A Wiener–Wintner theorem for 1/ f power spectra

Abstract

Wiener's generalized harmonic analysis (GHA) provides a theory of harmonic analysis for subspaces of tempered functions not accessible to the L 1, L 2, and Fourier series theories; and it does it in a way that is usually more quantitative than that provided by the theory of distributions. On the other hand, GHA does not yield an adequate spectral analysis of large classes of functions, including nonstationary processes and, in particular, 1/ f noise. In this paper we adapt GHA to deal with 1/ f noise by extending the Wiener–Wintner theorem to the case of 1/ f power spectra.