On the smoothness of positive definite and radial functions

We consider positive definite and radial functions. After giving general results concerning the smoothness of general positive definite and radial functions, we investigate the class of compactly supported, positive definite, and radial functions, where every function consists of a univariate polynomial within its support. Especially, we show that these functions necessarily possess an even number of continuous derivatives. Finally, we provide a general construction technique which we use to construct a new family of compactly supported basis functions of arbitrary smoothness.