 The theorem lacks an elementary proof.The original proof by Titchmarsh is based on the Phragmén–Lindelöf principle, Jensen's inequality, the Theorem of Carleman, and Theorem of Valiron.More proofs are contained in [Hörmander, Theorem 4.3.3] (harmonic analysis style), [Yosida, Chapter VI] (real analysis style), and [Levin, Lecture 16] (complex analysis style).