Titchmarch's theorem says that a complex function $f(z)$ is analytic on the (closed) upper half of the complex plane and decays rapidly as $|z| \to \infty$ iff its real and imaginary parts are Hilbert transforms of each other.
But just how fast does in need to decay as $|z| \to \infty$? This Wikipedia article says that we only need $\lim \limits_{|z| \to \infty} f(z) = 0$, but this one says that we need that $$\int_{-\infty}^\infty |f(x + i y)|^2\, dx < K.$$
I believe that this is a stronger requirement. For example, if $f(z) \sim 1/\sqrt{z}$ as $|z| \to \infty$, then the former requirement is met but the latter is violated.