New characterizations of completely monotone functions and Bernstein functions, a converse to Hausdorff’s moment characterization theorem

• Published in: Arab journal of mathematical sciences Vol. 25; no. 1; pp. 57 - 82
• Main Authors: Aguech, Rafik, Jedidi, Wissem
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2019; Emerald Publishing
• Subjects: Bernstein functions; Completely alternating functions; Completely alternating sequences; Completely monotone functions; Completely monotone sequences; Hausdorff moment problem; Hausdorff moment sequences; Self-decomposability; Self-decomposability; 30E05; 60B10; Hausdorff moment sequences; Completely alternating sequences; Completely monotone sequences; 44A60; 44A10; 47A57; Hausdorff moment problem; Completely monotone functions; Completely alternating functions; Bernstein functions; 60E07; 60E05
• ISSN: 1319-5166; 2588-9214
• DOI: 10.1016/j.ajmsc.2018.03.001

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on N0. We give a complete answer to the following question: Can we affirm that a functionfis completely monotone (resp. a Bernstein function) if we know that the sequencef(k)kis completely monotone (resp. alternating)? This approach constitutes a kind of converse to Hausdorff’s moment characterization theorem in the context of completely monotone sequences.