On weighted polynomial regression designs with minimum average veriance

Let the measurements of the function $\eta(x)=(w(x))^{1/2}\sum_{\alpha=0}^m\theta_\alpha x^\alpha$ at points $x_i$ $i=1,\dots,N$ give the values $y_i=\eta(x_i)+\nu_i$, $\nu_i$ being independent random variables, $\mathbf E\nu_i=0$, $\mathbf D\nu_i=\sigma^2$.
The design of the experiment can be described by a discrete probability measure $\varepsilon(x)$ which is the proportion of measurements at $x$. Let $d(x,\varepsilon)$ be the variance of the least-squares estimate $\widehat\eta(x)$ of the function $\eta(x)$.
The unique designs of the experiment minimizing
$$ a(\varepsilon)=\int_Xd(x,\varepsilon)\,dx $$
are found in the two cases: 1) $w(x)\equiv1$, $X=[-1,1]$ and 2) $w(x)=e^{-x^2}$, $X=(-\infty,\infty)$.