Weighted shifts induced by Hamburger moment sequences

We indicate how our subject emerges from the confluence of several streams of analysis, including the classical moment problems, the theory of positive matrices and subnormal operator theory. Some new properties H(n)H(n)(n=1,2,…)(n=1,2,…) and a Hamburger-type weighted shift are considered via a Hamburger moment sequence. We discuss examples to show the various H(n)H(n) are distinct; study flatness, backward n  -step extensions and perturbations of weighted shifts; and, given three initial weights α0,α1,α2α0,α1,α2 with α0≤α2<α1α0≤α2<α1, we produce a completion: a weighted shift of Hamburger type but not subnormal, extending a (subnormal) completion by Stampfli in the case α0<α1<α2α0<α1<α2.