Nonlinear Fourier series via Blaschke products

Classical Fourier series may be interpreted as repeatedly adding and removing roots in the origin of the complex plane. A natural modification, proposed by Coifman in the 1990s, is to remove all roots inside the unit disk - this can be done without having to find the roots and gives rise to a nonlinear form of Fourier series with many curious properties (among them: fast convergence). We explain convergence in $L^2$, convergence control quantitative control in Sobolev spaces and describe some of the nonlinear processes and applications to real-life data.

This is joint work with Raphy Coifman (Yale) and Hau-tieng Wu (Toronto).