Schoenberg’s Theory of Totally Positive Functions and the Riemann Zeta Function

Abstract

This chapter is a short survey of some aspects and applications of Schoenberg’s work on totally positive functions. Totally positive functions arise prominently in sampling theory in connection with optimal results. The main purpose is to advertise the connection between totally positive functions and the Riemann hypothesis. In particular, the Riemann hypothesis is equivalent to the total positivity of a transform of the Riemann zeta function.