ON CONFIGURATIONS OF POINTS ON THE SPHERE AND APPLICATIONS TO APPROXIMATION OF HOLOMORPHIC FUNCTIONS BY LAGRANGE INTERPOLANTS

ON CONFIGURATIONS OF POINTS ON THE SPHERE AND APPLICATIONS TO
APPROXIMATION OF HOLOMORPHIC FUNCTIONS BY LAGRANGE
INTERPOLANTS
PHUNG VAN MANH
A BSTRACT. We study certain configurations of points on the unit sphere in RN . As an application, we prove that the sequence of Lagrange interpolation polynomials of holomorphic
functions at certain Chung-Yao lattices converge uniformly to the interpolated functions.

1. I NTRODUCTION
Let Pd (CN ) denote the space of polynomials of degree at most d in N complex variables.
A subset A of CN that consists of N+d
distinct points is said to be unisolvent of degree d if,
d
for every function f defined on A, there exists a unique polynomials P ∈ Pd (CN ) such that
P(z) = f (z) for all z ∈ A. This polynomial is called the Lagrange polynomial interpolation
of f at A and is denoted by L[A; f ]. We are concerned with the problem of approximation of
holomorphic functions.
Problem 1. Let F be a subclass of entire functions in CN and Ad a unisolvent set of degree d
for d = 1, 2, . . .. Under what conditions does L[Ad ; f ] converge to f uniformly on every compact
subset of CN for every f ∈ F ?
It is well-known that if N = 1 then a sufficient condition is the boundedness of ∪∞
d=1 Ad . This
is an immediate consequence of the Hermite Remainder Formula (see [12, p. 59]). Moreover,
when the interpolation sets are unbounded, there exists a function f ∈ H(C) for which convergence does not hold. In this case, the problem is valid for a subclass F of H(C) in which the
modulus of the interpolation points are controlled by the order of f (see [1] for more details).
It is also proved in [1] that the same results also hold true for Kergin interpolation in CN , a
natural generalization of the univariate Lagrange interpolation. In contrast to the univariate
case, Bloom and Levenberg showed in [5] that the boundedness of the interpolation array (Ad )
does not guarantee the uniform convergence of every entire function as soon as N ≥ 2. The
trouble here is that the interpolation operator has bad behavior when interpolation points tend
to an algebraic hypersurface of degree d.
Problem 2. Let E be a compact subset of CN and F the class of functions which are holomorphic in a neighborhood of E (which can depend on the functions). Let Ad ⊂ CN be a unisolvent

set of degree d for d = 1, 2, . . .. Under what conditions does L[Ad ; f ] converge to f uniformly
on E for every f ∈ F ?
We mention that tools from (pluri)potential theory can be used to solve Problem 2. The
sufficient conditions are related to the Lebesgue constants, the transfinite diameter and global
extremal functions, etc. We refer the reader to [2, 4, 18] and the references therein.
The aim of this paper is to give answers to the above problems when Ad is a Chung-Yao lattice generated by hyperplanes in CN . Under natural conditions on (normal) vectors defining the
hyperplanes, we prove in Theorem 3.8 that the interpolation polynomials of any holomorphic
functions in a sufficiently large domain converge uniformly on a compact subset of CN to the
2000 Mathematics Subject Classification. Primary 41A05, 41A63, 52C35.
Key words and phrases. Lagrange interpolation, Chung-Yao lattices. Configurations on spheres.
1


interpolated functions, provided the boundedness of the interpolation points. From Theorem
3.8, we obtain a result regarding the uniform convergence of any entire function on every compact set of CN . Under the same assumptions on the vectors, we prove in Theorem 3.11 that the
convergence in Problem 1 is valid for entire functions of finite order and for the interpolation
points that may be unbounded. Moreover, the additional conditions regarding the location of
interpolation points are similar to those given in [1, Theorem 2.3]. In Theorem 3.6, we give
explicit Chung-Yao lattices satisfying Theorems 3.8 and 3.11. We mention the equally spaced
points in the simplex introduced in [3] and which also satisfy the above two theorems. More
generally, explicit interpolation points can be obtained by interwining sequences in the complex
plane, see for instance [18].
Now the problem turns to find of set of vectors which satisfy certain conditions. For simplicity, we work with unit vectors of real coordinates. They can be viewed as points on the
unit sphere SN−1 in RN . Let Vd = {n1 , . . . , nd } be a set of points on SN−1 . We associate Vd
to a number hVd that measures the thickness of Vd (see Section 2 for precise definitions). If
each n ∈ Vd is regarded as a normal vector of a hyperplane, then the condition on vectors in
Theorems 3.8 and 3.11 become lim infd→∞ hVd > 0. In Theorem 2.6, we construct a set Vd such
that hVd has this asymptotic behavior. We finally note that hVd is well-studied when N = 2. We
obtain precise estimates in this case.
Notations. The product of z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) in RN or CN is defined by z, w =

1
∑Nj=1 z j w j . Let us denote by z = (∑Nj=1 |z j |2 ) 2 the norm of z. The set B(z, R) (resp. B(z, R))
is the open ball (resp. closed ball) of center z ∈ CN and radius R > 0. Let A be a nonempty set.
Let An denote the class of subsets of A containing n elements.
2. C ONFIGURATIONS OF POINTS ON THE SPHERE
2.1. Distance on the sphere. In RN , N ≥ 2, each point can be regarded as a vector. Let
e := {e j = (0, . . . 0, 1, 0, . . . , 0) : j = 1, . . . , N} stand for the standard basis for RN . We denote
by SN−1 the unit sphere in RN ,
SN−1 = {x ∈ RN : x = 1}.
Let V = {n1 , . . . , nN−1 } be a set of linear independent unit vectors and nk = (nk1 , nk2 , . . . , nkN ),
1 ≤ k ≤ N − 1. We can easily check that the vector nV defined by


e1
e2
···
eN
 n11
n12 · · ·
n1N 

nV = det(e, n1 , . . . , nN−1 ) := det 
(2.1)
..
..
..
...


.

.
.
nN−11 nN−12 · · · nN−1N
is nonzero and orthogonal to V since n, nV = det(n, n1 , . . . , nN−1 ) for n ∈ RN . Here the
determinant in (2.1) is taken pointwisely according to the first row. Therefore, it is a normal
vector of the hyperplane HV which passes through V and the origin. We thus get HV = {x ∈
RN : nV , x = 0} and
| n, nV |
, n ∈ RN .
(2.2)
dist(n, HV ) =
nV
Note that dist(n, HV ) ≤ 1 for all n ∈ SN−1 .
Now let Vd = {n1 , . . . , nd } be a set of d ≥ N points on SN−1 . First, we make the assumption
that Vd is N-independent, that is, any N vectors in Vd are linearly independent, or equivalently
Vd
det(nk1 , nk2 , · · · , nkN ) = 0 for all 1 ≤ k1 , k2 , . . . , kN ≤ d pairwise distinct. For each V ∈ N−1
,a
2


subset consisting of N − 1 vectors of Vd , we write hV,Vd for the geometric mean of the distances
from all points in Vd \V to HV . Due to (2.2), we have
hV,Vd :=

∏

dist(n, HV )

1

d−N+1

n∈Vd \V

∏n∈Vd \V | n, nV |
=
nV

1
d−N+1

.

(2.3)

Let us define

Vd
.
(2.4)
N −1
On the other hand, if Vd is not N-independent, then we set hVd = 0.
By definition, hVd is independent of the ordering of the vectors in Vd . In the result below, we
fix an order of Vd and identify Vd with (n1 , . . . , nd ).
hVd := min hV,Vd : V ∈

Proposition 2.1. The map (n1 , . . . , nd ) → h{n1 ,...,nd } is continuous on (SN−1 )d .
Proof. We first prove the continuity of the map at a N-independent tuple Vd = (n1 , . . . , nd ).
By definition, we can choose a neighborhood Ω of Vd in (SN−1 )d such that every Xd ∈ Ω is
also N-independent. Thus formulas (2.3) and (2.4) are well-defined for Xd ∈ Ω and imply

the continuity. We now turn to the case in which Vd is not N-independent. Hence, there
exist N dependent vectors, say n1 , . . . , nN ∈ Vd . We can assume that nN = ∑N−1
j=1 a j n j where
a1 , . . . , aN−1 ∈ R. Since hVd = 0, we only need to verify that
lim hXd = 0,

Xd →Vd

where Xd is N-independent. Write Xd = (x1 , . . . , xd ). We denote by H the hyperplane
H{x1 ,...,xN−1 } . Using (2.2) we have
dist(nN , H ) ≤

N−1

∑

|a j |dist(n j , H ) ≤

j=1

N−1

∑ |a j |

nj −xj .

j=1

It is easily seen that
dist(xN , H ) ≤ nN − xN + dist(nN , H ) ≤ nN − xN +

N−1

∑ |a j |

nj −xj .

j=1

As xk goes to nk for k = 1, . . . , N, we have
dist(xN , H ) → 0.
Since all perpendicular distances appearing in the definition of hXd are bounded by 1, hXd ≤
dist(xN , H ), which implies the desired limit.
Let A be a subset of SN−1 . If A contains at least d ≥ N points, then we define
hd (A) := sup{hVd : Vd ⊂ A}.

(2.5)

Otherwise, we set hd (A) = 0. The supremum in (2.5) is attained when A is compact by PropoN−1 . We define the asymptotic
sition 2.1. Now, let V = (Vd )∞
d=N be an array of points on S
behavior of hVd as follows.
hV := lim inf hVd .
(2.6)
d→∞

We state a few simple properties which follow immediately from the definitions.
Proposition 2.2.
(1) 0 ≤ hVd , hd (A) ≤ 1 for all Vd and A.
(2) hVd and hd (A) are invariant under linear isometries of RN .

(3) hVd keeps its value when we replace some elements of Vd by their additive inverses.
(4) hd (A) = hd (A ∪ (−A)), where −A = {−a : a ∈ A}.
3


Vd
Proof. The first two assertions are trivial. Observe that for all V ∈ N−2
, HV coincides with
N
H(V \{n})∪{−n} for n ∈ X and dist(a, HV ) = dist(−a, HV ) for a ∈ R . The third assertion
follows. The last conclusion is an immediate consequence of the third one.

2.2. The two-dimensional case. We treat the case N = 2. Each point on S1 can be regarded as
a complex number. Let Vd = {n1 , . . . , nd }, d ≥ 2 and set nk = eiθk for 1 ≤ k ≤ d. If V = {nk },
then HV is the straight line passing through nk and the origin. Hence
dist(n j , H{nk } ) = | sin(θ j − θk )| and hnk ,Vd =

d

∏

| sin(θ j − θk )|

1
d−1

.

(2.7)

j=1, j=k

We now recall the notation of the d-th diameter and transfinite diameter of a plane compact set
A. For details we refer the reader to [16, p. 152-158]. The d-th diameter of A, denoted by
diamd (A), is defined by
diamd (A) := sup{

|xk − x j |

∏

2
d(d−1)

: x1 , . . . , xd ∈ A}.

(2.8)

1≤ j
A set {x1 , . . . , xd } ⊂ A for which the supremum is attained is called a d-Fekete system for A.
∞
It is known that the sequence diamd (A) d=2 is decreasing and its limit is called the transfinite
diameter of A,
lim diamd (A) = diam∞ (A).
(2.9)
d→∞

1
d−1

(S1 )

We have diamd
= d , diam∞ (S1 ) = 1. More generally, diam∞ (A) = sin(α/4) if A ⊂ S1
is an arc that subtends an angle α. A d-Fekete system for S1 always forms a regular d-polygon.
These facts can be found in [16, p. 135, p. 158].
˜ where
Proposition 2.3.
(1) If Γ is a compact subset of S1 , then hd (Γ) ≤ (1/2)diamd (Γ)
˜Γ = {z2 : z ∈ Γ}.
(2) If Γ ⊂ S1 is an arc that subtends an angle α ≥ π, then there exists a set Ud of d points
1
1
in Γ such that hUd = (1/2)d d−1 . In particular, hd (Γ) = (1/2)d d−1 .
(3) If Γ ⊂ S1 is an arc that subtends an angle α < π, then there exists a set Wd of d points
1
1
in Γ such that hWd ≥ (1/2)d d−1 sin α2 . In particular, hd (Γ) ≥ (1/2)d d−1 sin α2 .
Proof. (1) Let Vd ⊂ Γ and Vd = {n1 , . . . , nd }. We write nk = eiθk , 0 ≤ θk < 2π. Since |eiφ −
eiθ | = 2| sin φ −θ
2 |, in view of (2.7), we have
hnk ,Vd =
Hence

d

hVd ≤

∏ hnk ,Vd

k=1

1
d

=

1
2

1
2

d

|e2iθ j − e2iθk |

∏

1
d−1

.

(2.10)

j=1, j=k
d

∏

|e2iθ j − e2iθk |

1≤ j
2
d(d−1)

1
˜
≤ diamd (Γ),
2

(2.11)

where Γ˜ = {z2 : z ∈ Γ}. The first assertion follows.
(2) Let Γ be an arc that subtends an angle α ≥ π. Since hd (Γ) is invariant under rotation,
we can assume that Γ = {eiθ : 0 ≤ θ ≤ ϕ}, ϕ ≥ π. Then Γ˜ = S1 . We take Ud = {eiϕk :
ϕk = kπ/d, k = 0, . . . , d − 1}. Evidently, e2iϕk , k = 0, . . . , d − 1, are the vertices of a regular
d-polygon. Therefore they form a d-Fekete system for S1 . Moreover, in view of (2.10), we
realize that heiϕk ,Ud does not depend on k. It follows that the two inequalities in (2.11) are thus
equalities when Vd is replaced by Ud . This gives
1
1
˜ = 1 d d−1
hd (Γ) ≥ hUd = diamd (Γ)
.
(2.12)
2

2
4


The reverse inequality is a consequence of (1). Note that the proof of (2) also gives the following relation that we will use in the proof of (3),
d−1

| sin

∏
j=0, j=k

( j − k)π
| = hd−1
= hUd
eiϕk ,Ud
d

d−1

=

d
2d−1

,

0 ≤ k ≤ d − 1.

(2.13)

(3) Assume that Γ is an arc that subtends an angle at most π and Γ = {eiθ : −ω ≤ θ ≤ ω}
with 0 < ω < π2 . To prove the desired estimates, we will construct a set of points in Γ, say
1

Wd = {eiθ j : j = 1, . . . , d}, such that hWd ≥ (1/2)d d−1 sin ω. Our construction is motivated by
[8]. Let us set
φj =

(2 j − 1)π
,
2d

ξ j = cos φ j

and θ j = arcsin(ξ j sin ω),

j = 1, . . . , d.

(2.14)

Obviously, θ j ∈ (−ω, ω) and ξ j = −ξd+1− j for all 1 ≤ j ≤ d. For 1 ≤ k ≤ d we have
d

hd−1
eiθk ,Wd

=

d

| sin(θ j − θk )| =

∏
j=1, j=k

| sin arcsin(ξ j sin ω) − arcsin(ξk sin ω) |.

∏

(2.15)

j=1, j=k

Using the angle sum identity for the sin and the identity cos arcsin(ξ j sin ω) =
we get

1 − ξ j2 sin2 ω

d

hd−1
= (sin ω)d−1
eiθk ,W
d

∏

ξj

1 − ξk2 sin2 ω − ξk

1 − ξ j2 sin2 ω .

(2.16)

j=1, j=k

We denote by Pk the product of the d − 1 factors at the right hand side of (2.16) and by q j the
j-factor of Pk . In other words, Pk = ∏dj=1, j=k q j . It remains to prove that
d

Pk ≥

2d−1

.

(2.17)

Since ξ j = −ξd+1− j for all 1 ≤ j ≤ d, we have
q j qd+1− j = |ξ j2 − ξk2 | = | cos2 φ j − cos2 φk | = | sin(φk − φk ) sin(φ j + φk )|.

(2.18)

The proof of (2.17) will be divided into two cases.
Case 1. d + 1 = 2k (of course d must be odd ). By (2.18) we can write
Pk2

d

=

| sin(φ j − φk ) sin(φ j + φk )|

∏

(2.19)

j=1, j=k

Looking at the values of the φ j ’s given by (2.14) and using (2.13) we have
d

∏

| sin(φ j − φk )| =

j=1, j=k

d
2d−1

.

(2.20)

On the other hand, since φ j + φd+1− j = π, we get
| sin(φ j + φk )| = | sin(φd+1− j − φk )| = | sin(φd+1− j − φd+1−k )|.

(2.21)

Consequently,
d

∏
j=1, j=k

d

| sin(φ j + φk )| =

∏

| sin(φd+1− j − φd+1−k )| =

j=1, j=k
5

d
2d−1

.

(2.22)


Combining the above calculations we finally obtain Pk =
Case 2. d + 1 = 2k. Applying (2.18) again, we get

d
.
2d−1

d

Pk2 = q2d+1−k

| sin(φ j − φk ) sin(φ j + φk )|.

∏

(2.23)

j=1, j=k, j=d+1−k

On the other hand, since ξk + ξd+1−k = 0 and φk + φd+1−k = π,
qd+1−k = |2ξk

1 − ξk2 sin2 ω| ≥ | sin(2φk )| = | sin(φd+1−k − φk )|.

(2.24)

Combining (2.24) with (2.23) and using (2.21) we obtain
Pk2 ≥

d

∏

d

| sin(φ j − φk )|

j=1, j=k

| sin(φ j − φd+1−k )| =

∏
j=1, j=d+1−k

where we use (2.13) in the last equation. This gives Pk ≥

d
,
2d−1

d
2d−1

2

,

(2.25)

and the proof is complete.
1

Corollary 2.4. If Γ ⊂ S1 is an arc that subtends an angle α, then diamd (Γ) ≥ d d−1 sin α4 .

Proof. Suppose that Γ = {eiθ : 0 ≤ θ ≤ α} and γ = {eiθ : 0 ≤ θ ≤ α/2}. Proposition 2.3 now
yields
1
(1/2)diamd (Γ) ≥ hd (γ) ≥ (1/2)d d−1 sin(α/4),
and the corollary follows. Note that the properties of transfinite diameter give the weaker
estimate,
diamd (Γ) ≥ diam∞ (Γ) = sin(α/4).
Corollary 2.5. Let Γ ⊂ S1 be an arc that subtends an angle α. Then there exists an array (Vd )
of points in Γ such that
lim hVd = lim hd (Γ) =

d→∞

d→∞

1/2
(1/2) sin(α/2)

if α ≥ π
if α < π

Proof. If α ≥ π, the conclusion is trivial by Proposition 2.3(2). In case 0 < α < π, we assume
˜ = sin(α/2). From the
that Γ = {eiθ : 0 ≤ θ ≤ α}. Then Γ˜ = {eiθ : 0 ≤ θ ≤ 2α} and diam∞ (Γ)
first and third part of Proposition 2.3 we have
1
˜
(1/2)d d−1 sin(α/2) ≤ hWd ≤ hd (Γ) ≤ (1/2)diamd (Γ).

Letting d → ∞ we get the desired relations.

2.3. A lower bound for the general case. In higher dimensions, we will construct a precise
N−1 and get a lower bound for h . We need to compute the vecarray V = (Vd )∞
V
d=N on S
Vd
tor nV and the scalar product n j , nV for V ∈ N−1 and n j ∈ Vd . It can be done when the
computations reduce to univariate Vandermonde determinants which are recalled now. Let
F = (p1 , . . . , pm ) be a tuple of m univariate polynomials and T = {t1 , . . . ,tm } a set of m real
numbers. The Vandermonde determinant corresponding to F and T is defined as follows.
VDM(F ; T ) = det p j (tk ) .

(2.26)

N−1 such that
Theorem 2.6. There exists an array V = (Vd )∞
d=N of points on S

1
hV := lim inf hVd ≥ √
.
d→∞
N(2e)N−1
6

(2.27)


Corollary 2.7. We have
1
lim inf hd (SN−1 ) ≥ √

.
d→∞
N(2e)N−1
We divide the proof into a sequence of lemmas. The first lemma gives the formula for
generalized Vandermonde determinants. For convenience, we give its simple proof.
Lemma 2.8. Let m ≥ 1 and Fk = (1, . . . ,t k−1 ,t k+1 , . . . ,t m ), resulting by removing t k from
(1,t, . . . ,t m ), be tuples of monomials for k = 0, . . . , m. Then, for Tm = {t1 , . . . ,tm }, we have
VDM(Fk ; Tm ) = σm−k (t1 , . . . ,tm )

(t j − ti ),

∏

(2.28)

1≤i< j≤m

where σ0 = 1 and σ j is the elementary symmetric polynomial of degree j for 1 ≤ j ≤ m,
σ j (t1 , . . . ,tm ) =

tk1 · · ·tk j .

∑

1≤k1 <···
Proof. By continuity, we only prove the lemma for ti = t j , i = j. Let us recall the formula for
the univariate Vandermonde determinant,
VDM(Fm ; Tm ) =

(t j − ti ) = 0.

∏

(2.29)

1≤i< j≤m

We consider a polynomial of degree m defined as follows

1 t · · · tm
 1 t1 · · · t1m
p(t) = det 
 ... ... . . . ...
1 tm · · · tmm



.


(2.30)

∑ (−1)k VDM(Fk ; Tm)t k .

(2.31)

Expanding the determinant along the first row we obtain
m

p(t) =

k=0

It is clear that p(t j ) = 0 for all 1 ≤ j ≤ m. Hence
m

m

p(t) = (−1)m VDM(Fm ; Tm ) ∏ (t − tk ) = (−1)m VDM(Fm ; Tm ) ∑ (−1)m−k σm−kt k . (2.32)
k=1

k=0

Comparing the coefficients of t k in (2.31) and (2.32) we get the desired equalities.
Lemma 2.9. Let b > 0 and Td = {t1 , . . . ,td } ⊂ R satisfy |t j − tk | ≥
d

|t j − tk | ≥

∏
j=1, j=k

2 b
b 2e

d

,

b
d

for j = k. Then

∀1 ≤ k ≤ d.

Proof. Without loss of generality we assume that t1 < t2 < · · · < td . Then |t j − tk | ≥ | ( j−k)b
d | for
all 1 ≤ j, k ≤ d. Thus
d

∏
j=1, j=k

d

|t j − tk | ≥

∏
j=1, j=k

|

( j − k)b
b
|=
d
d

d−1

(k − 1)!(d − k)!.

(2.33)

The proof reduces to showing that
(k − 1)!(d − k)! ≥
7

2 d
d 2e

d

.

(2.34)


2

1 m 2m
,
m e

If d is even, say d = 2m, then (k − 1)!(d − k)! ≥ (m!)
m ≥
m m
m! ≥ e in the second estimate, and (2.34) follows.

If d is odd, say d = 2m + 1, then
m
e

(k − 1)!(d − k)! ≥ (m!)2 ≥

2m

2 d
d 2e

=

here we use the inequality

e

d

(1 +

≥

1 2m
2m )

2 d
d 2e

d

.

This finishes the proof.
Lemma 2.10. Let d ≥ N ≥ 2 and 2a ≥ b > 0. Let Td = {t1 , . . . ,td } be a set of real numbers in
[−a, a] such that |t j − tk | ≥ b/d for j = k. Consider the set of unit vectors
Vd = {n j = v j / v j : v j = (1,t j , . . . ,t N−1
), j = 1, . . . , d}.
j
Then there exists a positive constant C1 depending only on a, b and N such that
1

hVd ≥ C1d−N+1 C,
where C =

b
2e(a+1)

N−1

1
√
.
1+a2 +···+a2N−2

Proof. Note that Vd is N-independent. This property is a consequence of (2.40) below. It
suffices to show that
1
Vd
.

hV,Vd ≥ C1d−N+1 C for all V ∈
N −1
Without loss of generality we can assume that V = {n1 , n2 , . . . , nN−1 }. Looking at the formula
of det(e, n1 , . . . , nN−1 ) in (2.1) and expanding the determinant along the first row we can write
N−1

nV = det(e, n1 , . . . , nN−1 ) =

∑ (−1)k Dk ek+1,

(2.35)

k=0

where Dk is the determinant of (N − 1) × (N − 1) matrix obtained by deleting the first row and
N−1 1
the (k + 1)-th column of the corresponding matrix. Set δ = ∏s=1
vs . In view of (2.1) and the
definition of nk , we conclude from Lemma 2.8 that
Dk = δ VDM(Fk ; TN−1 ) = δ σN−1−k (TN−1 )

(tm − tn ),

∏

∀0 ≤ k ≤ N − 1, (2.36)

1≤n
where Fk = (1, . . . ,t k−1 ,t k+1 , . . . ,t N−1 ) and TN−1 = {t1 , . . . ,tN−1 }. Hence (2.35) and (2.36)

give
N−1

nV = δ

∏

∑ σk (TN−1)2

|tm − tn |

1≤n
1
2

.

(2.37)

k=0

Since σ0 = 1 and t j = tk for j = k, (2.37) shows that nV is a nonzero vector. To get an upper
bound for nV , we note that
N−1

∑

N−1

σk (TN−1 )2 ≤

k=0

∑
k=0

N −1
k

2

a2k < (a + 1)2(N−1)

(2.38)

|tm − tn |.

(2.39)

since TN−1 ⊂ [−a, a]. Hence
nV ≤ δ (a + 1)N−1

∏

1≤n8


By the definition of nV in (2.1) and the formula for a univariate Vandermonde determinant, we

have
| n j , nV | = | det(n j , n1 , . . . , nN−1 )|
=

N−1

δ
vj

∏ |tk − t j |

k=1

|tm − tn |,

∏

∀N ≤ j ≤ d.

(2.40)

1≤n
Combining (2.39) with (2.40) we obtain
N−1
| n j , nV |
1
∏N−1
k=1 |tk − t j |
≥

≥
(2.41)
∏ |tk − t j |, ∀N ≤ j ≤ d,
nV
(a + 1)N−1 v j
(a + 1)N−1 M k=1
√
where M := 1 + a2 + · · · + a2N−2 ≥ v j . Since |tk − t j | ≤ 2a for every 1 ≤ j, k ≤ d, (2.41)
the formula of hV,Vd shows that
d−N+1
≥
hV,V
d

≥

1
(a + 1)N−1 M

d−N+1 d N−1

1

1
(a + 1)N−1 M

(2a)(N−1)(N−2)

∏ ∏ |tk − t j |

j=N k=1

d−N+1 N−1

d

∏ ∏

|tk − t j |

k=1 j=1, j=k

≥ C1Cd−N+1 .
Here we use Lemma 2.9 in the last estimate. The constant C1 is given by
C1 =

1
(2a)(N−1)(N−2)

2
b
2
( )N−1 ( )(N−1) .
b
2e

We finally remark that when N = 2, the empty product ∏1≤nThe Lemma is proved.
Proof of Thereom 2.6. For each d ≥ N, let Vd be the tuple of unit vectors defined in Lemma
2.10. Set V = (Vd )∞

d=N . By the estimate in Lemma 2.10, we have
N−1

1
√
.
(2.42)
d→∞
1 + a2 + · · · + a2N−2
We will find the maximal value of the right hand side of (2.42). The AM-GM inequality gives
hV = lim inf hVd ≥

b
2e(a + 1)

N−1

b
2e(a + 1)

1
b
√
√
≤
2
2N−2
4e a
1+a +···+a

N−1

1
1
√
≤√
.
N−1
N(2e)N−1
Na

Note that the maximum is attained when a = 1 and b = 2, and the proof is complete.
Remark 2.11. Examining the proof of Lemma 2.10 and looking at (2.38), we have the better
estimate
1
b N−1
1
.
hVd ≥ C1d−N+1
2e
N−1 N−1 2 2k
N−1 2k
∑k=0 k a · ∑k=0 a
Taking a = 1 and b = 2, we obtain
lim inf hVd ≥
d→∞

1
eN−1

N

2N−2
N−1

.

Open question. Find the asymptotic behavior of hd (SN−1 ) as d → ∞ and an array of unit
vectors V = (Vd )∞
d=N such that hV attains the asymptotic behavior (if it exists).
9


3. A PPLICATIONS TO APPROXIMATION OF HOLOMORPHIC FUNCTIONS
In this section, we essentially follow the notational conventions presented in [7, 10].
3.1. Chung-Yao lattices. A hyperplane in CN is defined by an equation = {z ∈ CN :
n, z + c = 0}, |n = 1. From now on, we assume that the (normal) vector in the definition of
the hyperplane is a unit vector. For convenience, we write (z) = n, z + c and ˜(z) = n, z .
A set H of N hyperplanes in CN is said to be in general position if their intersection is a singleton, that is ∩Nj=1 j = {ϑH }. If j (z) = n j , z + c j , then H is in general position if and only
if det(n1 , . . . , nN ) = 0. Here and subsequently, we identify {ϑH } with ϑ H . More generally, a
family H = { 1 , . . . , d } of d ≥ N hyperplanes in CN is said to be in general position if
(1) Every H ∈ H
N , a subset of N hyperplanes of H, is in general position;
(2) The map H ∈ H
N → ϑH = ∩ ∈H is one-to-one.
d
The set ΘH = {ϑH : H ∈ H
N }, consisting of N points, is called a Chung-Yao lattice of order
d. It is well-known that ΘH is a unisolvent set of degree d − N. We can now state a theorem
due to Chung and Yao.

Theorem 3.1. Let H = { 1 , . . . , d } be a family of d ≥ N hyperplanes in general position in
CN . If j is given by j = {z ∈ CN : n j , z + c j = 0} and f is a function defined on ΘH , then
L[ΘH ; f ](z) =

f (ϑH )l(ΘH , ϑH ; z),

∑

(3.1)

H∈(H
N)

where the fundamental Lagrange interpolation polynomial (FLIP) is given by
l(ΘH , ϑH ; z) =

∏

∈H\H

(z)
,
(ϑH )

H∈

H
.
N

(3.2)

Now if K = { k1 , . . . , kN−1 } ⊂ H, k1 < k2 < · · · < kN−1 , then ∩K = ∩ ∈K is a complex line
in CN . It passes through a point a and is parallel to the vector nV = det(e, nk1 , . . . , nkN−1 ), where
V = {nk1 , . . . , nkN−1 } is the set of nomal vectors of hyperplanes in K. Hence we can write
∩K = {a + nV t : t ∈ C}.
From now on, we write nK instead of nV . With this notation, we have
˜j (nK ) = n j , nK = n j , nV .

(3.3)

Note that the last term is actually introduced in Section 2 in which it plays an important role.
3.2. The de Boor error formula. Let Ω be a convex open set in CN . We will denote by H(Ω)
the set of all holomorphic functions in Ω. To every A = {a0 , . . . , ak } ⊂ Ω and f ∈ H(Ω), we
associate a symetric k-linear form denoted by [A, ·]( f ) and defined by
[a0 , . . . , ak |u1 , . . . , uk ]( f ) =

Dk f (·)(u1 , . . . , uk ),

Du1 . . . Duk f =
[A]

u j ∈ CN ,

(3.4)

[A]

where

k

h(a0 + ∑ ξ j (a j − a0 ))dξ1 . . . dξk

h=
[A]

∆k

(3.5)

j=1

and ∆k = {ξ = (ξ1 , . . . , ξk ) : ξ j ≥ 0, ∑kj=1 ξ j ≤ 1}. This k-linear form is called the multivariate
divided difference of f at A. Using the above notations, we can state a beautiful remainder
formula for Lagrange polynomial at Chung-Yao lattices due to de Boor, (see [7, Theorem 3.1]).
10


Note that the original version deals with real variables. Examining the proof we realize that it
is still valid for the complex case.
Theorem 3.2 (de Boor [7]). Let H be a collection of d hyperplanes in CN in general position
and ΘH the corresponding Chung-Yao lattice. Let Ω be a convex neighborhood of ΘH and
f ∈ H(Ω). Then
f (z) = L[ΘH ; f ](z) +

PK [ΘK , z| nK , · · · , nK ]( f ),

∑
K∈(

H
N−1

)

(3.6)

d−N+1

with ΘK := ΘH ∩ ∩K, the points in ΘH on the line ∩K = ∩
PK (z) =

z ∈ Ω,

∏

∈H\K

∈K

, and

(z)
.
˜(nK )

3.3. Convergence of Lagrange interpolants at Chung-Yao lattices. Let f be a holomorphic
function in an open subset Ω of CN . The norm of the k-th total derivative of f at z ∈ Ω is
defined by

Dk f (z) = sup{|Dk f (z)(z1 , . . . , zk )| : z j ∈ CN , z j ≤ 1, j = 1, . . . , k}.
We set f E := supz∈E | f (z)| for E ⊂ Ω. The following lemma gives a bound of the remainder
for Lagrange interpolation. It is analogous to Theorem 4 in [14].
Lemma 3.3. Let H = { 1 , . . . , d } be a family of d ≥ N hyperplanes in CN in general position.
Suppose that
(1) ΘH ⊂ B(0, R);
˜ K )|
H
d−N+1 , ∀K ∈
(2) ∏ ∈H\K | (n
N−1 , where α > 0, β > 0
nK ≥ αβ
Let R1 > 0, max(R, R1 ) < R2 and f holomorphic in a neighborhood of B(0, R2 ). Then
f − L[ΘH ; f ]

B(0,R1 )

≤

d
1
α N −1

e(R + R1 )
β (R2 − max(R, R1 ))

d−N+1

f

B(0,R2 ) .

(3.7)

Proof. For simplicity we set m = d − N + 1. Assume that the hyperplane j is given by j (z) =
{z : n j , z + c j = 0}, n j = 1. Since j passes through a point ϑ ∈ ΘH and ϑ ≤ R, we have
|c j | = | n j , ϑ | ≤ R. Consequently, | j (z)| ≤ n j · z + |c j | ≤ R1 + R for all z ∈ B(0, R1 ).
Combining this estimate with the second hypothesis we obtain
PK

B(0,R1 ) =

sup
z∈B(0,R1 )

| (z)|
(R1 + R)m
≤
∏ | ˜(nK )| α(β nK )m ,
∈H\K

∀K ∈

H
.
N −1

(3.8)

H

, since ΘH ⊂ B(0, R), the convex hull of {z}∪ΘK is contained
Given z ∈ B(0, R1 ) and K ∈ N−1
in B(0, R3 ) with R3 = max(R, R1 ). It implies that

[ΘK , z|nK , · · · , nK ]( f )

≤

|DnK · · · DnK f | ≤
[ΘK ,z]

=

(supw∈B(0,R3 )

∆m
m
D f (w)

(

Dm f (w) ) nK

sup

w∈B(0,R3 )

Dm f (w) ≤

) nK

m

mm
f
(R2 − R3 )m

11

dξ1 . . . dξm

w∈B(0,R3 )

.
m!
According to the Cauchy inequality, see for instance [15, p. 23], we have
sup

m

B(0,R2 ) .

(3.9)

(3.10)


Combining (3.9) and (3.10) and using the fact that mm /m! ≤ em , we obtain
m
H

e nK
f B(0,R2 ) , ∀z ∈ B(0, R1 ), K ∈
. (3.11)
N −1
R2 − R3
In view of (3.8) and (3.11) we conclude from Theorem 3.2 that
1
d
e(R + R1 ) m
| f (z) − L[ΘH ; f ](z)| ≤
f B(0,R2 ) , ∀z ∈ B(0, R1 ),
α N − 1 β (R2 − R3 )
and the lemma follows.

[ΘK , z|nK , · · · , nK ] f ≤

N
Definition 3.4. An array (Θd )∞
d=N of Chung-Yao lattices in C is said to be regular (or more
∞
precisely, β -regular) if there exists an array (Hd )d=N of hyperplanes in CN such that Θd = ΘHd
for all d ≥ N and the following assumptions hold:
(1) Hd consists of d hyperplanes in general position in CN for all d ≥ N.
(2) lim infd→∞ hHd = β > 0, where
1
Hd
| ˜(nK )|
d−N+1
.
hHd = min

∏ nK : K ∈ N − 1
∈H \K
d

Remark 3.5. Looking at (3.3), we see that hHd is equal to hVd , where Vd is a set of normal
vectors of Hd . Roughly speaking, the condition lim infd→∞ hHd > 0 guarantees ΘHd not to tend
to a hyperplane. It is a quite natural assumption (see the discussions in Section 1). .
Theorem 3.6. Let N ≥ 2 and 2a ≥ b > 0. For each d ≥ N, let Td = {t1 , . . . ,td } be a set of
real numbers in [−a, a] such that |t j − tk | ≥ b/d for j = k. We denote by Θd the subset of CN
d
consisting of d−N
points and defined by
Θd = ϑU = (−1)N−1 σN (U), (−1)N−2 σN−1 (U), · · · , σ1 (U) : U ∈

Td
N

,

where σ j (U) is the j-th elementary symmetric polynomial of N elements in U. Then (Θd )∞
d=N
is a C-regular and bounded array of Chung-Yao lattices, where C is the constant defined in
Lemma 2.10.
Proof. For each d ≥ N, let Vd = {n1 , . . . , nd } be the set of unit vectors defined in Lemma 2.10,
that is
Vd = {n j = v j / v j : v j = (1,t j , . . . ,t N−1
), j = 1, . . . , d}.
j
We denote by Hd the set of hyperplanes in CN ,
Hd = {

k

: 1 ≤ k ≤ d}

where

tkN
.
k (z) = nk , z −
vk

We claim that Hd is in general position in CN and ΘHd = Θd . Indeed, for simplicity, we work
with the first N + 1 hyperplanes { 1 , . . . , N+1 }. Consider the system of linear equations
k (z) =

nk , z −

tkN
= 0,
vk

k = 1, 2, . . . , N,

(3.12)

or equivalently,
z1 + tk z2 + · · · + tkN−1 zN = tkN , k = 1, 2, . . . , N.
(3.13)
Evidently, the determinant of the coefficient matrix is equal to D = ∏1≤ j

different from 0. By Cramer’s rule the system has a unique solution z = (z1 , z2 , . . . , zN ) given
by
Dk
zk =
(3.14)
D
12


where Dk is the determinant of the matrix formed by replacing the k-th column of the coefficient
matrix by the column vector (t1N ,t2N , . . . ,tNN )T . Now Lemma 2.8 gives
Dk = (−1)N−k σN+1−k (t1 , . . . ,tN )

(tk − t j ).

(3.15)

k = 1, 2, . . . , N.

(3.16)

∏

1≤ j
It follows that
zk = (−1)N−k σN+1−k (t1 , . . . ,tN ),
Hence
z = (−1)N−1 σN (t1 , . . . ,tN ), (−1)N−2 σN−1 (t1 , . . . ,tN ), · · · , σ1 (t1 , . . . ,tN ) = ϑU ,
where U = {t1 , . . . ,tN } ∈

Td
N

. Consequently,
∩N
k=1

k

= z = ϑU ∈ Θd .

Next, we consider the system of N + 1 equations,
z1 + tk z2 + · · · + tkN−1 zN = tkN ,

k = 1, 2, . . . , N + 1.

(3.17)

Since the rank of the coefficient matrix is equal to N and the rank of its augmented matrix
equals N + 1, Kronecker-Capelli’s theorem shows that the system (3.17) has no solution. It
/ Consequently, Hd = { 1 , · · · , d } is in general position in CN and
implies that ∩N+1
k=1 k = 0.
ΘHd = Θd for d ≥ N.
By the definition, (3.3) shows that hVd = hHd . Hence Lemma 2.10 gives
lim inf hHd = lim inf hVd = C.
d→∞

d→∞

∞
It follows that (Θd )∞
d=N is C-regular. It remains to check the boundedness of (Θd )d=N . Since

Td ⊂ [−a, a], U ⊂ [−a, a] for all U ∈ TNd . This clearly forces ϑU ≤
Therefore,
Θd ⊂ B(0, M), ∀d ≥ N.

∑N
k=1

N 2 2k
k a

= M.

Remark 3.7. Other regular Chung-Yao lattices can be constructed as follows. Let V = (Vd )∞
d=N
be an array of points on SN−1 such that lim infd→∞ hVd = β > 0 and Vd = {n1 , . . . , nd } is Nindependent for every d. Let Hd be a family of hyperplanes in gereral position defined by
N
j = {z ∈ C : n j , z + c j = 0}, j = 1, . . . , d. By the definition, (3.3) gives hVd = hHd . Hence,
lim inf hHd = β .
d→∞

It follows that (ΘHd )∞
d=N is regular.
Given R > 0, let us define a new family of hyperplanes in general position,
Hεd = { εj (z) = n j , z + εc j , j = 1, . . . , d ,

ε > 0.

It is easily seen that Hεd is the image of Hd under the dilation of center 0 and ratio ε. Thus,
when ε > 0 is small enough, ΘHεd ⊂ B(0, R).
N
∞
Theorem 3.8. Let (Θd )∞
d=N be a β -regular array of Chung-Yao lattices in C and ∪d=N Θd ⊂
B(0, R). Let R1 > 0 and f holomorphic in a neighborhood of B(0, R2 ) with R2 = max(R, R1 ) +
e(R+R1 )
. Then the sequence (L[Θd ; f ])∞
d=N converges to f uniformly on B(0, R1 ).
β
13


Proof. By hypothesis, f is holomorphic in a neighborhood of B(0, R3 ) for some R3 > R2 . We
1)
choose 0 < ε < β small enough such that R3 > max(R, R1 ) + e(R+R
β −ε .
N
Let (Hd )∞
d=N be the array of hyperplanes in C defined as in Definition 3.4. Suppose that
Hd = { 1 , . . . , d }, j (z) = n j , z +c j , n j = 1 for j = 1, . . . , d. By hypothesis, lim infd→∞ hHd =
β . We can find α > 0 such that
hd−N+1
≥ α(β − ε)d−N+1 ,
Hd

∀d ≥ N.

(3.18)

Consequently,

j

| ˜j (nK )|
≥ α(β − ε)d−N+1 ,
∏
n
K
∈H \K

∀K ∈

d

Hd
, d ≥ N.
N −1

(3.19)

Now using Lemma 3.3 we obtain
f − L[Θd ; f ]

B(0,R1 )

≤

1
d
α N −1

e(R + R1 )
(β − ε)(R3 − max(R, R1 ))

The right hand side of (3.20) tends to 0 as d → ∞ since
follows.

d−N+1

f

e(R+R1 )
(β −ε)(R3 −max(R,R1 ))

B(0,R3 ) ,

(3.20)

< 1 and the theorem

Corollary 3.9. Under the same assumptions of Theorem 3.8, for any entire function f in CN ,
N
the sequence (L[Θd ; f ])∞
d=N converges to f uniformly on every compact subset of C .
Remark 3.10. 1) We do not know whether the radius R2 in Theorem 3.8 is optimal.
2) Roughly speaking, the boundedness of the array of interpolation points is a necessary condition for the convergence of entire functions. Indeed, suppose that N ≥ 2. Let Ad ⊂ CN be

unisolvent of degree d such that ∪∞
d=1 Ad has no limit points. By Theorem 7.2.11 in [13], there
exists an entire function f = 0 such that f (a) = 0 for all a ∈ ∪∞
d=1 Ad . It follows that L[Ad ; f ] = 0
for all d ≥ 1. Hence L[Ad ; f ](z) does not tend to f (z) whenever f (z) = 0.
We recall the following measures of the rate of growth of entire functions in CN , following
Boas [6, p. 8]. Let f be an entire function in CN . The order of f is defined by
log(log f

B(0,r) )

.
log r
The order is a non-negative real number or infinite. If 0 < µ < ∞, then we can define the type
of f as follows.
log f B(0,r)
σ = lim sup
.
rµ
r→∞
If f is of order 0 < µ < ∞ and type σ , then for every δ > 0 there exists r0 > 0 such that
µ = lim sup
r→∞

log f

B(0,r)

≤ (σ + δ )r µ ,

∀r > r0 .

Theorem 3.11. Let f be an entire function in CN of order µ ∈ (0, ∞). Let (Θd )∞
d=N be a regular
1
N
array of Chung-Yao lattices in C . Suppose that there exist λ ∈ (0, µ ) and 0 < ρd ≤ d λ ,
limd→∞ ρd = ∞ such that Θd ⊂ B(0, ρd ) for sufficiently large d. Then the sequence of Lagrange
N
polynomials (L[Θd ; f ])∞
d=N converges to f uniformly on every compact subset of C .
Proof. Suppose that (Θd )∞
d=N is β -regular. Then relation (3.19) still holds true. We fix R1 > 0.
Using Lemma 3.3 in which R and β are replaced by ρd and β − ε respectively we have
f − L[Θd ; f ]

B(0,R1 )

≤

1
d
α N −1

e(ρd + R1 )
(β − ε)(R2 − max(ρd , R1 ))
14

d−N+1

f

B(0,R2 ) ,

(3.21)


where R2 > max(R1 , ρd ) and 0 < ε < β . We want to show that there is a sequence R2 = R2 (d)
such that the logarithm of right hand side of (3.21) tends to −∞ as d → ∞. Given δ > 0, since
f is of order µ, we have
log f

B(0,r)

≤ (σ + δ )r µ ,

∀r > r0 ,

(3.22)

e
where σ is the type of f . Choose C > 1 such that log( (β −ε)(C−1)
) < 0 and then take R2 =
R2 (d) = C(R1 + ρd ). Obviously, R2 (d) → ∞ as d → ∞. Now we see at once that

e
e(ρd + R1 )
≤ log(
).
(β − ε)(R2 − max(ρd , R1 ))

(β − ε)(C − 1)
On the one hand, using (3.22) we obtain
log

log f

µ

B(0,R2 )

≤ (σ + δ )R2 = (σ + δ )C µ (R1 + ρd )µ
≤ (σ + δ )C µ (R1 + d λ )µ ≤ (σ + δ )(2C)µ d λ µ .

On the other hand, log
that
log f − L[Θd ; f ]

d
N−1

B(0,R1 )

≤ (N − 1) log d. From what has already proved, we may conclude
≤ − log α + (N − 1) log d
+ (d − N + 1) log(

e
) + (σ + δ )(2C)µ d λ µ (3.23)
(β − ε)(C − 1)

e
Since log( (β −ε)(C−1)
) < 0 and λ µ < 1, the right hand side of (3.23) tends to −∞ as d → ∞.
The proof is now complete.

Remark 3.12. 1) We have proved more, namely that the sequence of Lagrange polynomials in
Corollary 3.9 and Theorem 3.11 converges geometrically on every compact subset of CN to the
interpolated function, that is
lim

d→∞

f − L[Θd ; f ]

1
d−N

B(0,R)

< 1,

R > 0.

2) In [17], Sauer and Xu constructed beautiful bi-dimensional Chung-Yao lattices ΘH2d+1 located in the real unit disk in R2 . Moreover, Proposition 3 in [17] shows that the array (ΘH2d+1 )∞
d=1
is regular. Thus it gives an explicit array of points having the convergence property.
N
Remark 3.13. Let (Hd )∞
d=N be an array of regular Chung-Yao lattices in C in which the set
of normal vectors of Hd is Vd defined in Lemma 2.10 and ΘHd ⊂ B(0, ηd ), limd→∞ ηd = 0.

We write Hd = { 1 , . . . , d } where k (z) = nk , z + ck , nk = vk / vk , vk = (1,tk , . . . ,tkN−1 ).
Looking at the proof of Lemma 3.3 we have |ck | ≤ ηd for all 1 ≤ k ≤ d. On the other hand, for
H ∈ HNd , ϑH ∈ B(0, ηd ). It follows that

| k (ϑH )| ≤ nk · ϑH + |ck | ≤ 2ηd ,

∀1 ≤ k ≤ d, H ∈

Hd
.
N

Given R > 0, we have
k (R, 0, · · · , 0) =

R
R
+ ck ≥ − ηd ,
vk
C

∀1 ≤ k ≤ d,

2j
where we use the inequality vk ≤ ∑N−1
j=0 a := C in the last estimate. We conclude from the
formula of the FLIP in Theorem 3.1 that
| k (z)|
R/C − ηd d−N
sup |l(ΘHd , ϑH ; z)| = sup

≥
.
∏
2ηd
z∈B(0,R)
z∈B(0,R) ∈H \H | k (ϑH )|
k

d

15


Hence we can not find a real-valued continuous function h defined in B(0, R) with the following
property: Given ε > 0 there exists an integer d1 (depending on ε) such that, for all d ≥ d1 and
H ∈ HNd ,
1
log |l(ΘHd , ϑH ; z)| ≤ ε.
z∈B(0,R) d − N

h(ϑH ) + sup

The condition (2.7.1) in [2] does not hold. Therefore the assumptions in Theorem 3.8 are not
stronger than those given in [2].
Remark 3.14. We have proved that the convergence results hold if the arrays of Chung-Yao
lattices are regular in the sense of Definition 3.4. The following question arises naturally: Is
there any array of Chung-Yao lattices without the separation condition of hyperplanes but the
convergence still holds?
Acknowledgements. The author wishes to express his thanks to Professor Jean-Paul Calvi for
suggesting this problem and for stimulating conversations. The author would like to thank the

referees for a careful reading of the manuscript. This work has been partially done during a
visit of the author at the Vietnam Institute for Advanced Mathematics in 2014. He wishes to
thank this institution for financial support and the warm hospitality that he receive. This work
was supported by the NAFOSTED program.
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