On positive and conditionally negative definite functions with a singularity at zero, and their applications in potential theory

It is widely known that positive and conditionally negative definite functions take finite values at the origin. Nevertheless, there exist functions with a singularity at zero, arising naturally e.g.\ in potential theory or the study of (continuous) extremal measures, which still exhibit the general characteristics of positive or conditional negative definiteness. Taking a framework set up by Lionel Cooper as a motivation, we study the general properties of functions which are positive definite in an extended sense. We prove a Bochner-type theorem and, as a consequence, show how unbounded positive definite functions arise as limits of classical positive definite functions, as well as that their space is closed under convolution. Moreover, we provide criteria for a function to be positive definite in the extended sense, showing in particular that complete monotonicity in conjunction with local absolute integrability is sufficient. The celebrated Schoenberg theorem establishes a relation between positive definite and conditionally negative definite functions. By introducing a notion of conditional negative definiteness which accounts for the classical, non-singular conditionally negative definite functions, as well as functions which are unbounded at the origin, we extend this result to real-valued functions with a singularity at zero. Moreover, we demonstrate how singular conditionally negative definite functions arise as limits of classical conditionally negative definite functions and provide several examples of functions which are unbounded at the origin and conditionally negative definite in an extended sense. Finally, we study the convexity and minimisation of the energy associated with various singular, completely monotone functions, which have not previously been considered in the field of potential theory or experimental design and solve the corresponding energy problems by means of numerically computing approximations to the (optimal) minimising measures