Remarks on converse Carleman and Krein criteria for the classical moment problem

The key theme is converse forms of criteria for deciding determinateness in the classical moment problem. A method of proof due to Koosis is streamlin ed and generalized giving a convexity condition under which moments mu (n) = integral (infinity)(0) x(n) f (x) dx satisfying Sigma mu (-c/n)(n) < <inf inity> implies that integral (infinity)(x') x(-1-c) (-logf (x)) dx < <infin ity>, c a positive constant. A contrapositive version is proved under a rap id variation condition on f (x), generalizing a result of Lin. These result s are used to obtain converses of the Stieltjes versions of the Carleman an d Krein criteria. Hamburger versions are obtained which relax the symmetry assumption of Koosis and Lin, respectively. A sufficient condition for Stie ltjes determinateness of a discrete law is given in terms of its mass funct ion. These criteria are illustrated through several examples.