Does there exist a function $c:\mathbb R\to\mathbb C$ such that $$\psi(x)=\int_{\mathbb R}c(k)e^{ikx}dk$$ is square integrable with a nonzero integral, i.e. $0<\int_{\mathbb R}|\psi(x)|^2dk<\infty$ ?
There should probably be some regularity constraints on $\psi$, although I am not sure what. So hopefully $c(k)$ can be analytic. For more details please see below.
Background: One familiar with quantum mechanics should probably know that $e^{ikx}$ stands for a planar wave. Of course, a single planar wave is not normalizable. But according to Introduction to quantum mechanics by David Griffith, a combination of multiple planar waves (more specifically, an integral shaped like the $\psi$ above) could be normalizable due to interference. But I sort of doubt this conclusion, and I am looking for a normalizable example.
It is well known that the FT $\psi$ of $c$ satisfies $\int |\psi|^{2} <\infty$ whenever $\int |c|^{2} <\infty$. Also $\int |\psi|^{2}=0$ iff $c=0$ a.e.. So there are plenty of examples. $c(k)=e^{-|k|}$ is one specific example.