Extremal solutions of the strong Stieltjes moment problem

[摘要] A solution of the strong Stieltjes moment problem for the sequence {c(n): n = 0, +/- 1, +/- 2,...} is a finite positive measure mu on [0,infinity) such that c(n) = integral(0)(infinity) t(n) d mu(t) for all n, while a solution of the strong Hamburger moment problem for the same sequence is a finite positive measure mu on (-infinity, infinity) such that c(n) = integral(-infinity)(infinity) t(n) d mu(t) for all n. When the Hamburger problem is indeterminate, there exists a one-to-one correspondence between all solutions mu and all Nevanlinna functions phi, the constant infinity included. The correspondence is given by F(m)u(z) = -alpha(z)phi(z) - gamma(z)/beta(z)phi(z) - delta(z)' where alpha, beta, gamma, delta are certain functions holomorphic in C - {0}. The extremal solutions are the solutions mu(t) corresponding to the constant functions phi(z) = t, t is an element of R boolean OR {infinity}. The accumulation points of the (isolated) set Z(t) of zeros of beta(z)t - delta(z) consists of 0 and infinity. The support of the extremal solution mu(t) is the set Z(t) boolean OR {0}. There exists an interval [t((0)), t(infinity)] such that the extremal solutions of the Stieltjes problem are exactly those mu(t) for which t is an element of [t((0)), t((infinity))]. The measures mu(t(0)) and mu(t(infinity)) are natural solutions, and the only ones. If xi(k)((n)) denote the zeros of the orthogonal Laurent polynomials determined by {c(n)} ordered by size, then {xi(k)((n))} tends to 0 and {xi(n-k)((n))} tends to infinity for arbitrary constant k when n tends to infinity.

[发布日期] 1995-12-29

[效力级别]  Proceedings Paper

[关键词] strong Stieltjes moment problem;Nevanlinna functions;extremal solutions