A Generalization of Bernstein'S Theorem and a Differential Inversion Formula

摘要

A function phi(t) is called completely momotone (CM) on (o, infinity) if (-1) raised to the nth power phi(t) = or > o, n = 0, 1,... and t > o. Bernstein's theorem states that a function phi is CM if and only if phi is the Laplace transform of a measure on (o, infinity). Moreover the generating measure can be obtained from phi by applying a sequence of differential operators to phi. These two results are generalized to the case where (-1) raised to the nth power phi(t) = or > o, n = 0,1,... is replaced by an infinite sequence of differential inequalities of a special type. (Author)