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Zeitschrift für Analysis und ihre Anwendungen

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Volume 37, Issue 1, 2018, pp. 1–24
DOI: 10.4171/ZAA/1599

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