  In this article we prove a generalization of the Beurling--Malliavin Majorant
Theorem. In more detail, we establish a new sufficient condition for a function
to be a Beurling--Malliavin Majorant. Our result is strictly more general than
that of the Beurling--Malliavin Majorant Theorem. We also show that our result
is sharp in a number senses.
