Extensions of Bernstein's theorem on absolutely monotonic functions
type: Item
instance of: scholarly article
title: Extensions of Bernstein's theorem on absolutely monotonic functions
zbMATH Open document ID: 0594.26013
DOI: 10.1016/S0022-247X(86)80013-0
author: G. G. Bilodeau
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: 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 \(D^ n[\exp (knx)f(x)]>0.\)
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 26A48
Mathematics Subject Classification ID: 26E10
zbMATH DE Number: 3955284
zbMATH Keywords: absolutely monotonic functions
zbMATH Keywords: Bernstein's theorem
MaRDI profile type: Publication
cites work: Signs of Derivatives and Analytic Behavior
cites work: What is the Laplace Transform?
description: scientific article; zbMATH DE number 3955284