Before revision:  For sliding windows, we obtain a (1+\epsilon)-approximation to F_p for p\in(0,2 , resolving a longstanding question of Braverman and Ostrovsky (FOCS 2007). For example, for p = 2 we improve the dependence on \epsilon from \frac{1{\epsilon^4} to an optimal \frac{1}{\epsilon^2}.
After revision:  For sliding windows, we obtain a (1+\epsilon)-approximation to F_p for p\in(0,2 , suppressing logarithmic factors. For general functions the latter is known to be best-possible, by a result of Kaplan et. al. (CRYPTO 2021). We show how to bypass this impossibility result by developing data stream algorithms for a large class of streaming problems, with no overhead in the approximation factor. Our class of streaming problems includes the most well-studied problems such as the L_2-heavy hitters problem, F_p-moment estimation, as well as empirical entropy estimation. We substantially improve upon all prior work on these problems, giving the first optimal dependence on the approximation factor. As in previous work, we obtain a general transformation that applies to any non-robust streaming algorithm and depends on the dependence on \epsilon from \frac{1{\epsilon^4} to an optimal \frac{1}{\epsilon^2}.