On positive definite distributions with compact support (Q2341447)

The aim of the paper is to give a characterisation via the Cauchy transform when a compactly supported distribution is positive definite (recall that \(f\in\mathcal{D}'(\mathbb{R}^n)\) is said to be positive definite if \(\langle f,\varphi*\tilde{\varphi}\rangle\geq 0\), \(\forall \varphi\in\mathcal{D}(\mathbb{R}^n)\), where \(\tilde{\varphi}(x)=\overline{\varphi(-x)}\)). Given a smooth function \(\theta:\Gamma\rightarrow \mathbb{C}\), where \(\Gamma\subseteq \mathbb{R}^n\) is a regular cone, the author calls \(\theta\) completely monotonic when \((-1)^kD_{\gamma_1}\ldots D_{\gamma_k}\theta(y)\geq 0\), for all \(y\in\Gamma\), \(\gamma_1,\ldots,\gamma_k\in\Gamma\), \(k\in\mathbb{N}_0\), where \(D_{\gamma_j}\theta\) stands for the directional derivative of \(\theta\) along \(\gamma_j\in\Gamma\). If \(\{\Gamma_k\}_{k=1}^m\) is a set of regular cones such that \(\overline{\bigcup_k\Gamma_k}=\mathbb{R}^n\) and the measure of \(\overline{\Gamma_k}\cap\overline{\Gamma_j}\) is zero when \(j\neq k\), the main result of the article is that a compactly supported distribution \(f\) is positive definite if and only if the functions \(y\mapsto K_{\Gamma_k}(f)(iy)\), \(\Gamma_k\rightarrow \mathbb{C}\), are completely monotonic for each \(k=1,\dots,m\) (\(K_{\Gamma_k}(f)\) is the Cauchy transform of \(f\)).