Subharmonic Additions to the Beurling–Malliavin Theorems. I. On the Multiplier

The Beurling–Malliavin Theorem on the multiplier and its various versions give several variants of conditions for the function \(f\) on the real axis \(\mathbb{R}\) , under which this function can be multiplied by an entire, bounded on \(\mathbb{R}\) , function \(h\) of arbitrarily small exponential type \(>0\) so that the product of \(fh\) is bounded on \(\mathbb{R}\) . We consider a new version for the function \(f=\exp(u-M)\) , where \(u\) and \(M\) is a pair of subharmonic functions of finite type with finite logarithmic integrals over \(\mathbb{R}\) .