This result was stated in Beurling's complete works without proof and proved in Hörmander (the case d = 1 , N = 0 ) and Bonami, Demange, and Jaming for the general case. Note that Hörmander–Beurling's version implies the case ab > 1 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.