Given a function $f\left(t\right)\ge 0$ on $ℝ$ with ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2\left)\right)dt\<\infty$ and $\left|f\right(t)-f(t^{\text{'}}\left)\right|\le l|t-t^{\text{'}}|$, a procedure is exhibited for obtaining on $ℂ$ a (finite) superharmonic majorant of the function$$F\left(z\right):\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\left|𝔍z\right|}{|z-t{|}^2}f\left(t\right)dt-Al\left|𝔍z\right|,$$where $A$ is a certain (large) absolute constant. This leads to fairly constructive proofs of the two main multiplier theorems of Beurling and Malliavin. The principal tool used is a version of the following lemma going back almost surely to Beurling: suppose that $f\left(t\right)$, positive and bounded away from 0 on $ℝ$, is such that ${\int }_{-\infty }^{\infty }\left(f\right(t)/(1+t^2)dt\<\infty$ and denote, for any constant $\alpha \>0$ and each $x\in ℝ$, the unique...