Krein’s Strings, the Symmetric Moment Problem, and Extending a Real Positive Definite Function

The symmetric moment problem is to find a possibly unique, positive symmetric measure that will produce a given sequence of moments {Mn}. Let us assume that the (Hankel) condition for existence of a solution is satisfied, and let σn be the unique measure, supported on n points, whose first 2n moments agree with M0, . . . ,M2n−1. It is known that σ2n =⇒ σ0 (weak convergence) and σ2n+1 =⇒ σ∞, where σ0 and σ∞ are solutions to the full moment problem. Moreover, σ0 = σ∞ if and only if the problem has a unique solution. In this paper we present an analogue of this theorem for Krein’s problem of extending to R a real, even positive definite function originally defined on [−T,T ] where T < ∞. Our proof relies on the machinery of Krein’s strings. As we show, these strings help explain the connection between the moment and the extension problems. c © 1999 John Wiley & Sons, Inc.