AlgHOF · Algorithmic higher-order Fourier analysis

The main objective of this action is to develop and implement constructive tools in higher-order Fourier analysis and apply them in data analysis and artificial intelligence. Fourier analysis is an extremely powerful tool to analyze functions defined on compact abelian groups. During the past decades, advances in additive combinatorics and ergodic theory have led to the discovery of a new form of representation theory on compact abelian groups that generalizes Fourier analysis. This theory is known as higher-order Fourier analysis. Roughly speaking, while Fourier analysis deals with representing functions in terms of harmonics such as exp(2*pi*i*t*x), Higher Order Fourier analysis deals with representing functions in terms of higher order harmonics such as exp(2*pi*i*t*x^2). This theory has found applications in many areas of pure mathematics but not so many in applied mathematics due to the lack of an analogue of the Fourier Transform. The objectives of this action are thus:1) Develop an algorithm that decomposes a function in terms of higher-order harmonics. This decomposition will have a notion of approximate uniqueness analogous to that of classical Fourier analysis.2) Implement such an algorithm in Python and test it in data analysis and artificial intelligence problems where Fourier analysis plays a prominent role, such as time series continuation, signal compression, and machine learning.