Naimark Extensions for Indeterminacy in the Moment Problem. An Example

Using the possibility of extending a symmetric operator to a selfadjoint one, which goes beyond the Hilbert space, we construct an example (in fact a class of them) of an indeterminate Hamburger moment sequence admitting a family of representing measures such that: the support of each of them is in arithmetic progression, the supports of all the measures together partition ℝ, none of them is N-extremal, and all of them are of infinite order. In fact, our goal is twofold: to present an example and to propose a method which may be useful for obtaining other results of this kind.