Krein's strings, the symmetric moment problem, and extending a real positive definite function
✍ Scribed by Uri Keich
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 251 KB
- Volume
- 52
- Category
- Article
- ISSN
- 0010-3640
No coin nor oath required. For personal study only.
✦ Synopsis
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 M 0 ,...,M 2n-1 . It is known that σ 2n =⇒ σ 0 (weak convergence) and σ 2n+1 =⇒ σ ∞ , where σ 0 and σ ∞ are solutions to the full moment problem. Moreover, σ 0 = 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.