First question: how do you choose $a_j$?. Since I expect you want an operator $P_h : X \to L^2$, you must make the choice before hand. You are unlikely to find any better estimation, since you can take analytical functions designed specifically to break any $P_h$ you construct.
Your result so far is a natural consequence of the Sobolev embedding theorem (see Brézis' book). In dimension one, $H^1 ([a,b]) \subset \mathcal C([a,b])$. In this sense you have just proved the convergence of Riemann sums in the standard way.