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Krein's strings, the symmetric moment problem, and extending a real positive definite function
Author(s) -
Keich Uri
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199910)52:10<1315::aid-cpa7>3.0.co;2-o
Subject(s) - mathematics , moment problem , moment (physics) , positive definite matrix , measure (data warehouse) , sequence (biology) , function (biology) , extension (predicate logic) , combinatorics , convergence (economics) , connection (principal bundle) , symmetric function , mathematical analysis , pure mathematics , geometry , quantum mechanics , physics , statistics , eigenvalues and eigenvectors , database , evolutionary biology , biology , computer science , economics , principle of maximum entropy , genetics , programming language , economic growth
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 2 n moments agree with M 0 ,…, M 2 n −1 . It is known that σ 2 n ⇒ σ 0 (weak convergence) and σ 2 n +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. © 1999 John Wiley & Sons, Inc.