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Positive definite functions on semilattices (1804.03047v2)
Published 9 Apr 2018 in math.NT and math.CO
Abstract: We introduce a notion of positive definiteness for functions $f!:P\to\mathbb{R}$ defined on meet semilattices $(P,\preceq,\wedge)$ and prove several properties for these functions. In addition, we utilize the $LDL{\rm T}$ 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.