Universal Factorizations of Quasiperiodic Functions

TL;DR

This paper introduces a universal factorization framework for quasiperiodic functions, allowing classification based on their phase space and phase function, with implications for analyzing complex oscillatory signals.

Contribution

It establishes the existence and uniqueness of a universal factorization for quasiperiodic functions under certain conditions, extending the understanding of their structure.

Findings

Universal factorization of quasiperiodic functions is unique.

Quasiperiodic functions can be classified by their phase space and phase mapping.

The framework generalizes to replacing the circle with other phase spaces.

Abstract

Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circle with other phase spaces to obtain a general quasiperiodic function. This paper shows that under appropriate restrictions, each quasiperiodic function has a unique universal factorization. Quasiperiodic functions can therefore be classified based on their phase space and the phase function mapping into it.