STRICT DEFINITENESS OF INTEGRALS VIA COMPLETE MONOTONICITY OF DERIVATIVES

Let k be a nonnegative integer and let phi : (0,infinity) --> R be a C -infinity function with (-)(k).phi((k)) completely monotone and not co nstant. If sigma not equal 0 is a signed measure on any euclidean spac e R-d, with vanishing moments up to order k(-1), then the integral int egral(Rd) integral(Rd) phi(parallel to x-y parallel to(2)) d sigma(x)d sigma(y) is strictly positive whenever it exists. For general d no la rger class of continuous functions phi seems to admit the same conclus ion. Examples and applications are indicated. A section on ''bilinear integrability'' might be of independent interest.