Beurling-Malliavin multiplier theorem: The seventh proof

We present a new proof of the Beurling-Malliavin theorem, often called the ``multiplier theorem'', concerning the existence of a real-valued function on $\mathbb{R}$ with spectrum in a given (small) interval and with a given small majorant of the modulus. This proof pertains entirely to real analysis. It only involves elementary facts about the Hilbert transformation; neither complex variable methods nor potential theory is exploited. The heart of the proof is Theorem 2, which treats preservation of the Lipschitz property under the Hilbert transformation. We also include a short survey of earlier proofs of the Beurling--Malliavin theorem and its generalizations to model (coinvariant) subspaces of the Hardy space $H^2(\mathbb{R})$.