A friendly introduction to Fourier analysis on polytopes

TL;DR

This book introduces Fourier analysis on polytopes, covering foundational concepts and diverse applications such as tilings, volume computation, sphere packings, and sampling, aimed at students and professionals.

Contribution

It provides a comprehensive, accessible introduction to Fourier analysis on polytopes, including new formulations and applications in geometry and number theory.

Findings

Formulations for Fourier transform of polytopes

Applications to sphere packings and tilings

Methods for computing discrete volumes

Abstract

This book is an introduction to the nascent field of Fourier analysis on polytopes, and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field. Of the many applications of these techniques, we have chosen to focus on the following topics: (a) Formulations for the Fourier transform of a polytope (b) Minkowski and Siegel's theorems in the geometry of numbers (c) Tilings and multi-tilings of Euclidean space by translations of a polytope (d) Computing discrete volumes of polytopes, which are combinatorial approximations to the continuous volume (e) Sphere packings, and their packing density (f) Iterating the divergence theorem to give new formulations for the Fourier transform of a polytope, with applications (g) Shannon sampling, in several variables (h) More topics in the classical geometry of numbers We assume familiarity with Linear Algebra, with some Calculus and infinite series. Throughout, we introduce the topics gently, by giving many examples and exercises, so that this book is ideally suited for a course, or for self-study.