Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2018-01-08
Propagation of Regularity and Positive Definiteness: a Constructive Approach
Jorge Buescu[1], António Paixão[2] and Claudemir Oliveira[3] (1) Universidade de Lisboa, Portugal(2) Instituto Superior de Engenharia de Lisboa, Portugal
(3) Universidade Federal de Itajubá, Brazil
We show that, for positive definite kernels, if specific forms of regularity (continuity, $\mathcal{S}_n$-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.
Keywords: Positive definite kernels, positive definite functions, differentiability, holomorphy, constructive approximation, exponentially convex functions
Buescu Jorge, Paixão António, Oliveira Claudemir: Propagation of Regularity and Positive Definiteness: a Constructive Approach. Z. Anal. Anwend. 37 (2018), 1-24. doi: 10.4171/ZAA/1599