Fourier Analysis

The field that studies subjects of interest using Fourier series and Fourier transforms is called Fourier analysis^{Fourier analysis} or harmonic analysis^{harmonic analysis}.

Fourier Series

• The set of trigonometric functions is orthogonal
• The sum of mutually perpendicular trigonometric functions
• The set of exponential functions, trigonometric functions are an orthonormal basis of $L^{2}$
• Derivation of Fourier series
• Dirichlet kernel
• Complex Fourier series
• Fourier coefficients of derivatives
• Definite integral of Fourier series
• Fourier cosine series and sine series, Fourier coefficients of even and odd functions
• Fourier coefficients of half-wave symmetric functions
• The constant term of Fourier series is equal to the average of one period of the function
• Bessel’s inequality for Fourier series

Convergence

• Fourier series of Riemann-integrable functions converges
• Convergence of Fourier series at discontinuities
• The limit of Fourier coefficients is $0$
• Sufficient condition for the absolute and uniform convergence of a function’s Fourier series

Fourier Transform

• Derivation of Fourier transform
  • Various definitions and notations of Fourier transform
  • Various interpretations of Fourier transform
• Properties of Fourier transform
• Riemann-Lebesgue lemma
• Fourier inversion theorem
  • Fourier inversion theorem for smooth functions
• Plancherel’s theorem

Fourier Transform of Various Functions

• Fourier transform of Gaussian functions
• Fourier transform of characteristic functions
• Fourier transform of Dirac delta function

Discrete Fourier Transform

• Discrete Fourier Transform^DFT
• Properties of Discrete Fourier Transform
• Inverse Discrete Fourier Transform
• Fast Fourier Transform^FFT

Applications

• Differential Equations: Solving the heat equation
• Signal Analysis: Sampling theorem
• Quantum Mechanics: Heisenberg uncertainty principle

Convolution

• Definition of convolution
  • Properties
  • Convolution of multivariable functions
• Support of convolution
• Convolution convergence theorem
• Norm convergence theorem of convolution

Mellin Transform

• Mellin transform
• Convolution of Mellin transform

Wavelet Analysis

• Definition of wavelets
• Multiresolution analysis
• Scaling equation

Splines

• Splines, B-splines
• Properties
• Fourier transform
• Explicit formulas
• Regularity
• Central B-splines
• Scaling equation

References

• Gerald B. Folland, Fourier Analysis and Its Applications (1992)
• 최병선, Fourier 해석 입문 (2002)