Let
{
Q
n
}
n
=
1
∞
\{ {Q_n}\} _{n = 1}^\infty
denote a sequence of quadrature formulas,
Q
n
(
f
)
≡
∑
i
=
1
k
n
w
i
(
n
)
f
(
x
i
(
n
)
)
{Q_n}(f) \equiv \sum _{i = 1}^{{k_n}}w_i^{(n)}f(x_i^{(n)})
, such that
Q
n
(
f
)
→
∫
0
1
f
(
x
)
d
x
{Q_n}(f) \to \int _0^1 f (x)dx
for all
f
∈
C
[
0
,
1
]
f \in C[0,1]
. Let
0
>
ε
>
1
4
0 > \varepsilon > \frac {1}{4}
and a sequence
{
a
n
}
n
=
1
∞
\{ {a_n}\} _{n = 1}^\infty
be given, where
a
1
≧
a
2
≧
a
3
≧
.
.
.
{a_1} \geqq {a_2} \geqq {a_3} \geqq ...
, and where
a
n
→
0
{a_n} \to 0
as
n
→
∞
n \to \infty
. Then there exists a function
f
∈
C
[
0
,
1
]
f \in C[0,1]
and a sequence
{
n
k
}
k
=
1
∞
\{ {n_k}\} _{k = 1}^\infty
such that
|
f
(
x
)
|
≦
2
a
1
/
|
(
1
−
4
)
|
|f(x)| \leqq 2{a_1}/|(1 - 4)|
, and such that
∫
0
1
f
(
x
)
d
x
−
Q
n
k
(
f
)
=
a
k
,
k
=
1
,
2
,
3
,
.
.
.
\int _0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,...
.