Suppose I have function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that it's absolutely integrable: $\int_{\mathbb{R}}|f(x)|dx<\infty$.
I am sampling function $f(x)$ with some period $T_s$. I am interested whether
$\sum_{k=-\infty}^{k=\infty}|f(kT_s)|<\infty$
It seems to me that it's true, but I can't figure out how to prove that.
The reason I ask is that I know that if $f(x)$ is absolutely integrable, then its Fourier transform exists, and I am wondering if the sampled version is guaranteed to have a discrete Fourier transform. Sorry if this is a silly question.