Moment Problem

Moment Problem

Please inform me if you have an idea in the question stated below.

　We cite a paper :

Codimension of polynomial subspace in 　for discrete indeterminate measure _.

Andrew G.Bakan, Proceedings of the AMS,(2002)vol.130,no 12,3545-3553

Abstracts:

Let be the set of positive Borel measures on the real line 　with moments of all orders. A measure is said to be indeterminate if the set 　contains at least one measure

other than　 where 　is a set of measures such that

Theorem 1.Let

be any discrete indeterminate measure from the class .

Then the following statements hold.

(A)If ,

where codim 　denotes the codimension of the closures of the algebraic polynomialsin .

(B)If 　is an n-canonical measure for some nonnegative integer n, then　 there exist numbers , such that

I am here interested in a relation between the part (B) in this Theorem and the Cassie’s Proposition in the following.

Cassier has characterized n canonical measures in the following way:

Proposition 4. Soit 　une measure ayant des moments de tous les orders, 　un entire, 　des points distincts de 　n’apartement pas au support de , et 　des constantes strictement positives. On a alors les equivalences:

a) 　est N-extremale;

b) .

Measures Canoniques dan le Problem Classique des Momens.

Ann.Inst.Fourier,Grenoble,34,2(1984),45-52.

Question[1]:

Is Cassier’s Proposition　 a special case of the Bakan’s Theorem 1 ?

For example 　for n points of k. Does’nt this be permitted?

Question[2]:

is known to be a value at of some N-extremal measure.

Cassier’s　 proposition　 said that n-canonocal measures are larger than N-extremal measure for n supporting points. But Theorem 1 seems to describe that n canonical measure is less than N-extremal measure　　in .

Is this a contradiction?　 I think that it’s not, and　 that ’s　 should come from more than two N-extremal measures. Probably they come from the n N-extremal measures.