Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ is $O(h^2)$, where $S(a,h)$ denotes the sum $\sum_{n \in {\bf Z}} f(a+nh)$ and $I$ denotes the integral $\int_{-\infty}^{\infty} f(x) \: dx$? (Note that for fixed $h$, $S(a,h)$ is periodic in $a$ with period $h$, so the supremum over $0 \leq a < h$ is equivalently the supremum over all $a$ in ${\mathbb R}$.)

The class of functions $f(x)$ should include the functions $e^{-x^2}$, $e^{-|x|}$, $\max(1-x^2,0)$, and $\max(1-|x|,0)$.

I can prove this estimate for functions $f$ that (like the four functions listed above) have bounded first derivative and have bounded second derivative on the complement of a finite set of "bad" points, but this class of functions seems rather unnatural.

I would settle for a bound weaker than $O(h^2)$ as long as it's $o(h)$. (Note that the integrability of $f$ gives us $o(1)$ automatically.)

I'm including the fourier-analysis and harmonic-analysis tags because $S(a,h)$ looks like something that could be approached using harmonic analysis.