Strict definiteness of integrals via complete monotonicity of derivatives

Abstract:

Let $k$ be a nonnegative integer and let $\phi : (0,\infty ) \rightarrow \mathbb {R}$ be a $C^\infty$ function with $(-)^k\cdot \phi ^{(k)}$ completely monotone and not constant. If $\sigma \neq 0$ is a signed measure on any euclidean space $\mathbb {R}^d$, with vanishing moments up to order $k-1$, then the integral $\int _{\mathbb {R}^d} \int _{\mathbb {R}^d} \phi ( \|x-y\|^2 ) d\sigma (x) d\sigma (y)$ is strictly positive whenever it exists. For general $d$ no larger class of continuous functions $\phi$ seems to admit the same conclusion. Examples and applications are indicated. A section on ”bilinear integrability” might be of independent interest.