Positive definite functions on semilattices

Abstract

We introduce a notion of positive definiteness for functions $f:P\to \mathbb{R}$ defined on meet semilattices $(P,⪯,\wedge )$ and prove several properties for these functions. In addition, we utilize the $LD{L}^{\mathrm{T}}$ decomposition of meet matrices in order to explore the properties of multivariate positive definite arithmetic functions $f:{\mathbb{Z}}_{+}^d\to \mathbb{R}$. Finally, we give a series of examples and counterexamples of positive definite functions.

12 pages, 1 figure