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
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
(
∗
)
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:
\[
min
1
≤
i
≠
j
≤
N
arccos
(
z
i
⋅
z
j
)
≥
c
w
,
d
N
−
1
d
−
1
>
0.
\min _{1\leq i\neq j\leq N}\arccos (z_i\cdot z_j)\geq c_{w,d} N^{-\frac 1{d-1}}>0.
\]
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
\[
N
n
(
w
d
σ
d
)
∼
max
x
∈
S
d
−
1
1
w
(
B
(
x
,
n
−
1
)
)
,
n
=
1
,
2
,
…
.
\mathcal {N}_n (wd\sigma _d) \sim \max _{x\in {\mathbb {S}^{d-1}}} \frac 1 {w(B(x, n^{-1}))},\qquad n=1,2,\ldots .
\]
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.