We show that, for positive definite kernels, if specific forms of regularity (continuity, -differentiability or holomorphy) hold locally on the diagonal, then they must hold globally on the whole domain of positive-definiteness. This local-to-global propagation of regularity is constructively shown to be a consequence of the algebraic structure induced by the non-negativity of the associated bilinear forms up to order~5. Consequences of these results for topological groups and for positive definite and exponentially convex functions are explored.
Cite this article
Jorge Buescu, António Paixão, Claudemir Oliveira, Propagation of Regularity and Positive Definiteness: a Constructive Approach. Z. Anal. Anwend. 37 (2018), no. 1, pp. 1–24DOI 10.4171/ZAA/1599