Spirals in Hilbert space: With an application in information theory

A (logarithmic) spiral of order α∈R is defined as a continuous path t⟼x(t) in a real Hilbert space such that∥x(t1+t)-x(t2+t)∥=eαt∥x(t1)-x(t2)∥,t,t1,t2∈R.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 R+ considered in information theory are negative definite, and hence are squares of metrics on R+, is confirmed.