A note on the Carleman condition for determinacy of moment problems

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

“…https://doi.org/10.1017/S1446788700002731 [17] Converse Carleman and Krein criteria 97 work relevant to our theme. Sjodin [14] shows that HC(^) is 'sharp in a certain sense'. Let f5 2n = sup ;>n /M^1 /2; ' .…”

“…Another criterion of Carleman is that F is H-det if ]C n P\n -oo. Sjodin [14] shows there is a non-determining moment sequence {k n } satisfying lim n _ 0O (A.2 n //x 2n ) l/ " = 0 if and only if £ Pin < oo. This assertion implies that YJ Phl l2n < oo, but since Carleman's condition is not necessary, it can be that [li n } is determining.…”

“…Before discussing applications of Theorem 1 and Theorem 2 to the Hamburger problem we summarize some known work relevant to our theme. Sjodin [14] shows that HC(^) is 'sharp in a certain sense'. Let f5 2n = sup ;>n /M^1 /2; ' .…”

Representation of measures with simultaneous polynomial denseness in Lp(R, dμ), 1≤p&lt;∞

“…The set of Carleman measures will be denoted by M~(R). They have originally been introduced in the context of the so-called Hamburger and Stieltjes determinacy problem (see [1], [17 22], [25], [28] and [29]), and it has been known for some time (see [6] and [27]) that the deternfinacy of measures in M*(R) is related to the polynomial density in certain L2 spaces. For us the following result by C. Berg and J. P. R. Christensen [6] (see also [5]) is the most relevant.…”