Propagation of regularity and positive definiteness: a constructive approach

TL;DR

This paper demonstrates that local regularity properties of positive definite kernels, such as continuity and differentiability, propagate globally due to their algebraic structure, with implications for topological groups and convex functions.

Contribution

It provides a constructive proof that local regularity implies global regularity for positive definite kernels based on their algebraic structure.

Findings

Local regularity propagates globally for positive definite kernels.

Constructive method relies on algebraic structure up to order 5.

Applications to topological groups and convex functions.

Abstract

We show that, for positive definite kernels, if specific forms of regularity (continuity, Sn-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.