Fourier Analysis An Introduction

Table of Contents

• Foreword
• Preface
• Chapter 1. The Genesis of Fourier Analysis
  • 1 The vibrating string
    • 1.1 Derivation of the wave equation
    • 1.2 Solution to the wave equation
    • 1.3 Example: the plucked string
  • 2 The heat equation
    • 2.1 Derivation of the heat equation
    • 2.2 Steady-state heat equation in the disc
  • 3 Exercises
  • 4 Problem
• Chapter 2. Basic Properties of Fourier Series
  • 1 Examples and formulation of the problem
    • 1.1 Main definitions and some examples
  • 2 Uniqueness of Fourier series
  • 3 Convolutions
  • 4 Good kernels
  • 5 Cesàro and Abel summability: applications to Fourier series
    • 5.1 Cesàro means and summation
    • 5.2 Fejér’s theorem
    • 5.3 Abel means and summation
    • 5.4 The Poisson kernel and Dirichlet’s problem in the unit disc
  • 6 Exercises
  • 7 Problems
• Chapter 3. Convergence of Fourier Series
  • 1 Mean-square convergence of Fourier series
    • 1.1 Vector spaces and inner products
    • 1.2 Proof of mean-square convergence
  • 2 Return to pointwise convergence
    • 2.1 A local result
    • 2.2 A continuous function with diverging Fourier series
  • 3 Exercises
  • 4 Problems
• Chapter 4. Some Applications of Fourier Series
  • 1 The isoperimetric inequality
  • 2 Weyl’s equidistribution theorem
  • 3 A continuous but nowhere differentiable function
  • 4 The heat equation on the circle
  • 5 Exercises
  • 6 Problems
• Chapter 5. The Fourier Transform on ℝ
  • 1 Elementary theory of the Fourier transform
    • 1.1 Integration of functions on the real line
    • 1.2 Definition of the Fourier transform
    • 1.3 The Schwartz space
    • 1.4 The Fourier transform on
    • 1.5 The Fourier inversion
    • 1.6 The Plancherel formula
    • 1.7 Extension to functions of moderate decrease
    • 1.8 The Weierstrass approximation theorem
  • 2 Applications to some partial differential equations
    • 2.1 The time-dependent heat equation on the real line
    • 2.2 The steady-state heat equation in the upper half-plane
  • 3 The Poisson summation formula
    • 3.1 Theta and zeta functions
    • 3.2 Heat kernels
    • 3.3 Poisson kernels
  • 4 The Heisenberg uncertainty principle
  • 5 Exercises
  • 6 Problems
• Chapter 6. The Fourier Transform on ℝd
  • 1 Preliminaries
    • 1.1 Symmetries
    • 1.2 Integration on ℝd
  • 2 Elementary theory of the Fourier transform
  • 3 The wave equation in ℝd × ℝ
    • 3.1 Solution in terms of Fourier transforms
    • 3.2 The wave equation in ℝ3 × ℝ
    • 3.3 The wave equation in ℝ2 × ℝ: descent
  • 4 Radial symmetry and Bessel functions
  • 5 The Radon transform and some of its applications
    • 5.1 The X-ray transform in ℝ2
    • 5.2 The Radon transform in ℝ3
    • 5.3 A note about plane waves
  • 6 Exercises
  • 7 Problems
• Chapter 7. Finite Fourier Analysis
  • 1 Fourier analysis on ℤ(N)
    • 1.1 The group ℤ(N)
    • 1.2 Fourier inversion theorem and Plancherel identity on ℤ(N)
    • 1.3 The fast Fourier transform
  • 2 Fourier analysis on finite abelian groups
    • 2.1 Abelian groups
    • 2.2 Characters
    • 2.3 The orthogonality relations
    • 2.4 Characters as a total family
    • 2.5 Fourier inversion and Plancherel formula
  • 3 Exercises
  • 4 Problems
• Chapter 8. Dirichlet’s Theorem
  • 1 A little elementary number theory
    • 1.1 The fundamental theorem of arithmetic
    • 1.2 The infinitude of primes
  • 2 Dirichlet’s theorem
    • 2.1 Fourier analysis, Dirichlet characters, and reduction of the theorem
    • 2.2 Dirichlet L-functions
  • 3 Proof of the theorem
    • 3.1 Logarithms
    • 3.2 L-functions
    • 3.3 Non-vanishing of the L-function
  • 4 Exercises
  • 5 Problems
• Appendix: Integration
  • 1 Definition of the Riemann integral
    • 1.1 Basic properties
    • 1.2 Sets of measure zero and discontinuities of integrable functions
  • 2 Multiple integrals
    • 2.1 The Riemann integral in ℝd
    • 2.2 Repeated integrals
    • 2.3 The change of variables formula
    • 2.4 Spherical coordinates
  • 3 Improper integrals. Integration over ℝd
    • 3.1 Integration of functions of moderate decrease
    • 3.2 Repeated integrals
    • 3.3 Spherical coordinates
• Notes and References
• Bibliography
• Symbol Glossary
• Index