Extensions of Bernstein's theorem on absolutely monotonic functions (Q1076828)

Bernstein's theorem is that if \(D^ nf(x)\geq 0\) for all n, \(a\leq x\leq b\), then f is real analytic. The author replaces \(D^ nf(x)\geq 0\) by \(D^ n[\exp (a_ nx)f(x)]\geq 0.\) His main result is that if \(\{a_ n\}\) is a nondecreasing sequence and \(\lim \sup a_ n/n<\infty\) then f is analytic; \(a_ n/n\) cannot be replaced by \(a_ n/n^ a\) with \(a>1\). As a partial converse, the author shows that if f is analytic and positive then there is a constant k for which \(D^ n[\exp (knx)f(x)]>0.\)