1999/06/18 by Alexandr Borisov, Borisov, Alexandr · 3 citations
Mathematics · Medicine · #Abelian group #FOS: Mathematics #Functional Analysis (math.FA) #Locally compact space #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Mathematics #Medicine #Physics #Positive-definite matrix #Pure mathematics #advanced mathematical theories #math.FA
paper · pdf · doi:10.48550/arxiv.math/9906126
published in arXiv (Cornell University) (Cornell University) · 12 pages
arxiv created 1999/06/18 · openalex publication_date 1999/06/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The author was recently able to provide a cohomological interpretation of Tate's Riemann-Roch formula for number fields using some new harmonic analysis objects, ghost-spaces. When trying to investigate these objects in general, we realized the importance of functions and measures on locally compact abelian groups that are both positive and positive-definite at the same time. It looks like this class of functions and measures was not systematically studied before. The goal of this paper is to partially fill in this gap. We answer some of the natural questions involving these functions and measures, especially those that satisfy some extra integrability conditions. We also study some operations and constructions involving these functions and measures. There are several very interesting open questions, that we are only able to point out at this moment. In particular, the structure of the cone of such functions is not clear even when the group is just \R.