This result was stated in Beurling's complete works without proof and proved in Hörmander (the case ) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case in Hardy's Theorem while the version by Bonami–Demange–Jaming covers the full strength of Hardy's Theorem. A different proof of Beurling's theorem based on Liouville's theorem appeared in ref.
this result was stated in beurling's complete works without proof and proved in hörmander (the case ) and bonami demange and jaming for the general case note that hörmander–beurling's version implies the case in hardy's theorem while the version by bonami–demange–jaming covers the full strength of hardy's theorem a different proof of beurling's theorem based on liouville's theorem appeared in ref