HoFa documentation

Version: 0.1.0

Useful links: Installation, Source repository

HoFa is a Python library for Higher-order Fourier analysis.

Higher-order Fourier analysis is a theory that generalizes Fourier analysis. Roughly speaking, while Fourier analysis deals with representing functions in terms of Fourier characters, e.g. functions of the form \(\exp(2\pi i \xi\cdot x)\), higher-order Fourier analysis deals with representing functions in terms of higher-order Fourier characters, including functions of the form \(\exp(2\pi i P(x))\) for polynomials \(P\). Such higher-order representations can capture more subtle structural aspects of the function that can often be missed by looking only at the function’s dominant Fourier characters. These insights have already had major impact in pure mathematics, and the present project aims to facilitate exploration and application of these insights in more applied settings.

Note

This project is under active development.

Getting started

Here you will find an accessible introduction to higher-order Fourier analysis. You will also learn the basics of how to use this theory for applications via the HoFa package.

To the starting guide

User guide

This guide delivers comprehensive coverage of the core concepts of HoFa, accompanied by valuable context and detailed explanations. This section assumes concepts from the starting guide.

To the user guide

API reference

A detailed description of all methods and classes in HoFa. This section assumes familiarity with the key concepts and ideas of the starting guide and the user guide.

To the reference guide

Developer’s guide

Help us develop HoFa!

To the developer’s guide

HoFa is developed in GitHub but the official archival version (DOI) is hosted in the CSIC Institutional Repository.

About the original authors

This package was originally developed by the following team:

Pablo Candela	Diego González Sánchez	Balázs Szegedy
Affiliation: Instituto de Ciencias Matemáticas (ICMAT), CSIC, Madrid, Spain Website	Affiliation: Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France Website	Affiliation: HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary Website

Funding

This work was supported by funding from project PID2024-156180NB-I00 (MICIU/AEI and the European Union). The second-named author is funded by HORIZON-MSCA-2024-PF-01, AlgHOF 101202161 funded by the European Union. The third-named author was partially supported by the Hungarian Ministry of Innovation and Technology NRDI Office within the framework of the Artificial Intelligence National Laboratory Program (MILAB, RRF-2.3.1-21-2022-00004).