Chebyshev-type cubature formulas for doubling weighted spheres, balls, and simplexes
Authors: Dai Feng , Feng Han 
Published: July 2019

This paper shows that, given a doubling weight

w
w

on the unit sphere

S

d
−
1

\mathbb {S}^{d-1}

of

R

d

\mathbb {R}^d

, there exists a positive constant

K

w
,
d

K_{w,d}

such that, for each positive integer

n
n

and each integer
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to max Underscript x element-of double-struck upper S Superscript d minus 1 Baseline Endscripts StartFraction upper K Subscript w comma d Baseline Over w left-parenthesis upper B left-parenthesis x comma n Superscript negative 1 Baseline right-parenthesis right-parenthesis EndFraction">

N
≥

max

x
∈

S

d
−
1

K

w
,
d

w
(
B
(
x
,

n

−
1

)
)

N\ge \max _{x\in {\mathbb {S}^{d-1}}} \frac {K_{w,d}} {w(B(x, n^{-1}))}

, there exists a set of

N
N

distinct nodes

z
1

z_1

, …,

z
N

z_N

on

S

d
−
1

\mathbb {S}^{d-1}

for which
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel StartFraction 1 Over w left-parenthesis double-struck upper S Superscript d minus 1 Baseline right-parenthesis EndFraction integral Underscript double-struck upper S Superscript d minus 1 Baseline Endscripts f left-parenthesis x right-parenthesis w left-parenthesis x right-parenthesis d sigma Subscript d Baseline left-parenthesis x right-parenthesis equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts f left-parenthesis z Subscript j Baseline right-parenthesis comma for-all f element-of normal upper Pi Subscript n Superscript d Baseline comma EndLayout">

(

∗

)

1

w
(

S

d
−
1

)

∫

S

d
−
1

f
(
x
)
w
(
x
)

d

σ
d

(
x
)
=

1
N

∑

j
=
1

N

f
(

z
j

)
,

∀
f
∈

Π
n
d

,

\begin{equation} \tag {$\ast $} \frac {1}{w({\mathbb {S}^{d-1}})} \int _{{\mathbb {S}^{d-1}}} f(x) w(x)\, d\sigma _d(x)=\frac 1N \sum _{j=1}^N f(z_j),\qquad \forall f\in \Pi _n^d, \end{equation}

where

d

σ
d

d\sigma _d

,

B
(
x
,
r
)

B(x,r)

, and

Π
n
d

\Pi _n^d

denote the surface Lebesgue measure on

S

d
−
1

{\mathbb {S}^{d-1}}

, the spherical cap with center

x
∈

S

d
−
1

x\in \mathbb {S}^{d-1}

and radius

r
>
0

r>0

, and the space of all spherical polynomials of degree at most

n
n

on

S

d
−
1

{\mathbb {S}^{d-1}}

, respectively, and

w
(
E
)
=

∫
E

w
(
x
)

d

σ
d

(
x
)

w(E)=\int _E w(x) \, d\sigma _d(x)

for

E
⊂

S

d
−
1

E\subset {\mathbb {S}^{d-1}}

. If, in addition,

w
∈

L
∞

(

S

d
−
1

)

w\in L^\infty ({\mathbb {S}^{d-1}})

, then the above set of nodes can be chosen to be well separated:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min Underscript 1 less-than-or-equal-to i not-equals j less-than-or-equal-to upper N Endscripts arc cosine left-parenthesis z Subscript i Baseline dot z Subscript j Baseline right-parenthesis greater-than-or-equal-to c Subscript w comma d Baseline upper N Superscript minus StartFraction 1 Over d minus 1 EndFraction Baseline greater-than 0 period">

min

1
≤
i
≠
j
≤
N

arccos
⁡
(

z
i

⋅

z
j

)
≥

c

w
,
d

N

−

1

d
−
1

>
0.

\min _{1\leq i\neq j\leq N}\arccos (z_i\cdot z_j)\geq c_{w,d} N^{-\frac 1{d-1}}>0.

\]
It is further proved that the minimal number of nodes

N

n

(
w
d

σ
d

)

\mathcal {N}_{n} (wd\sigma _d)

required in (

∗
\ast

) for a doubling weight

w
w

on

S

d
−
1

{\mathbb {S}^{d-1}}

satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper N Subscript n Baseline left-parenthesis w d sigma Subscript d Baseline right-parenthesis tilde max Underscript x element-of double-struck upper S Superscript d minus 1 Baseline Endscripts StartFraction 1 Over w left-parenthesis upper B left-parenthesis x comma n Superscript negative 1 Baseline right-parenthesis right-parenthesis EndFraction comma n equals 1 comma 2 comma ellipsis period">

N

n

(
w
d

σ
d

)
∼

max

x
∈

S

d
−
1

1

w
(
B
(
x
,

n

−
1

)
)

,

n
=
1
,
2
,
…
.

\mathcal {N}_n (wd\sigma _d) \sim \max _{x\in {\mathbb {S}^{d-1}}} \frac 1 {w(B(x, n^{-1}))},\qquad n=1,2,\ldots .

\]
Proofs of these results rely on new convex partitions of

S

d
−
1

{\mathbb {S}^{d-1}}

that are regular with respect to a given weight

w
w

and integer

N
N

. Similar results are also established on the unit ball and the standard simplex of

R

d

\mathbb {R}^d

.

Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs.