I was working on a problem for a Harmonic Analysis class, and have developed a heuristic which I think I could make rigorous, except that I would need the following technical statement to work out:
Let $f$ be a Schwartz class function $\Bbb R\to\Bbb R$, and let $x_n$ be the collection of points where $f'(x_n)=0$ (with index set $\Bbb Z$, say, having $x_n<x_m$ iff $n<m$). Then the series $$ \sum_n f(x_n)^2$$ converges.
My intuition is that this statement has a reasonable chance of being true: since $f''$ must decay to zero quickly, this implies that $f'$ must eventually oscillate relatively slowly. Therefore, the $x_n$'s should eventually be "not too close together", and so the $L^2$ convergence of $f$ would hopefully imply the convergence of the sum.
I'm almost certain that there is an easier way to do my homework problem, but I'm now curious about this statement itself. Can anyone prove it, or suggest references for sufficient conditions that allow passing from sequences to integrals to check (absolute) convergence?
(EDIT: I just realized that perhaps $f'$ could be zero on an interval, which kills any hope. I'm not sure if there is a more subtle modification that I'd want than "don't consider those functions")