Multiscale decompositions of Hardy spaces.

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. Our goal is to follow their seminal work and introduce recent developments in nonlinear analysis. In particular we sketch methods extending conventional Fourier analysis, exploiting both phase and amplitudes of holomorphic functions. The Blaschke factors are a key ingredient, in building analytic tools, starting with the Malmquist Takenaka orthonormal bases of the Hardy space, continuing with "best" adapted bases obtained through phase unwinding, and concluding with relations to composition of Blaschke products and their dynamics. We also remark that the phase of a Blaschke product is a one layer neural net with arctan as an activation sigmoid and that the composition is a "Deep Neural Net" whose depth is the number of compositions. Our results provide a wealth of related library of orthonormal bases.