Hi

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.

Francesco