Spirals in Hilbert space: With an application in information theory

A (logarithmic) spiral of order is defined as a continuous path t x(t) in a real Hilbert space such that For α=0 the spiral becomes a helix. The elegant proof by P. Masani of the spectral characterization of helices, due to Kolmogorov and to von Neumann and Schoenberg, is adapted here to spirals. As an application a conjecture by F. Topsøe that certain kernels on considered in information theory are negative definite, and hence are squares of metrics on , is confirmed.