  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.
