Special N-extremal solutions to indeterminate moment problems

Abstract

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