António Paixão | Searcharxiv

arXiv2018

Propagation of regularity and positive definiteness: a constructive approach

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.

math.CV↗

arXiv2018

Positive-definiteness and integral representations for special functions

We characterize a holomorphic positive definite function $f$ defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure on $\mathbb{R}$. The classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become special cases of this characterization: they are respectively the real and pure imaginary sections of the complex integral representation. We apply this representation to special cases, including the $Γ$, $ζ$ and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions: in the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.

math.CV↗