Extension of the Schoenberg theorem to integrally conditionally positive definite functions.

Abstract The celebrated Schoenberg theorem establishes a relation between positive definite and conditionally positive definite functions. In this paper, we consider the classes of real-valued functions P(J) and CP(J), which are positive definite and respectively, conditionally positive definite, with respect to a given class of test functions J. For suitably chosen J , the classes P(J) and CP(J) contain classically positive definite (respectively, conditionally positive definite) functions, as well as functions which are singular at the origin. The main result of the paper is a generalization of Schoenberg's theorem to such function classes. [ABSTRACT FROM AUTHOR]