Author's Preface
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Preface by the editor

introduction

Chapter 1: The Origins of Fourier Analysis
1 string vibration
1.1 Derivation of the wave equation
1.2 Solution to the wave equation
1.3 Example: Tungin string
2-column equation
2.1 Derivation of heat equation
2.2 Steady-state heat equation in a disk
Practice problems
Advanced problems

Chapter 2 Basic Properties of Fourier Series
1. Clarification of the Example and Problem
1.1 Key definitions and some examples
2 Uniqueness of Fourier series
3 Convolution
4 good nuclei
5 Chesaro summation and abelian summation possibilities: application to Fourier series
5.1 Average and sum by Chesaro
5.2 Peser's theorem
5.3 Abelian Mean and Abelian Sum
5.4 Dirichlet problem on Poisson core and unit disk
Practice problems
Advanced problems

Chapter 3 Convergence of Fourier Series
1 Mean square convergence of Fourier series
1.1 Vector space and inner product
1.2 Proof of mean square convergence
Back to point-by-point convergence
2.1 Local Results
2.2 Continuous functions with diverging Fourier series
Practice problems
Advanced problems

Chapter 4 Applications of Fourier Series
1. Rankine Inequality
2 Weyl's uniform distribution theorem
3 A function that is continuous but not differentiable at all points
Heat equation on a circle
Practice problems
Advanced problems

Chapter 5 Fourier Transform in R
1 Basic theory of Fourier transform
1.1 Integral of functions defined in real numbers
1.2 Definition of Fourier Transform
1.3 Schwarz space
Fourier transform at 1.4 S
1.5 Inverse Fourier Transform
1.6 Francherel formula
1.7 Extension to a suitably decreasing function
1.8 Weierstrass approximation theorem
2 Applications in partial differential equations
2.1 Time-dependent heat equation in real numbers
2.2 Steady-state heat equation in the upper half-plane
3 Poisson sum formula
3.1 Theta and Zeta Functions
3.2 Thermonuclear
3.3 Poisson kernel
4 Heisenberg uncertainty principle
Practice problems
Advanced problems

Chapter 6 Fourier Transform in R^d
1 Basic knowledge
1.1 Symmetry
1.2 Integration in R^d
2 Basic theory of Fourier transform
Wave equation in 3 R^d×R
3.1 Solution from the Fourier transform perspective
3.2 Wave equation in R^3×R
3.3 Wave equation in R^2×R: descent method
4 Radial symmetry and Bessel functions
5 Radon Transform and Some Applications
5.1 X-ray transformation in R^2
5.2 Radon Transform in R^3
5.3 Note on Plane Waves
Practice problems
Advanced problems

Chapter 7 Finite Fourier Analysis
Fourier analysis in 1 Z(N)
1.1 Group Z(N)
1.2 Inverse Fourier transform theorem and Francherel identity in Z(N)
1.3 Fast Fourier Transform
2 Fourier analysis in finite Abelian groups
2.1 Abelian group
2.2 Indicators
2.3 Orthogonal relationship
2.4 Indicators as complete sets
2.5 Inverse Fourier transform formula and Francherel formula
Practice problems
Advanced problems

Chapter 8 Dirichlet's Theorem
1 Basic Number Theory
1.1 Fundamental Theorems of Arithmetic
1.2 Infinity of prime numbers
2 Dirichlet's theorem
2.1 Fourier analysis, Dirichlet indicators, and reduction of the theorem
2.2 Dirichlet L-function
Proof of the 3rd theorem
3.1 Logarithmic function
3.2 L-function
3.3 L-functions and nontrivial Dirichlet indicators
Practice problems
Advanced problems

Appendix Integration
1 Riemann integral
1.1 Basic properties
1.2 Sets with zero measure and discontinuities of integrable functions
2 Double integrals
2.1 Riemann integral in R^d
2.2 Iterated integration
2.3 Substitution integration method
2.4 Spherical coordinates
Ideal integral in 3 R^d
3.1 Integral of a suitably decreasing function
3.2 Iterated integration
3.3 Spherical coordinates

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