A friendly invitation to Fourier analysis on polytopes
arXiv (Cornell University)(2021)
摘要
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. We assume a familiarity with Linear Algebra, and
some Calculus. Of the many applications, we have chosen to focus on: (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) Optimizing sphere packings and their densities, and (f) use iterations of
the divergence theorem to give new formulations for the Fourier transform of a
polytope, with an application. Throughout, we give many examples and exercises,
so that this book is also appropriate for a course, or for self-study.
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关键词
polytopes,fourier analysis
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