as reconstruction operator to gain pressure-
robust methods in the spirit of [7]. Boundary layers or edge singularities in these problems re-
quire the use of anisotropic, i.e., highly stretched elements, so interpolation error estimates for
such settings are required, see [3]. In [2], these estimates were shown for simplicial elements in
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
438 Volker Kempf
terms of the L2-norm, and [5] contains estimates for anisotropic parallelotopes in terms of Lp -
norms.
The main focus in [5] is on estimates in the H div-norm and while the technique for these proofs
can also be used for simplices, it is not applicable for the case of prisms as defined in [8], as here
the commuting diagram property [5, (3)] is not satisfied. Interestingly, it is neither satisfied on
cubes as defined in [8]; here [5] uses the original definition of the BDM elements, see, e.g., [4].
In the following sections we provide a generalization of the results from [2] to Lp spaces,
1 ≤ p ≤ ∞, and show new interpolation error estimates for anisotropic triangular prisms. The
results with detailed proofs are contained in the author’s PhD thesis [6] that is to appear.
Notation
Vectors, vector valued functions and the spaces of such functions are set in bold and we use
In = {1, . . . , n} for index sets. The spatial dimension is denoted by d and the expression a . b
means there is a positive constant C so that a ≤ C b. The norms of the Sobolev spaces W m,p (G)
are denoted by ‖·‖m,p,G , multi-indices by α. The directional derivative in direction l is denoted
by ∂∂l .
2. Error estimates on anisotropic simplices
We state the anisotropic interpolation error estimates on simplices in terms of Lp -norms, which
is a generalization of the results from [2], where only p = 2 was considered. As the proofs are
largely analogous to those in [2], we omit them here. The principal idea of the proof is to first
show stability estimates on the reference elements and then transfer them in two steps first to an
element of a reference family using an affine transformation with a diagonal matrix and then to
the general element.
Recall that a simplex is said to satisfy the maximum angle condition if all angles within and
between facets are bounded by a constant φ̄<π. In addition, it satisfies the regular vertex property
if for one vertex there is a constant c̄ > 0 so that |det N | ≥ c̄ holds for the matrix N whose columns
consist of the unit vectors l i , i ∈ Id , along the outgoing edges from this vertex. The lengths of
the edges corresponding to the vectors l i are the element size parameters and are denoted by hi .
The degrees of freedom for the BDM interpolation operator I BDMk of order k on simplices can be
found in [8, p. 59].
The error estimates for the BDM interpolation depending on the geometric regularity are given
by the two following theorems, cf. [2, Theorems 4.3, 4.4] and [1, Theorems 6.2, 6.3].
Theorem 1. Let a simplicial element T satisfy the regular vertex property with constant c̄. Then
for k ≥ 1, 0 ≤ m ≤ k and v ∈W m+1,p (T ), 1 ≤ p ≤∞, the estimate∥∥v − I BDMk v
∥∥
0,p,T .
∑
|α|=m+1
hα
∥∥Dαl v
∥∥
0,p,T +hT
∑
|α|=m
hα
∥∥Dαl div v
∥∥
0,p,T
holds, the constant only depends on c̄ and k. Here
hT = diamT,hα =
∏
j ∈ Id
h
α j
j and D
α
l =
∂|α|
∂
α1
l 1
· · ·∂αdl d
.
Theorem 2. Let a simplicial element T satisfy the maximum angle condition with constant φ̄.
Then for k ≥ 1, 0 ≤ m ≤ k and v ∈W m+1,p (T ), 1 ≤ p ≤∞, the estimate∥∥v − I BDMk v
∥∥
0,p,T . h
m+1
T
∥∥Dm+1v
∥∥
0,p,T
holds, the constant only depends on φ̄ and k. The notation Dn means the sum of the absolute values
of all derivatives of order n.
Volker Kempf 439
1
1
1
p1
p2
p3
p1 p
2
p3
P̂
x̂1
x̂2
x̂3
JP̃
h1
h2
h3
P̃
x̃1
x̃2
x̃3
Figure 1. Reference prism P̂ with vertex numbering, and a transformed prism of the
reference family RP .
3. Error estimates on anisotropic prisms
Similar results can be achieved for triangular prisms with some modifications. The prismatic
reference element P̂ with the notation for the vertices is given on the left hand side of Figure 1.
The element P̂ is transformed to an element P̃ of the reference family RP by the transforma-
tion x̃ = JP̃ x̂ , where
JP̃ =


h1 0 0
0 h2 0
0 0 h3

 , (1)
see Figure 1. The vertical facet of P̂ opposite of the vertices p i and p
i is denoted by ei , the
horizontal facet at x̂3 = 0 by eb and the one at x̂3 = 1 by et . The facet normals of P̂ are thus
given by
nP̂ ,e1 =


−1
0
0

 , nP̂ ,e2 =


0
−1
0

 , nP̂ ,e3 =
1p
2


1
1
0

 , nP̂ ,eb =


0
0
−1

 , nP̂ ,et =


0
0
1

 .
The local BDM interpolation operator I BDMk of order k on a prism P maps into the space
P k,k (P ) of vector valued polynomials of total degree k in x1 and x2 and degree k in x3. It is defined
by the functionals, see [8, p. 64],∫
ei
I BDMk v ·nP,ei z ds =
∫
ei
v ·nP,ei z ds ∀ z ∈ Pk (ei ), i ∈ {b, t }, (2a)
∫
ei
I BDMk v ·nP,ei z ds =
∫
ei
v ·nP,ei z ds ∀ z ∈Qk (ei ), i ∈ I3, (2b)
∫
P
(
I BDMk v
)
3 z3 dx =
∫
P
v3z3 dx ∀ z3 ∈ Pk,k−2(P ), (2c)
∫
P
(
I BDMk v
)
1 z1 +
(
I BDMk v
)
2 z2 dx =
∫
P
v1z1 + v2z2 dx ∀ (z1, z2) ∈Pk−1,k (P ), (2d)
where Pk (e) is the space of polynomials with maximal degree k, Qk (e) is the space of polynomials
with degree k in each of the dimensions of the facet e, Pm,n(P ) is the space of polynomials
with total degree m in x1 and x2, and degree n in x3. The space Pm,n(P ) consists of pairs of
polynomials with degree n in x3, which are for fixed x3, i.e., on the triangle Tx3 , in the space
N m(Tx3 ) = P m−1(Tx3 )⊕Sm(Tx3 ), where Sm(Tx3 ) = {p ∈ P m(Tx3 ) : p(x) · x = 0 ∀ x ∈ Tx3 }.
We first require the analog of [2, Lemma 3.1] for the prism reference element. The proof is
omitted here for brevity since it follows along the same steps as the proof for the analogous lemma
for simplices.
440 Volker Kempf
Lemma 3. Let P̂ be the reference element from Figure 1, f̂ j ∈ Lp (e j ), j ∈ I2, f̂ 3 ∈ Lp (eb), 1 ≤ p ≤∞,
and
û(x̂) = ( f̂ 1(x̂2, x̂3),0,0
)T
, v̂ (x̂) = (0, f̂ 2(x̂1, x̂3),0
)T
, ŵ (x̂) = (0,0, f̂ 3(x̂1, x̂2)
)T
.
Then there are functions q̂ j ∈ Pk (e j ), j ∈ I2, q̂3 ∈ Pk (eb), so that
Î BDMk û(x̂) =
(
q̂1(x̂2, x̂3),0,0
)T
, Î BDMk v̂ (x̂) =
(
0, q̂2(x̂1, x̂3),0
)T
, Î BDMk ŵ (x̂) =
(
0,0, q̂3(x̂1, x̂2)
)T
.
With this lemma, we get the stability estimate on the reference element.
Lemma 4. Let û ∈W 1,p (P̂ ), 1 ≤ p ≤∞. Then we have the estimates
∥∥(Î BDMk û
)
i
∥∥
0,p,P̂
. ‖ûi‖1,p,P̂ +
∥∥∥d̂iv û
∥∥∥
0,p,P̂
+
∥∥∥∥
∂û3
∂x̂3
∥∥∥∥
0,p,P̂
, i ∈ I2,
∥∥(Î BDMk û
)
3
∥∥
0,p,P̂
. ‖û3‖1,p,P̂ +
∥∥∥d̂iv û
∥∥∥
0,p,P̂
.
Proof. The proofs for i ∈ I2 are analogous so we only show the details for the first and third
components, starting with i = 1. Let
û∗ =


0
û2(x̂1,0, x̂3)
û3(x̂1, x̂2,0)

 , v̂ = û − û∗ =


û1
û2 − û2(x̂1,0, x̂3)
û3 − û3(x̂1, x̂2,0)

 .
The previous Lemma 3 thus yields (Î BDMk v̂ )1 = (Î BDMk û)1, and it holds d̂iv v̂ = d̂iv û−d̂iv û∗ = d̂iv û.
We define the two functions
v̂∗ =


0
x̂2q̂2
x̂3q̂3

 , ŵ = v̂ − v̂∗ =


v̂1
v̂2 − x̂2q̂2
v̂3 − x̂3q̂3

 ,
where q̂2 ∈ Pk−1,k (P̂ ), q̂3 ∈ Pk,k−1(P̂ ), so that∫
P̂
ŵ2z dx =
∫
P̂
(v̂2 − x̂2q̂2)z dx = 0 ∀ z ∈ Pk−1,k (P̂ ),
∫
P̂
ŵ3z dx =
∫
P̂
(v̂3 − x̂3q̂3)z dx = 0 ∀ z ∈ Pk,k−1(P̂ ).
This means that the functions q̂2 and q̂3 are well defined. Since v̂∗ ∈ P k,k (P̂ ) it follows that
Î BDMk v̂∗ = v̂∗ and thus
(
Î BDMk ŵ
)
1 =
(
Î BDMk v̂
)
1 =
(
Î BDMk û
)
1 .
The interpolated function Î BDMk ŵ = t̂ = (t̂ 1, t̂ 2, t̂ 3)T is then defined by the relations, see (2),∫
eb
t̂ 3z ds =
∫
eb
ŵ3z ds = 0 ∀ z ∈ Pk (eb),
∫
et
t̂ 3z ds =
∫
et
ŵ3z ds =
∫
et
ŵ3z ds −
∫
eb
ŵ3z ds =
∫
P̂
∂ŵ3
∂x̂3
z dx ∀ z ∈ Pk (et ),
∫
e1
t̂ 1z ds =
∫
e1
ŵ1z ds ∀ z ∈Qk (e1),
∫
e2
t̂ 2z ds =
∫
e2
ŵ2z ds = 0 ∀ z ∈Qk (e2),
∫
e3
(t̂ 1 + t̂ 2)z ds =
∫
e3
(ŵ1 + ŵ2)z ds ∀ z ∈Qk (e3),
∫
P̂
t̂ 3z3 dx =
∫
P̂
ŵ3z3 dx =
∫
P̂
(v̂3 − x̂3q̂3)z3 dx = 0 ∀ z3 ∈ Pk,k−2(P̂ ),
∫
P̂
t̂ 1z1 + t̂ 2z2 dx =
∫
P̂
ŵ1z1 + ŵ2z2 dx =
∫
P̂
ŵ1z1 dx ∀ (z1, z2) ∈Pk−1,k (P̂ ),
Volker Kempf 441
where the definitions of q̂2, q̂3, and that ŵ2|e2 ≡ 0, ŵ3|eb ≡ 0 were used. Additional computations
for the relation on e3 yield
1p
2
∫
e3
(t̂ 1 + t̂ 2)z ds = 1p
2
∫
e3
(ŵ1 + ŵ2)z ds =
∫
e3
ŵ ·ne3 ds
=
∫
P̂
(
∂ŵ1
∂x̂1
+ ∂ŵ2
∂x̂2
)
z dx +
∫
P̂
ŵ1
∂z
∂x̂1
dx −
∫
∂P̂\e3
ŵ ·n∂P̂ z ds
=
∫
P̂
(
d̂iv ŵ −2∂ŵ3
∂x̂3
)
dx +
∫
P̂
ŵ1
∂z
∂x̂1
dx −
∫
e1
ŵ1z ds .
Thus, the terms∫
e1
ŵ1z ds =
∫
e1
û1z ds ,
∫
P̂
ŵ1z dx =
∫
P̂
û1z dx ,
∫
P̂
∂ŵ3
∂x̂3
z dx =
∫
P̂
∂û3 − x̂3q̂3
∂x̂3
z dx ,
∫
P̂
(d̂iv ŵ )z dx =
∫
P̂
d̂iv(û − v̂∗)z dx
define the interpolant. We get the desired estimate
∥∥(Î BDMk û
)
1
∥∥
0,p,P̂
. ‖û1‖1,p,P̂ +
∥∥∥d̂iv û
∥∥∥
0,p,P̂
+
∥∥∥∥
∂û3
∂x̂3
∥∥∥∥
0,p,P̂
using a trace theorem and steps similar to those in the proof of [1, Lemma 3.3] to estimate the
terms involving x̂3q̂3 and d̂iv v̂∗. For the third component, with the definitions
v̂ = û − û∗ =


û1 − û1(0, x̂2, x̂3)
û2 − û2(x̂1,0, x̂3)
û3

 , v̂∗ =


x̂1q̂1
x̂2q̂2
0

 , ŵ = v̂ − v̂∗ =


v̂1 − x̂1q̂1
v̂2 − x̂2q̂2
v̂3

 ,
where the functions q̂ j ∈ Pk−1,k (P̂ ), j ∈ I2, are now defined by∫
P̂
ŵ j z dx =
∫
P̂
(
v̂ j − x̂ j q̂ j
)
z dx = 0 ∀ z ∈ Pk−1,k (P̂ ),
the relevant terms of the interpolation relations are∫
eb
ŵ3z ds =
∫
eb
û3z ds ,
∫
et
ŵ3z ds =
∫
et
û3z ds ,
∫
P̂
ŵ3z3 dx =
∫
P̂
û3z3 dx ,
∫
P̂
d̂iv ŵ z dx =
∫
P̂
d̂iv(û − v̂∗)z dx .
From here the same steps as for the first component yield the desired estimate. 
Using the transformation (1) we bring the stability estimate to an element of the reference
family.
Lemma 5. Let P̃ = JP̃ P̂+x0, x0 ∈R3, and ṽ ∈W 1,p (P̃ ), 1 ≤ p ≤∞. Then on the prism P̃ the estimate
∥∥Ĩ BDMk ṽ
∥∥
0,p,P̃ .
∑
|α|≤1
hα
∥∥Dαṽ
∥∥
0,p,P̃ +hP̃
∥∥∥d̃iv ṽ
∥∥∥
0,p,P̃
+ (h1 +h2)
∥∥∥∥
∂ṽ3
∂x̃3
∥∥∥∥
0,p,P̃
holds, where hP̃ = max{h1,h2,h3}.
Proof. Using Lemma 4 and the relations
‖w̃‖0,p,P̃ =
(∫
P̃
∑
i ∈ Id
|w̃ i |p dx
)1/p
≤ (det JP̃
)1/p ∑
i ∈ Id
i h
−1
(∫
P̂
|ŵ i |p dx
)1/p
= (det JP̃
)1/p ∑
i ∈ Id
i h
−1‖ŵ i‖0,p,P̂ ,
(det JP̃ )
1/p‖v̂ i‖1,p,P̂ = i h
∑
|α|≤1
hα
∥∥Dαṽ i
∥∥
0,p,P̃ ,
442 Volker Kempf
where i h =∏ j ∈ I3\{i } h j , we compute∥∥Ĩ BDMk ṽ
∥∥
0,p,P̃ ≤
(
det JP̃
)1/p ∑
i ∈ I3
i h
−1∥∥(Î BDMk v̂
)
i
∥∥
0,p,P̂
.
(
det JP̃
)1/p
[ ∑
i ∈ I3
i h
−1
(
‖v̂ i‖1,p,P̂ +
∥∥∥d̂iv v̂
∥∥∥
0,p,P̂
)
+ h1 +h2
h1h2h3
∥∥∥∥
∂v̂3
∂x̂3
∥∥∥∥
0,p,P̂
]
= ∑
i ∈ I3
( ∑
|α|≤1
hα
∥∥Dαṽ i
∥∥
0,p,P̃ +hi
∥∥∥d̃iv ṽ
∥∥∥
0,p,P̃
)
+ (h1 +h2)
∥∥∥∥
∂ṽ3
∂x̃3
∥∥∥∥
0,p,P̃
.
∑
|α|≤1
hα
∥∥Dαṽ
∥∥
0,p,P̃ +hP̃
∥∥∥d̃iv ṽ
∥∥∥
0,p,P̃
+ (h1 +h2)
∥∥∥∥
∂ṽ3
∂x̃3
∥∥∥∥
0,p,P̃
. 
The estimate from the previous lemma can be brought to the general prism P where the
transformation is assumed to be reasonable in a certain sense. The stability estimate on P can
be used to get the interpolation error estimate by a Bramble–Hilbert type argument. The proofs
for the stability and interpolation error estimates on the element P follow the same steps as their
analogs on simplices, see [2, Theorems 3.5, 4.3] and cf. [1, Theorems 3.1, 6.2], which is why they
are omitted for brevity.
Theorem 6. Let P be a prism element that emerges by the affine transformation x = JP x̃ , with
‖JP‖∞,
∥∥J−1P
∥∥
∞ ≤C , of the element P̃ ∈RP . Then for v ∈W 1,p (P ), 1 ≤ p ≤∞, the estimate
∥∥I BDMk v
∥∥
0,p,P . ‖v‖0,p,P +
∑
j ∈ I3
h j
∥∥∥∥
∂v
∂l j
∥∥∥∥
0,p,P
+hP‖div v‖0,p,P + (h1 +h2)
∥∥∥∥
∂v3
∂l 3
∥∥∥∥
0,p,P
is satisfied. The vectors l j are the outgoing unit vectors along the edges adjacent to the transformed
vertex p i .
Theorem 7. Let a prism P satisfy the same condition as in Theorem 6. Then for k ≥ 1, 0 ≤ m ≤ k
and v ∈W m+1,p (P ), 1 ≤ p ≤∞, the estimate
∥∥v − I BDMk v
∥∥
0,p,P .
∑
|α|=m+1
hα
∥∥Dαl v
∥∥
0,p,P +hP
∑
|α|=m
hα
∥∥Dαl div v
∥∥
0,p,P
+ (h1 +h2)
∑
|α|=m
hα
∥∥∥∥∥
∂m+1v3
∂lα11 ∂l
α2
2 ∂l
α3+1
3
∥∥∥∥∥
0,p,P
holds and the constant only depends on k.
Remark 8. A limitation of our results is that the transformation from the general element to
the reference element is affine, which excludes settings where, e.g., the rectangular faces are not
flat or the triangular faces are not parallel in the general element. Such settings require further
investigation.
References
[1] G. Acosta, T. Apel, R. G. Durán, A. L. Lombardi, “Error estimates for Raviart–Thomas interpolation of any order on
anisotropic tetrahedra”, Math. Comput. 80 (2011), no. 273, p. 141-163.
[2] T. Apel, V. Kempf, “Brezzi–Douglas–Marini interpolation of any order on anisotropic triangles and tetrahedra”, SIAM
J. Numer. Anal. 58 (2020), no. 3, p. 1696-1718.
[3] ——— , “Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and
anisotropic mesh grading”, Calcolo 58 (2021), no. 2, article no. 15.
[4] D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications, Springer Series in Computational
Mathematics, vol. 44, Springer, 2013.
[5] S. Franz, “Anisotropic Hdi v -norm error estimates for rectangular Hdi v -elements”, Appl. Math. Lett. 121 (2021), article
no. 107453.
Volker Kempf 443
[6] V. Kempf, “Pressure-robust discretizations for incompressible flow problems on anisotropic meshes”, Doctoral thesis,
Universität der Bundeswehr München, Deutschland, 2022.
[7] A. Linke, “On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new
variational crime”, Comput. Methods Appl. Mech. Eng. 268 (2014), p. 782-800.
[8] J.-C. Nédélec, “A new family of mixed finite elements in R3”, Numer. Math. 50 (1986), no. 1, p. 57-81.


Brezzi–Douglas–Marini interpolation on anisotropic simplices and prisms

Yu, Gang

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

Marie, Emeryck

Continuité des racines d’après Rabinoff

Caillet, Thibault

The five gradients inequality for non quadratic costs

Coron, Jean-Michel

Ghys, Étienne

Saint-Raymond, Laure

Expression of concern from the editors in chief: “A $\Gamma $-convergence result for optimal design problems”

Bazao, Vanderléa R.

de Oliveira, César R.

Diaz, Pablo A.

Comptes Rendus
Mathématique
Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
On the Birman–Krein Theorem
Volume 361 (2023), p. 1081-1086
Published online: 24 October 2023
https://doi.org/10.5802/crmath.473
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. 1081-1086
https://doi.org/10.5802/crmath.473
Functional analysis / Analyse fonctionnelle
On the Birman–Krein Theorem
Sur le théorème de Birman–Krein
Vanderléa R. Bazaoa, César R. de Oliveira∗, b and Pablo A. Diazb
a Faculdade de Ciências Exatas e Tecnologias, UFGD, Dourados, MS, 79804-970 Brazil
b Departamento de Matemática, UFSCar, São Carlos, SP, 13560-970 Brazil
E-mails: vanderleabazao@ufgd.edu.br, oliveira@dm.ufscar.br, pdaraos@gmail.com
Abstract. It is shown that if X is a unitary operator so that a singular subspace of U is unitarily equivalent
to a singular subspace of U X (or XU ), for each unitary operator U , then X is the identity operator. In other
words, there is no nontrivial generalization of Birman–Krein Theorem that includes the preservation of a
singular spectral subspace in this context.
Résumé. On montre que si X est un opérateur unitaire tel qu’un sous-espace singulier de U est unitairement
équivalent à un sous-espace singulier de U X (ou XU ), pour chaque opérateur unitaire U , alors X est
l’opérateur d’identité. En d’autres termes, il n’y a pas de généralisation non triviale du théorème de Birman–
Krein qui inclut la préservation d’un sous-espace spectral singulier dans ce contexte.
2020 Mathematics Subject Classification. 47A55, 47A10, 81Q10.
Funding. CRdO thanks the partial support by CNPq (under contract number 303689/2021-8), and PAD
thanks the support by CAPES (Brazilian agencies).
Manuscript received 14 June 2022, accepted 31 January 2023.
1. Introduction
We are interested in preservations of spectral types of unitary operators U , on a Hilbert space H ,
under multiplicative (compositions, in fact) perturbations, that is, if X is another unitary opera-
tor, the perturbation has the form
U 7→U X . (1)
This is a right perturbation, and U 7→ XU is a left perturbation. Note that both XU and U X
are again unitary operators. Some notation: we shall denote the identity operator by 1 and
T1 ∼= T2 means that the linear operators T1 and T2 are unitarily equivalent; µ ⊥ ν means that
the measures µ and ν are mutually singular; T= {c ∈C | |c| = 1} is the one-dimensional torus and
the letter c will always denote an element of T.
The main question to be investigated is what kind of spectral types of (any) U are preserved
under such nontrivial perturbations X ̸= 1, and this study was motivated by the corresponding
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1082 Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
question, addressed by Howland [7], in the context of additive perturbations of self-adjoint op-
erators. Howland has concluded that there is no nontrivial generalization of Kato–Rosenblum
Theorem, that is, only absolutely continuous spectra (with respect to Lebesgue measure) can be
(always) preserved under some nonzero perturbation (for instance, under trace class perturba-
tions). Our main conclusion is similar and appear in Theorem 6, that is, there is no generaliza-
tion of Birman–Krein Theorem [1], so that only the absolutely continuous spectral component of
(any) U can be preserved under certain multiplicative unitary perturbation X ̸=1 (either right or
left perturbations).
Our arguments follow the general lines of [7], and do not go along with the Cayley transform
since it does not take sums of self-adjoint operators to the multiplicative form (4) of unitary ones,
and since there were some nonobvious choices in the construction here, we have found the result
interesting enough to justify publication.
As an additional motivation, it is worth mentioning that multiplicative perturbations (1) of
unitary operators naturally appear in quantum versions of time-periodic kicked systems [3, 5].
Let A be a self-adjoint operator describing certain quantum system (a free one) that, at instants
of time jτ, j ∈Z, undergoes kicked perturbations by a self-adjoint operator B ; the formal operator
describing such system is
A + B ∑
j∈Z
δ(t − jτ),
whose time evolution between two consecutive kicks (the Floquet operator) is given by [3]
e−i B e−i Aτ, (2)
that is, multiplicative perturbations (e−i B ) of unitary operators (e−i Aτ). A historically important
model [3] is the so-called kicked rotator with A = −d2/dθ2 and B = κcosθ, 0 ≤ θ < 2π, and a
real parameter κ > 0; it is expected that, for “most” periods τ > 0, the corresponding Floquet
operator (2) has pure point spectrum and a localized dynamics, specially for κ > 1 (it is a region
where classical and quantum physics would disagree), but currently, there is a mathematical
proof [2] of quantum localized dynamics only for small enough κ.
A basic general discussion of multiplicative perturbations is given in Section 2, including
a precise statement of preservation of spectral types and of our main result, whose proof is
concluded in Section 3.
2. Multiplicative perturbations
Sometimes it is convenient to write the perturbation in the form X = e i Y , with Y a bounded self-
adjoint operator, so that
X = e i Y =
∞∑
j=0
(i Y ) j
j !
=1+
∞∑
j=1
(i Y ) j
j !
=1+W, (3)
with W =∑∞j=1 (i Y ) j / j !. In this way, we can write
U X =U (1+W ) =U +UW (4)
and if W is a trace class operator, then UW is also trace class and, by Birman–Krein Theorem, the
Lebesgue absolutely continuous subspaces of UW and U are unitarily equivalent for all U (the
same for left perturbations XU ).
An example is the case of multiplicative perturbations of rank one: given φ ∈H with ||φ|| = 1,
let Pφ be the projector operator onto the subspace Lin{φ} generated by φ, i.e.,
Pφξ=
〈
φ,ξ
〉
φ , ∀ ξ ∈H ,
Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz 1083
which is self-adjoint and idempotent. The corresponding perturbing unitary operator, with
“intensity” λ ∈R, is
Xλ = Xλ,φ := e iλPφ , (5)
and by writing e iλPϕ =1+W , it follows by (3) that
W ξ= (e iλ−1)Pφξ,
and the (right) perturbed operator has the action
Uλ :=U Xλ =U1+U (e iλ−1)Pφ, (6)
with U0 = U . To simplify statements, denote by µλψ the spectral measure of the pair (Uλ,ψ),
ψ ∈H , and usually one supposes that φ is cyclic for U , that is, the closure
Lin
{
U jφ | j ∈Z}=H ,
with U 0 =1. Note that ifφ is cyclic for U , then U kφ is also is cyclic for U , for all k ∈Z; furthermore,
if φ is cyclic for U , then it is also cyclic for Uλ for all λ ∈R.
A particularly important result, that will be employed ahead, is the following version, for
unitary operators, of the Aronszajn–Donoghue Theorem for self-adjoint ones (for proofs see [6,
Proposition 8.3] and [4, Proposition 9.1.14]):
Theorem 1. Let φ be cyclic for U . If λ1 −λ2 ̸= 2πn, for all n ∈ Z, then the singular parts of Uλ1
and Uλ2 are mutually singular, i.e., for all 0 ̸=ψ ∈ H , the singular parts of the spectral measures
µ
λ1
ψ and µ
λ2
ψ are mutually singular.
In order to discuss more general spectral subspaces, for ψ ∈ H , denote by µUψ the spectral
measure of the pair (U ,ψ) and, given a (nonzero) finite Borel measure µ on T, let
Hµ(U ) :=
{
ψ ∈H |µUψ ≪µ
}
,
which is a closed subspace of the Hilbert space, whose orthogonal complement is{
ψ ∈H |µUψ ⊥µ
}
.
Finally, denote by [U ]µ the restriction of U to Hµ(U ), i.e., [U ]µ :=U |Hµ(U ).
Definition 2. A unitary operator X preserves µ on the right if [U X ]µ ∼= [U ]µ, for all unitary
operators U . And X preserves µ on the left if [XU ]µ ∼= [U ]µ, for all unitary operators U .
Remark 3.
(a) The simple case X = c1 translates the spectra in T and so does not preserve measures
if c ̸= 1.
(b) As already mentioned, by Birman–Krein Theorem, if X = 1+W with trace class W , then
X preserves the Lebesgue measure ℓ on T.
(c) If ν≪µ, then
[[U ]µ]ν = [U ]ν. (7)
(d) For real t , define the translated (in T) measure
µt (·) :=µ
(
e i t ·
)
. (8)
Then
[U ]µt = e i t [U ]µ. (9)
It is enough to discuss right or left preservation of a measure, as Proposition 4 ensures. After
this proposition we will just say “X preserves µ” (as already employed above).
1084 Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
Proposition 4. X preserves µ on the right if and only if X preserves µ on the left.
Proof. Let X ∗ denotes the adjoint of X . If X preservesµ on the right, then pick X ∗U (which is also
a unitary operator); by hypothesis [X ∗U X ]µ ∼= [X ∗U ]µ, and since X is unitary, [X ∗U X ]µ ∼= [U ]µ
and one obtains that [X ∗U ] ∼= [U ]µ, and so X ∗ preserves µ on the left. Now one has [U ]µ =
[X ∗XU ]µ ∼= [XU ]µ, and so X preserves µ on the left.
Similarly, one concludes the reciprocal. □
Corollary 5. X preserves µ if and only if X ∗ preserves µ.
Theorems 6 states the main result of this note.
Theorem 6. If the unitary operator X ̸=1 preserves µ, then µ≪ ℓ.
The completion of the proof of this theorem is the subject of Section 3; in the following we
present some basic and useful properties.
Proposition 7. Suppose that X preserves µ. Then:
(a) If ν≪µ, then X preserves ν.
(b) X preserves µt , for all t ∈R.
(c) If X ∼= Y , then Y preserves µ.
(d) If PE is an orthogonal reducing projection for X (projection onto the closed subspace
E ⊂H ), then PE X preserves µ on PE (H ) = E. Recall that E⊥ will also be reducing for X .
(e) If Y also preserves µ, then X Y preserves µ.
Proof. (a). [U X ]ν ∼= [[U X ]µ]ν ∼= [Uµ]ν ∼= [U ]ν.
(b). [U X ]µt = e i t [U X ]µ = [e i tU X ]µ and since e i tU is a unitary operator, we have [e i tU X ]µ ∼=
[e i tU ]µ =Uµt ; hence [U X ]µt = [U ]µt .
(c). If X ∼= Y , that is, Y =V X V ∗ for some unitary operator V , then
[U Y ]µ = [UV X V ∗]µ = [V (V ∗UV X V ∗V )V ∗]µ
∼= [V ∗UV X V ∗V ]µ = [V ∗UV X ]µ ,
and since V ∗UV is unitary and X preserves µ, one obtains
[V ∗UV X ]µ ∼= [V ∗UV ]µ ∼= [U ]µ,
that is, [U Y ]µ ∼= [U ]µ.
(d). Write H = E ⊕E⊥; then X = X |E ⊕X |E⊥ , which can be written as
X =
(
X |E 0
0 X |E⊥
)
.
Now, if Ũ and Ṽ are unitary operators acting on E and E⊥, respectively, then
U =
(
Ũ 0
0 Ṽ
)
is unitary on H . Thereby,
[U ]µ ∼= [U X ]µ =
[(
Ũ 0
0 Ṽ
)(
X |E 0
0 X |E⊥
)]
µ
=
(
[Ũ X |E ]µ 0
0 [Ṽ X |E⊥ ]µ
)
,
and since [U ]µ =
(
[Ũ ]µ 0
0 [Ṽ ]µ
)
, by equating the first components, one gets [Ũ X |E ]µ ∼= Ũµ, and so X |E
preserves µ on E = PE (H ).
(e). Indeed, if U is a unitary operator, then U X is also unitary and
[U X Y ]µ ∼= [U X ]µ ∼= [U ]µ. □
Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz 1085
3. Proof of Theorem 6
It is supposed that X preserves µ; the goal is to show that µ≪ ℓ. By Remark 3(a), we may suppose
that X ̸= c1. By item (c) of Proposition 7, R∗X R also preserves µ, for any unitary operator R. Pick
the operator R = Xλ from (5), with λ = π and some normalized vector φ (that will be selected
ahead), that is,
Rξ= Xπξ= ξ+ (e iπ−1)
〈
φ,ξ
〉
φ= ξ−2〈φ,ξ〉φ,
for all ξ ∈H .
By Proposition 7, items (c) and (e), and Corollary 5, the operators R∗X R and
Z := X ∗R∗X R
preserve µ as well. Explicitly, one has
R∗X Rξ=R∗X (ξ−2〈φ,ξ〉φ)
=R∗ (X ξ−2〈φ,ξ〉 Xφ)
=R∗(X ξ)−R∗ (−2〈φ,ξ〉 Xφ)
=R∗(X ξ)−2〈φ,ξ〉R∗ (Xφ)
=X ξ−2〈φ, X ξ〉φ−2〈φ,ξ〉(Xφ−2〈φ, Xφ〉φ)
=X ξ+ (4〈φ,ξ〉〈φ, Xφ〉−2〈φ, X ξ〉)φ−2〈φ,ξ〉 Xφ,
and if ψ= X ∗φ,
Zξ= ξ+ [4〈φ,ξ〉〈φ, Xφ〉−2〈φ, X ξ〉]ψ−2〈φ,ξ〉φ =: (1+W )ξ,
with
W ξ= [4〈φ,ξ〉〈φ, Xφ〉−2〈φ, X ξ〉]ψ−2〈φ,ξ〉φ .
Since X ̸= c1, one may choose φ so that the set {φ,ψ} is linearly independent and so the
operator W has rank precisely 2 (note that if X = c1, then W = 0, as expected). Let e1 and e2 be
two normalized and independent eigenvectors of W and note that the corresponding eigenvalues
do not vanish, otherwise W would have rank smaller than 2. Observe that e1 and e2 are also
eigenvectors of the unitary operator Z , hence (one may suppose that) e1 ⊥ e2. If E is the
orthogonal complement of e1, which reduces Z , it follows, by Proposition 7(d), that the operator
Z2 := Z |E also preserves µ on E , and since we now have a nonzero rank one perturbation, Z2 has
the form (5), that is,
Z2 =1+ (e iλ−1)Pe2 ,
for some λ ̸= 2πk,k ∈Z.
Finally, pick a unitary operator U̇ on E with e2 one of its cyclic vectors; hence
U̇ Z2 = U̇ +U̇ (e iλ−1)Pe2
has the form (6). Since Z2 preserves µ,
[U̇ Z2]µ ∼= [U̇ ]µ
and, by Theorem 1, if µ has a (nonzero) singular component, Z2 could not preserve µ, and one
concludes that µ is absolutely continuous with respect to Lebesgue measure.
1086 Vanderléa R. Bazao, César R. de Oliveira and Pablo A. Diaz
References
[1] M. S. Birman, M. G. Kreı̆n, “On the theory of wave operators and scattering operators”, Dokl. Akad. Nauk SSSR 144
(1962), p. 475-478, English translation: Soviet Math. 3 (1962), p. 740–744.
[2] J. Bourgain, “Estimates on Green’s functions localization and the quantum kicked rotor model”, Ann. Math. 156
(2002), no. 1, p. 249-294.
[3] G. Casati, L. Molinari, “Quantum Chaos with time-periodic Hamiltonians”, Prog. Theor. Phys., Suppl. 98 (1989), p. 287-
322.
[4] J. A. Cima, A. L. Matheson, W. T. Ross, The Cauchy Transform, Mathematical Surveys and Monographs, vol. 125,
American Mathematical Society, 2006.
[5] M. Combescure, “Spectral properties of a periodically kicked quantum Hamiltonian”, J. Stat. Phys. 59 (1990), no. 3-4,
p. 679-690.
[6] S. R. Garcia, W. T. Ross, “Model spaces: a survey”, in Invariant subspaces of the shift operator. CRM workshop (Montréal
2013), Contemporary Mathematics, vol. 638, American Mathematical Society, 2015, p. 197-245.
[7] J. S. Howland, “On the Kato-Rosenblum Theorem”, Pac. J. Math. 123 (1986), p. 329-335.


On the Birman–Krein Theorem

Rosas, Rudy

Comptes Rendus
Mathématique
Rudy Rosas
Transversely product singularities of foliations in projective spaces
Volume 361 (2023), p. 1785-1787
Published online: 21 December 2023
https://doi.org/10.5802/crmath.528
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. 1785-1787
https://doi.org/10.5802/crmath.528
Functional analysis / Analyse fonctionnelle
Transversely product singularities of foliations
in projective spaces
Rudy Rosas a
a Departamento de Ciencias, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, Lima, Perú
E-mail: rudy.rosas@pucp.pe
Abstract. We prove that a transversely product component of the singular set of a holomorphic foliation on
Pn is necessarily a Kupka component.
Funding. The author was supported by Vicerrectorado de Investigación de la Pontificia Universidad Católica
del Perú.
Manuscript received 4 May 2023, revised and accepted 23 June 2023.
1. Introduction
Let U be an open set of a complex manifold M and let k ∈N. Let η be a holomorphic k-form on
U and let Singη := {p ∈ U : η(p) = 0} denote the singular set of η. We say that η is integrable if
each point p ∈ U \Singη has a neighborhood V supporting holomorphic 1-forms ξ1, . . . ,ξk with
η|V = ξ1 ∧·· ·∧ξk , such that dξ j ∧η= 0 for each j = 1, . . . ,k. In this case the distribution
Dη : Dη(p) = {v ∈ Tp M : ivη(p) = 0}, p ∈U \Singη
defines a holomorphic foliation of codimension k on U \Singη. A singular holomorphic foliation
F of codimension k on M can be defined by an open covering (U j ) j∈J of M and a collection
of integrable k-forms η j ∈ Ωk (U j ) such that ηi = gi jη j for some gi j ∈ O∗(Ui ∩U j ) whenever
Ui ∩U j ̸= ;. The singular set SingF is the proper analytic subset of M given by the union of the
sets Singη j . From now on we only consider foliations F such that SingF has no component of
codimension one.
Given a singular holomorphic foliation F of codimension k on M as above, the Kupka singular
set of F , denoted by K (F ), is the union of the sets
K (η j ) = {p ∈U j : η j (p) = 0,dη j (p) ̸= 0}.
This set does not depend on the collection (η j ) of k-forms used to define F . It is well known
(see [7, 9]) that, given p ∈ K (F ), the germ of F at p is holomorphically equivalent to the product
of a one-dimensional foliation with an isolated singularity by a regular foliation of dimension
(dimF − 1). More precisely, if dim M = k + m + 1, there exist a holomorphic vector field X =
X1∂x1 + ·· · + Xk+1∂xk+1 on Dk+1 with a unique singularity at the origin, a neighborhood V of p
in M and a biholomorphism ψ : V →Dk+1 ×Dm , ψ(p) = 0, which conjugates F with the foliation
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1786 Rudy Rosas
FX of Dk+1 ×Dm generated by the commuting vector fields X ,∂y1 , . . . ,∂ym , where y = (y1, . . . , ym)
are the coordinates in Dm . If µ= dx1 ∧·· ·∧dxk+1, the foliation FX is also defined by the k-form
ω= iXµ and the Kupka condition dω(0) ̸= 0 is equivalent to the inequality div X (0) ̸= 0.
Following [8], we say that F is a transversely product at p ∈ SingF if as above there exist
a holomorphic vector field X and a biholomorphism ψ : V → Dk+1 ×Dm conjugating F with
FX , except that it is not assumed that div X (0) ̸= 0. We say that Γ is a local transversely product
component of SingF if Γ is a compact irreducible component of SingF and F is a transversely
product at each p ∈ Γ. In particular, if Γ⊂ K (F ) we say that Γ is a Kupka component — for more
information about Kupka singularities and Kupka components we refer the reader to [1–7]. If Γ is
a transversely product component of SingF , we can cover Γ by finitely many normal coordinates
like ψ, with the same vector field X : that is, there exist a holomorphic vector field X on Dk+1 with
a unique singularity at the origin and a covering of Γ by open sets (Vα)α∈A such that each Vα
supports a biholomorphism ψα : Vα→Dk+1 ×Dm that maps Γ∩Vα onto {0}×Dm and conjugates
F with the foliation FX . The sets (Vα) can be chosen arbitrarily close to Γ.
In [8], the author proves that a local transversely product component of a codimension one
foliation on Pn is necessarily a Kupka component. The goal of the present paper is to generalize
this theorem to foliations of any codimension.
Theorem 1. Let F a holomorphic foliation of dimension ≥ 2 and codimension ≥ 1 on Pn . Let Γ be
a transversely product component of SingF . Then Γ is a Kupka component.
This theorem is a corollary of the following result.
Theorem 2. Let F a holomorphic foliation of dimension ≥ 2 and codimension k ≥ 1 on a complex
manifold M. Suppose that F is defined by an open covering (U j ) j∈J of M and a collection of k-
forms η j ∈ Ωk (U j ). Let L be the line bundle defined by the cocycle (gi j ) such that ηi = gi jη j ,
gi j ∈ O∗(Ui ∩U j ). Let Γ be a transversely product component of SingF that is not a Kupka
component. Then c1(L|Γ) = 0.
2. Proof of the results
Proof of Theorem 2. Let dim M = k + m + 1. As explained in the introduction, there exist a
holomorphic vector field X on Dk+1 with a unique singularity at the origin and a covering of
Γ by open sets (Vα)α∈A such that each Vα is contained in V and supports a biholomorphism
ψα : Vα → Dk+1 ×Dm that maps Γ∩Vα onto {0}×Dm and conjugates F with the foliation FX
generated by the commuting vector fields X ,∂y1 , . . . ,∂ym . Notice that div(X )(0) = 0, because Γ is
not a Kupka component. Since FX is defined by the k-form ω= iXµ, where µ= dx1 ∧·· ·∧dxk+1,
we have that F |Vα is defined by the k-form ψ∗α(ω). If Vα∩Vβ ̸= ;, there exists θαβ ∈ O∗(Vα∩Vβ)
such that
ψ∗α(ω) = θαβψ∗β(ω). (1)
Therefore the cocycle (θαβ) define the line bundle L restricted to some neighborhood of Γ. Thus,
in order to prove that c1(L|Γ) = 0 it is enough to show that each θαβ|Γ is locally constant. Fix
some α,β ∈ A such that Vα∩Vβ ̸= ;. If we set ψ =ψα ◦ψ−1β and θ = θαβ ◦ψ−1β , from (1) we have
that ψ∗(ω) = θω, which means that ψ preserves the foliation FX . It suffices to prove that the
derivatives θy1 (p), . . . ,θym (p) vanish if p ∈ {0}×Dm . Since ∂y1 is tangent to FX , then the vector
field ψ∗(∂y1 ) is tangent to FX and so we can express
ψ∗(∂y1 ) =λX +λ1∂y1 +·· ·+λm∂ym ,
where λ,λ1, . . . ,λm are holomorphic. Then
Lψ∗(∂y1 )ω=LλXω=λLXω+dλ∧ iXω=λLXω=λdiv(X )ω,
Rudy Rosas 1787
where the last equality follows from the identity ω= iXµ. Thus, since
ψ∗
(
Lψ∗(∂y1 )ω
)
=L∂y1ψ
∗ω=L∂y1 (θω) = θy1ω,
we obtain that
θy1ω=ψ∗ (λdiv(X )ω) =λ(ψ)div(X )(ψ)θω
and therefore θy1 (p) = 0 if p ∈ {0}×Dm , because div(X ) vanishes along {0}×Dm . In the same way
we prove that θy2 (p) = ·· · = θym (p) = 0 if p ∈ {0}×Dm , which finishes the proof. □
Proof of Theorem 1. Suppose that Γ is not a Kupka component. Let L be the line bundle associ-
ated to F as in the statement of Theorem 2. We notice that c1(L) ̸= 0, otherwise F will be defined
by a global k-form on Pn , which is impossible. Then, if we take an algebraic curve C ⊂ Γ, we have
c1(L) ·C ̸= 0. Therefore, ifΩ is a 2-form on Pn in the class c1(L) and V is a tubular neighborhood
of Γ,
c1(L|Γ) ·C =
∫
C
Ω|Γ =
∫
C
Ω= c1(L) ·C ̸= 0,
which contradicts Theorem 2. □
References
[1] M. Brunella, “Sur les feuilletages de l’espace projective ayant une composante de Kupka”, Enseign. Math. 55 (2009),
no. 3-4, p. 227-234.
[2] O. Calvo-Andrade, “Foliations with a Kupka component on Algebraic Manifolds”, Bol. Soc. Bras. Mat. 30 (1999), no. 2,
p. 183-197.
[3] ——— , “Foliations of codimension greater than one with a Kupka component”, Qual. Theory Dyn. Syst. 8 (2009), no. 2,
p. 241-253.
[4] ——— , “Foliations with a radial Kupka set on projective spaces”, Bull. Braz. Math. Soc. (N.S.) 47 (2016), no. 4, p. 1071-
1083.
[5] O. Calvo-Andrade, M. Corrêa, A. Fernández-Pérez, “Higher codimensional foliations with Kupka singularities”, Int. J.
Math. 28 (2017), no. 3, article no. 1750019 (29 pages).
[6] D. Cerveau, A. Lins-Neto, “Codimension-one foliations in CP n , n ≥ 3, with Kupka components”, in Complex analytic
methods in dynamical systems, Astérisque, vol. 222, Société Mathématique de France, 1994, p. 93-132.
[7] I. Kupka, “The singularities of integrable structurally stable pfaffian forms”, Proc. Natl. Acad. Sci. USA 52 (1964),
p. 1431-1432.
[8] A. Lins-Neto, “Local transversely product singularities”, Ann. Henri Lebesgue 4 (2021), p. 485-502.
[9] A. S. de Medeiros, “Structural stability of integrable differential forms”, in Geometry and Topology, Lecture Notes in
Mathematics, Springer, 1977, p. 395-428.


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