Fourier analysis on abelian groups; theory and applications

Abstract

Fourier analysis expresses a function as a weighted sum of complex exponentials. The Fourier machinery can be applied when a function is defined on a locally compact abelian group (LCA). The groups R, T = R / Z, Z and Z_n are all LCAs of great interest, but numerical computations are almost always done on the finite group Z_n using the Fast Fourier Transform. To reduce a general problem to a numerical computation, sampling and periodization is necessary. In this thesis we present a new software library which facilitates Fourier analysis on elementary LCAs. The software allows the user to work directly with abstract mathematical objects, perform numerical computations and handle the relationship between discrete and continuous domains in a natural way. The specific combination of mathematical objects and operations available in the software developed is to our knowledge not found elsewhere. Efforts have been made to efficiently open-source, document and distribute the software library, which is now available to every user of the Python programming language.