Positive Positive-definite Functions and Measures on Locally Compact Abelian Groups

In the paper [1] we gave a cohomological interpretation of Tate’s Riemann-Roch formula 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. Please refer to section 5 where this and other open problems are discussed. The paper is organized as follows. In section 2 the positive positivedefinite functions and measures are defined, some of their properties are discussed, and some simple constructions involving them are carried out. In sections 3 and 4 we restrict our attention to such functions and measures that satisfy some extra integrability conditions. Finally, in section 5 we point out some natural open questions that we were unable to answer. Acknowledgments. The author thanks Jeff Lagarias, Barry Mazur and Christopher Deninger whose interest in [1] motivated the author to continue his work in this direction. The author also thanks Joaquim Ortega-Cerda for the references to Hardy’s theorem and its generalizations.