  Given an increasing function $H:[0,1)\to [0,\infty)$ and $$
A_n(H):=\inf_{\tau\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i}
(t_i-t)H^2(t)dt)^{{1/2}}, $$ where $\mathcal{T}_n:=\{\tau=(t_i)_{i=0}^n:
0=t_0<t_1<...<t_n=1\}$, we characterize the property $A_n(H)\leq
\frac{c}{\sqrt{n}}$, and give conditions for $A_n(H)\leq
\frac{c}{\sqrt{n^\beta}}$ and $A_n(H)\geq \frac{1}{c\sqrt{n^\beta}}$ for
$\beta\in (0,1)$, both in terms of integrability properties of $H$. These
results are applied to the approximation of certain stochastic integrals.
