Stability of N-extremal measures - Woracek, H.; Langer, M. (Methods of Functional Analysis and Topology, 2015)
A positive Borel measure μ on R, which possesses all power moments, is N-extremal if the space of all polynomials is dense in L^2(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={n^σn: nN} with some σ>2, then shifts of mass points behaving asymptotically like, e.g. n^σ-2[n]^-2 are permitted.