Let $f\in L_2(\mathbb R)$ be a function. Is $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\infty $?

Question

Let $f\in L_2(\mathbb R)$ be a function. Is $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\infty $ ?

I am trying to prove the relation $\displaystyle\sum_{k\in\mathbb{Z}}|f(k)|^2<\int_{\mathbb R}|f(x)|^2dx$.

I am not able to produce examples to say above relation is false, so I guess that the above relation is true. But, I am not able to prove that.

Please help me!

Answer 1

This result is false. Let $ f: \mathbb{R} \to \mathbb{R} $ via $ f(x) = x \chi_\mathbb{Z}(x) $, where $ \chi_\mathbb{Z}$ is the characteristic function of the integers. Then $ f $ is zero almost everywhere, so $ f \in L^2(\mathbb{R}) $. However, $ \sum_{k \in \mathbb{Z}} |f(k)|^2 = \sum_{k \in \mathbb{Z}} k^2 $, which is certainly not finite.