This question is somehow related to my previous post but asks for somewhat stronger conditions.
I am trying to find an approximation of the characteristic function of given interval satisfying the following conditions, but unfortunately I haven't been able to come up with anything.
For $\epsilon>0$ and $t\in[0,T]$ let $K_t^{\epsilon}:[0,T]\to \mathbb R$ be a deterministic function satisfying:
- $K_t^{\epsilon}\longrightarrow \mathbf{1}_{[0,t]}$ in $L^2[0,T]$ as $\epsilon\to 0$;
- There exists a constant $C$ such that \begin{align*} \sup_{0\leq s,t\leq T}\left(|\partial_tK_t^{\epsilon}(s)|+|\partial_t^2K_t^{\epsilon}(s)|\right)\leq C; \end{align*}
- $\frac{d}{dt}\|K_t^{\epsilon}\|^2_{L^2[0,T]}\longrightarrow 1$ as $\epsilon\to 0$, $\star$-weakly in $L^{\infty}[0,T]$.
I would really appreciate if you come up with an example, or at least some thought.