New criteria on positive-definite distributions

We establish several sufficient conditions under which a locally integrable function $f:{\mathbb{R}}^n\to \mathbb{R}$ represents a positive-definite distribution. In particular we consider functions of the form $f(\mathrm{\parallel }x\mathrm{\parallel })$ where $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is a fixed norm in ${\mathbb{R}}^n$.