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.



Seeking a class of functions for which sums approximate integrals well
One class of functions you could consider is those $f$ for which $|f(x)|+|{\hat f}(x)| \le C(1+|x|)^{-1-\delta}$ for some constant $C$ and some $\delta>0$. Here ${\hat f}(x) = \int\_{-\infty}^{\infty} f(t) e^{-2\pi i tx} dt$ is the Fourier transform. For this class of functions, one can use the Poisson summation formula, which gives 
$$ 
h\sum\_{n\in {\Bbb Z}} f(a+nh) = \sum\_{k\in {\Bbb Z}} {\hat f}(k/h) e^{2\pi i ak/h}.
$$ 
Since $I={\hat f}(0)$, 
$$ 
|hS(a,h) - I| \le \sum\_{k\neq 0} |{\hat f}(k/h)| \le 2C \zeta(1+\delta) h^{1+\delta},
$$ 
using our assumption on $f$. See e.g. Stein & Weiss, or the wikipedia article on Poisson summation. The functions you listed would fall into this category.


