Multiscale decompositions of Hardy spaces

We would like to elaborate on a program of analysis pursued by Alex Grossmann and his collaborators on the analytic utilization of the phase of Hardy functions, as a multiscale signal processing tool. An inspiration at the origin of ”wavelet” analysis (when Grossmann, Morlet, Meyer and collaborators were interacting and exploring versions of multiscale representations) was provided, by the analysis of holomorphic signals, for which, the images of the phase of Cauchy wavelets were remarkable in their ability to reveal intricate singularities or dynamic structures, such as instantaneous frequency jumps, in musical recordings. This work which was pursued by Grossmann, Kronland Martinet et al [11] exploiting phase and amplitude variability of holomorphic signals was challenged by computational complexity as well as by the lack of simple, efficient, mathematical processing, and generalizations to higher dimensional signals. It was mostly bypassed by the orthogonal wavelet transforms. We aim to show that these ideas are powerful nonlinear subtle tools. Our goal here is to follow their seminal work and introduce recent developments in nonlinear analysis. In particular we will sketch methods extending conventional Fourier analysis, exploiting both phase and amplitudes of holomorphic functions.