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I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that: \begin{equation} \sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+. \end{equation}
In other words I am trying to find conditions on $f$ such that the above sum is bounded for $h$ small enough. Alternatively, can I impose conditions of $f$ that make $\sum_{k\in\mathbb{Z}}f(kh)h \to_{h\to0} \int_R f (x)dx$? Many thanks.


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This looks too much like a homework problem and therefore inappropriate for this site. You might want to try math.stackexchange.com. But if it really is homework, you really should try to figure this out yourself using the definition of an integral as a limit of Riemann sums, as well as the usual $\epsilon-\delta$ type of arguments. –  Deane Yang Aug 10 '12 at 20:52
It is not a homework problem. And the interval of integration is not closed and bounded, Riemann sums need not converge to the respective integral, even when the latter is finite... –  Francesco Mina Aug 11 '12 at 17:05

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