Estimates on the derivatives and analyticity of positive definite functions on Rm

In this paper we obtain some results about derivatives of positive definite functions in a"e (m) , using known properties of positive definite kernels. We prove, by purely algebraic methods, that certain derivatives of such functions are also positive definite and we show that simple conditions on their even order derivatives at the origin strongly determine their global properties. In particular, one can obtain an estimate for f and its derivatives at any point and a condition for real analyticity, using only the value of these derivatives at the origin.