Special N-extremal solutions to indeterminate moment problems

Abstract

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