Stability of N-extremal Measures

A positive Borel measure μ on R, which possesses all power moments, is N-extremal if the space of all polynomials is dense in L2(μ). If, in addition, μ generates an indeterminate Hamburger moment problem, then it is discrete. It is known that the class of N-extremal measures that generate an indeterminate moment problem is preserved when a finite number of mass points are moved (not “removed”!). We show that this class is preserved even under change of infinitely many mass points if the perturbations are asymptotically small. Thereby “asymptotically small” is understood relative to the distribution of suppμ; for example, if suppμ = {nσ log n : n ∈ N} with some σ > 2, then shifts of mass points behaving asymptotically like, e.g. nσ−2[log logn]−2 are permitted. A sequence ~s = (sn) ∞ n=0 of real numbers is called a Hamburger moment sequence if there exists a positive Borel measure μ on R which has ~s as its sequence of power moments (1) sn = ∫ R x dμ(x), n = 0, 1, 2, . . . If ~s is a Hamburger moment sequence, we denote by V~s the set of all positive Borel measures μ on R such that (1) holds. Hamburger moment sequences can be characterized by a determinant criterion (see, e.g. [1, Theorem 2.1.1]). If ~s is a Hamburger moment sequence and V~s contains only one element, the sequence ~s is called determinate; otherwise, it is called indeterminate. Which of these alternatives takes place can also be characterized by a determinant criterion (see, e.g. [1, Addendum 9 in Chapter 2]). A measure μ that leads to an indeterminate moment sequence ~s via (1) is also called indeterminate; similarly, one speaks of a determinate measure. For an indeterminate sequence ~s, the set V~s is infinite and can be described as follows. The Nevanlinna parameterization. Let ~s be an indeterminate Hamburger moment sequence. Then there exist four entire functions A,B,C,D of minimal exponential type (here and in the following we understand by this a function of order one and type zero or a function of order less than one) such that the formula (2) ∫ R dμ(x) x− z = A(z)τ(z) +B(z) C(z)τ(z) +D(z) , z ∈ C \ R, establishes a bijective correspondence μ ↔ τ between V~s and the set N := { τ : τ analytic in C \ R, τ(z) = τ(z), Im z · Im τ(z) ≥ 0 } ∪ {∞}. For τ = ∞, the right-hand side of (2) is interpreted as A(z)/C(z). ♦ A measure μ, which possesses all moments, is called N-extremal if the space of all polynomials is dense in L(μ). By a theorem of M. Riesz, a measure, which possesses all 2000 Mathematics Subject Classification. Primary 30E05; Secondary 30D15.