Special N-Extremal Solutions to Indeterminate Moment Problems

For an N-extremal solution μ μ to an indeterminate moment problem it is known by a theorem of M. Riesz that the measure (1+x²)⁻¹dμ (x) ( 1 + x 2 ) - 1 d μ ( x ) is determinate. For 0<α <1 0 < α < 1 we show by contradiction that there exist indeterminate N-extremal solutions μ μ such that (1+x²)-αdμ (x) ( 1 + x 2 ) - α d μ ( x ) is determinate, and there exist also indeterminate N-extremal solutions μ μ such that (1+x²)-αdμ (x) ( 1 + x 2 ) - α d μ ( x ) is indeterminate. Explicit examples of such measures are so far only known when α =1/2 α = 1 / 2 . For indeterminate Stieltjes moment problems and for N-extremal solutions μ μ , we show that (1+x²)-1/2dμ (x) ( 1 + x 2 ) - 1 / 2 d μ ( x ) is indeterminate except when $$μ =μ