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On unbounded positive definite functions

Phillips, Tomos and Schmidt, Karl Michael 2018. On unbounded positive definite functions. Mathematica Pannonica 26 (2) , pp. 33-51.

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It is well known that positive definite functions are bounded, taking their maximum absolute value at 0. Nevertheless, there are unbounded functions, arising e.g. in potential theory or the study of (continuous) extremal measures, which still exhibit the general characteristics of positive definiteness. Taking a framework set up by Lionel Cooper as a motivation, we study the general properties of such functions which are positive definite in an extended sense. We prove a Bochner-type theorem and, as a consequence, show how unbounded positive definite functions arise as limits of classical positive definite functions, as well as that their space is closed under convolution. Moreover, we provide criteria for a function to be positive definite in the extended sense, showing in particular that complete monotonicity in conjunction with absolute integrability is sufficient.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Editorial Board of Mathematica Pannonica
ISSN: 0865-2090
Date of First Compliant Deposit: 5 October 2018
Date of Acceptance: 4 October 2018
Last Modified: 12 Mar 2020 02:24

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