Schoenberg-Type Results for Positive and Negative Definite Functions

Abstract

We have already mentioned the three fundamental papers of Schoenberg (1938a, b, 1942), all of which are very closely related to positive and negative definite kernels. The main purpose in (1938b) was to show the close connection between (real-valued) negative definite kernels and Hilbert metrics, see Chapter 3, §3. In his first mentioned paper (1938a), entitled “Metric spaces and completely monotone functions”, Schoenberg raises the question about the connection between the class of Fourier transforms of (finite, nonnegative) measures in euclidean spaces and the class of Laplace transforms of (finite, nonnegative) measures on the half-line ℝ₊.