In this note we are concerned with several questions on positive definite functions over a Hausdorff locally compact group. The main result, Theorem A, gives some necessary and sufficient conditions for to be a positive definite function when μ is a (complex Radon) measure. In particular, is a positive definite function if and only if μ ∊ L2, and Theorem B then follows by giving a complete characterization of functions of the type , where f ∊ L2. Perhaps the most interesting aspect of these results is that they provide further examples of results over a non-abelian, non-compact group, which otherwise are simple consequences (with μ, a bounded measure in Theorem A) of the theorems of Plancherel and Bochner.

Unless otherwise specified, all notation and definitions will follow [1;2]. The underlying group will always be G, a Hausdorff locally compact group with identity e, and with left Haar measure dx.