Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero?

In case the answer is affirmative (and please skip the obvious implication):

**1.** How far can we replace the condition "for every value of $h \in \mathbb{R}$" by a (much) weaker condition "for every value of $h \in A \subset \mathbb{R}$"? Will, for example, some countable or finite set $A$ suffice?

**2.** Is there anything special with the function $\sin^2(x)$ or can we replace it with (many) other functions?

The topmost question arose from a strategy to prove the following little nut:

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that the function $\Delta_h f(x) := f(x + h) - f(x)\ $ is smooth for every value of $h$, then $f$ is itself smooth.

This problem is in turn related to this previous MO problem.

**Edit June 2, 2014.** A colleague just "reminded me" that the OP is a consequence of the

**Cantor-Lebesgue Theorem.** *Let $A_n(x) := c_n e^{inx} + c_{-n}e^{-inx}$. If
$A_n(x)$ tends to zero as $n$ tends to infinity for all $x$ in some set of positive measure, then $c_n$ and $c_{-n}$ tend to zero as $n$ tends to infinity.*

See this paper by R.L. Cooke for a nice presentation.