Fourier Analysis on Polytopes and the Geometry of Numbers - Part I: A Friendly Introduction

Copyright
Contents
Acknowledgments
Preface
Introduction
	1. Initial ideas
	2. The Poisson summation formula
Chapter 1. Motivational problem: Tiling a rectangle with rectangles
	1. Intuition
	2. Nice rectangles
	3. Conventions and some definitions
	Notes
	Exercises
Chapter 2. Examples nourish the theory
	1. Intuition
	2. Dimension 1—the classical sinc function
	3. The Fourier transform of ???? as a complete invariant
	4. Bernoulli polynomials
	5. The cube and its Fourier transform
	6. The standard simplex and its Fourier transform
	7. Convex sets and polytopes
	8. Any triangle and its Fourier transform
	9. Stretching and translating
	10. The parallelepiped and its Fourier transform
	11. The cross-polytope
	12. Observations and questions
	Notes
	Exercises
Chapter 3. The basics of Fourier analysis
	1. Intuition
	2. Introducing the Fourier transform on ????¹(ℝ^{????})
	3. The triangle inequality for integrals
	4. The Riemann–Lebesgue lemma
	5. The inverse Fourier transform
	6. The torus ℝ^{????}/ℤ^{????}
	7. Piecewise smooth functions have convergent Fourier series
	8. As ???? gets smoother, ????̂ decays faster
	9. How fast do Fourier coefficients decay?
	10. The Schwartz space
	11. Poisson summation I
	12. Useful convergence lemmas in preparation for Poisson summation II
	13. Poisson summation II: Á la Poisson
	14. An initial taste of general lattices in anticipation of Chapter 5
	15. Poisson summation III: For general lattices
	16. The convolution operation
	17. More relations between ????¹(ℝ^{????}) and ????²(ℝ^{????})
	18. The Dirichlet kernel
	19. The extension of the Fourier transform to ????²: Plancherel
	20. Approximate identities
	21. Poisson summation IV: A practical Poisson summation formula
	22. The Fourier transform of the ball
	23. Uncertainty principles
	Notes
	Exercises
Chapter 4. Geometry of numbers Part I: Minkowski meets Siegel
	1. Intuition
	2. Minkowski’s first convex body theorem
	3. A Fourier transform identity for convex bodies
	4. Tiling and multi-tiling Euclidean space by translations of polytopes
	5. Extremal bodies
	6. Zonotopes and centrally symmetric polytopes
	7. Sums of two squares via Minkowski’s theorem
	8. The volume of the ball and the sphere
	9. Classical geometric inequalities
	10. Minkowski’s theorems on linear forms
	11. Poisson summation as the trace of a compact linear operator
	Notes
	Exercises
Chapter 5. An introduction to Euclidean lattices
	1. Intuition
	2. Introduction to lattices
	3. Sublattices
	4. Discrete subgroups: An alternate definition of a lattice
	5. Lattices defined by congruences
	6. The Gram matrix
	7. Dual lattices
	8. Some important lattices
	9. The Hermite normal form
	10. The Voronoi cell of a lattice
	11. Characters of lattices
	Notes
	Exercises
Chapter 6. Geometry of numbers Part II: Blichfeldt’s theorems
	1. Intuition
	2. Blichfeldt’s theorem
	3. Van der Corput’s inequality for convex bodies
	Notes
	Exercises
Chapter 7. The Fourier transform of a polytope via its vertex description: Brion’s theorem
	1. Intuition
	2. Cones, simple polytopes, and simplicial polytopes
	3. Tangent cones
	4. The Brianchon–Gram identity
	5. Brion’s formula for the Fourier transform of a simple polytope
	6. Proof of Theorem 7.12, the Fourier transform of a simple polytope
	7. The Fourier transform of any real polytope
	8. Fourier–Laplace transforms of cones
	9. The Fourier transform of a polygon
	10. Each polytope has its moments
	11. The zero set of the Fourier transform
	Notes
	Exercises
Chapter 8. What is an angle in higher dimensions?
	1. Intuition
	2. Defining an angle in higher dimensions
	3. Local solid angles for a polytope and Gaussian smoothing
	4. 1-dimensional polytopes: Their angle polynomial
	5. Pick’s formula and Nosarzewska’s inequality
	6. The Gram relations for solid angles
	7. Bounds for solid angles
	8. The classical Euler–Maclaurin summation formula
	9. Further topics
	Notes
	Exercises
Appendix A. Solutions and hints to selected problems
	Solutions to Chapter 1
	Solutions to Chapter 2
	Solutions to Chapter 3
	Solutions to Chapter 4
	Solutions to Chapter 5
	Solutions to Chapter 6
	Solutions to Chapter 7
Appendix B. The dominated convergence theorem and other goodies
	1. The dominated convergence theorem
	2. Big-O and little-o
	3. Various forms of convergence
Credits for photographs and pictures
Bibliography
Index