Stefan Steinerberger, Yale University, New Haven, CT , Nonlinear phase unwinding of functions
Sprache des Titels:
A method proposed almost 20 years ago analyzes an analytic signal by applying iterative Blaschke factorization and writing a function $F$ as
$$F = a_0 + a_1B_1 + a_2B_1B_2 + a_3 B_1B_2B_3 + \dots,$$
where the $a_i$ are numbers and the $B_i$ are Blaschke products. This can be understood as a nonlinear generalization of Fourier series. The method is numerically feasible (Guido and Mary Weiss, 1963), has been demonstrated to work extremely well in practice (Michel Nahon, PhD thesis Yale 2000), investigated with respect to stability (Letelier \& Saito) and shown to be highly effective in measuring phenomena related to the Doppler effect (Healy). It has recently been independently rediscovered and exhaustively used by (Qeng Tao and collaborators, 10+ papers, 2010-2015). We give the first rigorous mathematical analysis of the method, prove convergence of the formal series and discuss some curious arising phenomena. This is joint work with Raphy Coifman.