The author continues his previous work on estimating a polynomial \(P(z)\) from a knowledge of \[ \eta = \sum^\infty_{- \infty} \log^+ \bigl |P(n) \bigr |(1 + n^2). \tag{1} \] He is now able to show, rather simply, that (1) implies \(|P(z) |< C_\eta e^{k \eta |z |}\) for all complex \(z\), where \(C_\eta\) is independent of \(P\). Using this result the author gives a proof of the multiplier Theorem of \textit{A. Beurling}, \textit{P. Malliavin}, and the author [Acta Math. 116, 223-277 (1966; Zbl 0152.05403)]. The author's proof is a very considerable simplification of the original proof.