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 {M_n}. Let us assume that the (Hankel) condition for existence of a solution is satisfied, and let σ_n be the unique measure, supported on n points, whose first 2n moments agree with M₀,…,M_2n−1. It is known that σ_2n ⇒ σ₀ (weak convergence) and σ_2n+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. © 1999 John Wiley & Sons, Inc.