The universality of some compositions on short intervals

We obtain approximation theorems for analytic functions by the shifts $F(\zeta(s+i\tau))$ with $\tau \in {\Bbb R}$, where $\zeta(s)$ is the Riemann $\zeta$-function, while $F$ is some operator on the space of analytic functions, on short intervals $[T,T+H]$ with $T^{1/3}(\log T)^{26/15}\leq H\leq T$ as $T\to\infty$.