Krein's strings,the symmetric moment problem,and extending a real positive definite function |
| |
Authors: | Uri Keich |
| |
Abstract: | 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. © 1999 John Wiley & Sons, Inc. |
| |
Keywords: | |
|
|