Journal of Mathematical Sciences, Vol. 173, No. 6, 2011

THE VENKOV INEQUALITY WITH WEIGHTS ANDWEIGHTED SPHERICAL HALF-DESIGNS

N. O. Kotelina

Syktyvkar State University55, Oktyabr’skii pr., Syktyvkar 167001, Russia

nad7175@yandex.ru

A. B. Pevnyi ∗

Syktyvkar State University55, Oktyabr’skii pr., Syktyvkar 167001, Russia

pevnyi@syktsu.ru UDC 517.95

We prove the Venkov inequality with weights and introduce a weighted spherical half-

design. We show that the inequality becomes equality on and only on spherical half-

designs. We establish a criterion for a weighted spherical half-design in the language of

cubature formulas and give an example of such a half-design. Bibliography: 8 titles.

1 The Main Notions and Results

We use the inner product 〈x, y〉 = x1y1 + . . . + xnyn of vectors x, y ∈ Rn and the norm ‖x‖ =√〈x, x〉. Denote by Sn−1 = {x ∈ R

n : ‖x‖ = 1} the unit sphere in Rn. The following theorem is

a generalization of an inequality due to B. B. Venkov [1] to the case of arbitrary weights. The

theorem will be proved in Section 5.

Theorem 1.1. Suppose that t is an even number, t � 2, Φ = {ϕ1, ϕ2, . . . , ϕm} ⊂ Sn−1, and

W = (W1, . . . ,Wm) is such that

Wi > 0, i ∈ 1 : m,

m∑

i=1

Wi = 1. (1.1)

Thenm∑

i,j=1

WiWj(〈ϕi, ϕj〉)t � ct, (1.2)

where

ct =(t− 1)!!

n(n+ 2) . . . (n+ t− 2). (1.3)

∗ To whom the correspondence should be addressed.

Translated from Problems in Mathematical Analysis 55, March 2011, pp. 29–36.

1072-3374/11/1736-0674 c© 2011 Springer Science+Business Media, Inc.

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Corollary. If Wi = 1/m for every i, then the inequality (1.2) is the classical Venkov in-

equality [1]m∑

i,j=1

(〈ϕi, ϕj〉)t � ctm2. (1.4)

Theorem 1.2. The inequality (1.2) becomes equality if and only if

m∑

i=1

Wi(〈ϕi, x〉)t = ct‖x‖t, x ∈ Rn. (1.5)

Because of Theorem 1.2, it is natural to introduce the following definition.

Definition 1.1. A pair (Φ,W ), where Φ = {ϕ1, ϕ2, . . . , ϕm} ⊂ Sn−1 andW = (W1, . . . ,Wm)

satisfies (1.1), is called a weighted spherical half-design of order t if the identity (1.5) holds.

Theorem 1.2 can be reformulated as follows: The inequality (1.2) becomes equality on and

only on weighted spherical half-designs of order t.

With every weighted spherical half-design of order t we associate a cubature formula for

computing integrals over the sphere Sn−1. This formula is exact on homogeneous polynomials

of degree t. We use the multiindex tool to denote polynomials in n variables x = (x1, . . . , xn).

Let i = (i1, . . . , in) be a vector with nonnegative integer components, |i| = i1 + . . . + in, and

xi = xi11 xi22 . . . xinn . Then a homogeneous polynomial Q(x) of degree t is written as

Q(x) =∑

|i|=t

q(i)xi, (1.6)

where q(i) are arbitrary real coefficients.

Theorem 1.3. A pair (Φ,W ), where Φ = {ϕ1, . . . , ϕm} ⊂ Sn−1 and W = (W1, . . . ,Wm)

satisfies (1.1), is a weighted spherical half-design of order t if and only if

1

σn

∫

Sn−1

Q(x)dS =m∑

i=1

WiQ(ϕi) (1.7)

for all homogeneous polynomials Q(x) of the form (1.6). Here, σn = 2πn/2/Γ(n2 ) is the area of

the sphere Sn−1.

2 Property of Spherical Half-Designs

Let t be an even number, t � 2.

Lemma 2.1. Let (Φ,W ) be a weighted spherical half-design of order t. Then (Φ,W ) is a

weighted spherical half-design of order p for any p = 2, 4, . . . , t.

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Proof. We denote

Sk(x) =

m∑

i=1

Wi(〈ϕi, x〉)k, ωk(x) = ‖x‖k, k = 2, 4, . . . , t.

Applying the Laplace operator Δ to both sides of the equality (1.5), we obtain

t(t− 1)St−2(x) = ctt(t+ n− 2)ωt−2(x),

which implies

St−2(x) = ctt+ n− 2

t− 1ωt−2(x). (2.1)

Thus, (Φ,W ) is a weighted spherical half-design of order t− 2 with constant

ct−2 = ct(t+ n− 2)

t− 1=

(t− 3)!!

n(n+ 2) . . . (n+ t− 4).

Applying the Laplace operator to (2.1), we see that (Φ,W ) is a weighted spherical half-design

of order t− 4, t− 6, . . . , 2.

3 Connections between Spherical Half-Designs and Spherical

Designs

Definition 3.1. A pair (Φ,W ), where Φ = {ϕ1, ϕ2, . . . , ϕm} ⊂ Sn−1 andW = (W1, . . . ,Wm)

satisfies (1.1), is called a (weighted) spherical design of order t if

m∑

i=1

Wi(〈ϕi, x〉)k =

{ck‖x‖k, k is even, k � t,

0, k is odd, k � t.(3.1)

A similar definition in the case of equal weights is given in [2].

Definition 3.2. A spherical design with even m = 2μ is said to be symmetric if

ϕμ+i = −ϕi, i ∈ 1 : μ; Wμ+i = Wi, i ∈ 1 : μ. (3.2)

It is easy to clarify a connection between Definitions 1.1 and 3.1. If Φ is a symmetric spherical

design of order t+ 1, where t is an even number, then the pair

Φ(μ) = {ϕ1, . . . , ϕμ}, W (μ) = (2W1, . . . , 2Wμ),

μ∑

i=1

2Wi = 1, (3.3)

is a weighted spherical half-design of order t. Indeed,

μ∑

i=1

2Wi(〈ϕi, x〉)t =m∑

i=1

Wi(〈ϕi, x〉)t = ct‖x‖t, x ∈ Rn.

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Taking into account Lemma 2.1, we obtain the converse assertion: if (3.3) is a weighted

spherical half-design of even order t, then, setting ϕμ+i = −ϕi, Wμ+i = Wi, i ∈ 1 : μ, we

obtain a weighted spherical design of order t+ 1.

The points ϕ1, . . . , ϕ2μ can be regarded as nodes in a cubature formula for computing over the

sphere Sn−1. Then the numbers W1, . . . ,W2μ play role of weights (coefficients) in this cubature

formula.

Proposition 3.1. Let the pair (3.3) be a weighted spherical half-design of even order t � 2,

and let

ϕμ+i = −ϕi, Wμ+i = Wi, i ∈ 1 : μ.

Then

1

σn

∫

Sn−1

Q(x)dS =

2μ∑

i=1

WiQ(ϕi) (3.4)

for any polynomial Q(x) in n variables x = (x1, . . . , xn) of degree at most t+ 1.

Proof. The polynomial Q(x) can be written in the form

Q(x) = Q0(x) +Q1(x) + . . . +Qt+1(x),

where Qp(x) is a homogeneous polynomial of degree p. For Q0(x) = C = const the equality (3.4)

takes the form C = C. By Lemma 2.1, the spherical half-design {Φ(μ),W (μ)} is a spherical half-

design of any order p = 2, 4, . . . , t. By Theorem 1.3, the equality (3.4) is valid for Q2, Q4, . . . , Qt.

For Q1, Q3, . . . , Qt+1 the equality (3.4) is understood as 0 = 0.

We give an example related with minimal vectors of the Korkin–Zolotarev lattice E8. These

240 minimal vectors are listed in the monograph [3] and the article [4]. The vectors are divided

into two groups:

1. 112 vectors of the form ξ = (±12, 06) containing 2 components ±1 and 6 zeros. It is

obvious that ‖ξ‖2 = 2.

2. 128 vectors of the form ξ =(±1

2 ,±12 , . . . ,±1

2

)with an even number of pluses. Here,

‖ξ‖2 = 2.

All the vectors ϕi =1√2ξi, i ∈ 1 : 240, have the unit norm. We consider the system

Φ = {ϕ1, . . . , ϕ240} ⊂ S7. It is obvious that Φ is a symmetric set and is representable as

Φ = Φ0 ∪ −Φ0, Φ0 ∩Φ0 = ∅, |Φ0| = 120.

Proposition 3.2. The following assertions hold:

(i) Φ is a spherical design of order 7 with weights Wi =1

240,

(ii) Φ0 is a spherical half-design of order 6 with weights W 0i =

1

120.

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Proof. By arguments at the beginning of this section, assertion (ii) follows from (i). Let

us verify the conditions in Definition 3.1. Since Φ is a symmetric set, the identity (3.1) holds

automatically for odd k. It remains to prove the identity

1

240

240∑

i=1

(〈ϕi, x〉)6 = c6‖x‖6, x ∈ R8, (3.5)

where c6 =1

64. Substituting ϕi =

1√2ξi into (3.5), we obtain the equivalent identity

Q6(x) = 30ω6(x), x ∈ R8, (3.6)

where

Q6(x) =

240∑

i=1

(〈ξi, x〉)6, ω6(x) = ‖x‖6 = (x21 + . . .+ x28)3.

The polynomial ω6(x) can be written in the form

ω6(x) =

8∑

i=1

x6i + 3∑

i �=j

x4i x2j + 6

∑

i<j<k

x2ix2jx

2k. (3.7)

The polynomial Q6(x) contains the same monomials x6i , x4i x

2j , x

2i x

2jx

2k and no monomials of odd

degree xi because of the symmetry of Φ. We have

Q6(x) = Q1(x) +Q2(x),

where

Q1(x) =∑

i<j

(±xi ± xj)6,

Q2(x) =∑(

±x12

± x22

± . . .± x82

)6.

In the polynomial Q2(x), each expression in the brackets contains an even number of pluses,

and the total number of summands is equal to 128. The coefficient at x6i is equal to 28 in Q1(x)

and(12

)6 · 128 = 2 in Q2(x). Therefore, the coefficient at x6i on both sides of (3.6) is equal to

30.

The coefficient at x4i x2j in Q1(x) is equal to the coefficient at x4i x

2j in the polynomial

(xi + xj)6 + (−xi + xj)

6 + (xi − xj)6 + (−xi − xj)

6,

i.e., it is equal to 4C26 = 60. The coefficient at x4i x

2j in Q2(x) is equal to

128 ·(1

2

)6

C26 = 30.

Therefore, the coefficient at x4i x2j on both sides of the equality (3.6) is equal to 90.

The coefficient at x2i x2jx

2k vanishes in the polynomial Q1(x), is equal to

128

(1

2

)6 6!

2!2!2!= 2 · 90 = 180

in the polynomial Q2(x), and is equal to 180 in the polynomial 30ω6(x).

The equality (3.6) and thereby the proposition is proved.

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Remarks. 1. The constructed design Φ = {ϕ1, . . . , ϕ240} can be taken for nodes in the

cubature formula. By Propositions 3.2 and 3.1,

1

σ8

∫

S7

Q(x)dS =1

240

240∑

i=1

Q(ϕi)

for any polynomial Q(x) of degree at most 7. The authors did not find this formula in the list

of known cubature formulas for a sphere (cf. [5, pp. 311–324]).

2. By the Delsarte estimate (cf. [6]), the number of elements of a spherical design Φ ⊂ Sn−1

of order t = 2s + 1 is at least 2Csn+s−1. In this case, n = 8, s = 3, and 2C3

10 = 240. Hence the

constructed design Φ has minimal possible number of elements.

4 Auxiliaries

Let t be even, t � 2. We consider the space Pn,t of homogeneous polynomials of degree t

(written in the form (1.6)). For f, g ∈ Pn,t,

f(x) =∑

|i|=t

a(i)xi, g(x) =∑

|i|=t

b(i)xi,

we introduce the inner product

[f, g] =∑

|i|=t

a(i)b(i)

c(i), where c(i) =

t!

i1!i2! . . . in!. (4.1)

Equipped with this inner product, Pn,t becomes a Hilbert space. For an arbitrary ϕ ∈ Rn we

introduce the polynomial

ρϕ(x) = (〈ϕ, x〉)t = (ϕ1x1 + . . . + ϕnxn)t.

It is easy to verify that for any ϕ ∈ Rn and f ∈ Pn,t

[ρϕ, f ] = f(ϕ). (4.2)

Indeed, by the polynomial formula,

ρϕ(x) =∑

|i|=t

c(i)ϕixi. (4.3)

Hence for f ∈ Pn,t

[ρϕ, f ] =∑

|i|=t

c(i)ϕia(i)

c(i)= f(ϕ).

The equality (4.2) means that ρϕ(x) is the reproducing kernel of the space Pn,t.

Lemma 4.1. For any x ∈ Rn

∫

Sn−1

(〈ϕ, x〉)tdSϕ = ctσn‖x‖t, (4.4)

where ct is defined by formula (1.3).

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In [7, p. 105], the assertion of Lemma 4.1 is referred to as the Hilbert lemma. The integral

(4.4) is a particular case of the more general integral computed in [8, p. 347].

Lemma 4.2. For any Q ∈ Pn,t

1

σn

∫

Sn−1

Q(ϕ)dSϕ = ct[ωt, Q], (4.5)

where ct is a constant of the form (1.3) and ωt(x) = ‖x‖t.

Proof. We take the integral over the sphere Sn−1 of both sides of the equality (4.3):

∫ρϕ(x)dSϕ =

∑

|i|=t

c(i)M(i)xi,

where

M(i) =

∫ϕidSϕ.

We take the inner product of this equality and Q ∈ Pn,t :

[∫ρϕ(x)dSϕ, Q

]=

∑

|i|=t

M(i)q(i) =

∫Q(ϕ)dSϕ. (4.6)

On the other hand, by Lemma 4.1,

∫ρϕ(x)dSϕ = ctσnωt(x), x ∈ R

n.

Substituting into (4.6), we obtain the required equality.

If Q = ωt, then we have ωt(ϕ) = ‖ϕ‖t = 1 for ϕ ∈ Sn−1. Hence (4.5) implies

[ωt, ωt] =1

ct. (4.7)

5 Proof of Theorems

Proof of Theorem 1.1 is based on an idea due to B. B. Venkov [1]. Using the inner product

[f, g] defined by (4.1), we write the inequality

A :=

⎡

⎣m∑

i=1

Wiρϕi − ctωt,m∑

j=1

Wjρϕj − ctωt

⎤

⎦ � 0. (5.1)

As above, ωt(x) = ‖x‖t is an even number. We have

A =m∑

i,j=1

WiWj

[ρϕi , ρϕj

]− 2ct

[m∑

i=1

Wiρϕi , ωt

]

+ c2t [ωt, ωt].

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By (4.2), [ρϕi , ρϕj

]= ρϕj (ϕi) = (〈ϕi, ϕj〉)t.

Furthermore, [m∑

i=1

Wiρϕi , ωt

]

=

m∑

i=1

Wiωt(ϕi) =

m∑

i=1

Wi‖ϕi‖t = 1

since ‖ϕi‖ = 1 and {Wi} satisfies (1.1). Consequently, taking into account (4.3), we have

A =m∑

i,j=1

WiWj(〈ϕi, ϕj〉)t − 2ct + c2t ·1

ct� 0, (5.2)

which implies the inequality (1.2). �

Proof of Theorem 1.2. Necessity. Assume that for some pair (Φ,W ) the inequality (1.2)

becomes equality. Then the inequalities (5.1) and (5.2) are also equalities. Hence

m∑

i=1

Wiρϕi(x)− ctωt = 0

or, which is the same,m∑

i=1

Wi(〈ϕi, x〉)t = ct‖x‖t, x ∈ Rn. (5.3)

Sufficiency. Let the equality (5.3) hold. We substitute x = ϕj . Since ‖ϕj‖ = 1, we have

m∑

i=1

Wi(〈ϕi, ϕj〉)t = ct, j ∈ 1 : m. (5.4)

Multiplying (5.4) by Wj and taking the sum with respect to j ∈ 1 : m, we obtain the equality

m∑

i, j=1

WiWj(〈ϕi, ϕj〉)t = ct.

The theorem is proved.

Proof of Theorem 1.3. Necessity. Let (Φ,W ) be a weighted spherical half-design of order

t. Then the identity (1.5) holds. With the help of the reproducing kernel ρϕ(x) introduced in

Section 4, we can write this identity in the form

m∑

i=1

Wiρϕi(x) = ctωt(x), x ∈ Rn. (5.5)

We take the inner product of (5.5) and an arbitrary polynomial Q ∈ Pn,t :

m∑

i=1

Wi[ρϕi , Q] = ct[ωt, Q]. (5.6)

We have

[ρϕi , Q] = Q(ϕi), i ∈ 1 : m.

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By formula (4.5),

ct[ωt, Q] =1

σn

∫

Sn−1

Q(x)dS. (5.7)

From (5.7) and (5.6) we obtain the required equality (1.7).

Sufficiency. Let the equality (1.7) hold for any polynomial Q ∈ Pn,t. It can be written in

the form1

σn

∫

Sn−1

Q(ϕ)dSϕ =

m∑

i=1

Wi[ρϕi , Q].

Using (4.5), we findm∑

i=1

Wi[ρϕi , Q] = ct[ωt, Q].

Since this equality holds for any Q ∈ Pn,t, we conclude that

m∑

i=1

Wiρϕi(x) = ctωt(x), x ∈ Rn,

which is equivalent to the equality (1.5).

References

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2. V. A. Yudin, “Rotations of spherical designs” [in Russian], Probl. Peredachi Inf. 36, No.3, 39–45 (2000). English transl.: Probl. Inf. Transm. 36, No. 3, 230–236 (2000);

3. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, NewYork etc. (1988).

4. N. N. Andreev and V. A. Yudin, “Arithmetic minimum of quadratic form and sphericalcodes” [in Russian], Mat. Prosv., Ser. 3, No. 2, 133–140 (1998).

5. I. P. Mysovskikh, Interpolation Cubature Formulas [in Russian], Nauka, Moscow (1981).

6. P. Delsarte, J. M. Goetals, and J. J. Seidel, “Spherical codes and designs,” Geom. Dedicata6, 363–388 (1977).

7. B. Reznick, “Sums of even powers of real linear forms,” Mem. Am. Math. Soc. 96, No. 463,1–155 (1992).

8. P. J. Davis and P. Rabinovitz, Methods of Numerical Integration, Academic Press, NewYork (1984).

Submitted on January 31, 2011

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