Chebyshev-type quadrature for the weight functions
\[
w
a
(
t
)
=
1
−
a
t
π
1
−
t
2
,
−
1
>
t
>
1
,
−
1
>
a
>
1
,
{w_a}(t) = \frac {{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 > t > 1,\quad - 1 > a > 1,
\]
is related to a problem concerning partial sums of the exponential series, namely the problem to extend the nth partial sum to a polynomial of degree 2N having all zeros on the circle
|
z
|
=
|
a
|
N
|z| = |a|N
. Using this connection, we show that the minimal number N of nodes needed for Chebyshev-type quadrature of degree n for
w
a
(
t
)
{w_a}(t)
satisfies an inequality
C
1
n
≤
N
≤
C
2
n
{C_1}n \leq N \leq {C_2}n
with positive constants
C
1
,
C
2
{C_1},{C_2}
. As an application we prove that the minimal number N of nodes for Chebyshev-type quadrature of degree n on a torus embedded in
R
3
{{\mathbf {R}}^3}
satisfies an inequality
C
1
n
2
≤
N
≤
C
2
n
2
{C_1}{n^2} \leq N \leq {C_2}{n^2}
.