Special N-extremal solutions to indeterminate moment problems

Abstract

For an N-extremal solution $\hat{I}¼$ to an indeterminate moment problem it is known by a theorem of M. Riesz that the measure $(1+x^2{)}^{-1}d\hat{I}¼(x)$ is determinate. For $0<\hat{I}\pm <1$ we show by contradiction that there exist indeterminate N-extremal solutions $\hat{I}¼$ such that $(1+x^2{)}^{-\hat{I}\pm }d\hat{I}¼(x)$ is determinate, and there exist also indeterminate N-extremal solutions $\hat{I}¼$ such that $(1+x^2{)}^{-\hat{I}\pm }d\hat{I}¼(x)$ is indeterminate. Explicit examples of such measures are so far only known when $\hat{I}\pm =1/2$. For indeterminate Stieltjes moment problems and for N-extremal solutions $\hat{I}¼$, we show that $(1+x^2{)}^{-1/2}d\hat{I}¼(x)$ is indeterminate except when $\hat{I}¼=\hat{I}{¼}_F$ is the Friedrichs solution in case of which $(1+x^2{)}^{-1/2}d\hat{I}{¼}_F(x)$ is determinate. We identify the Friedrichs and Krein solutions for some indeterminate Stieltjes moment problems.

20 pages