What is a simple example of a set of functions $\{e_i\}$ in $L^2_w$ which form an orthonormal basis and satisfy the property that: \begin{equation} \langle e_i(t)\int_{-\infty}^t e_j(s)ds , e_k(t) \rangle_{L^2_w} = \delta_{i=k \mbox{ or } j=k} \end{equation}
(If this is too much to ask, change the righthand-side to $\delta_{i=j=k}$. )
Where $\int_{-\infty}^{\infty}w(s)ds =1$ and $L^2_w$ is the $L^2$-space weighted by $w$.