Fourier Analysis
The field that studies subjects of interest using Fourier series and Fourier transforms is called Fourier analysisFourier analysis or harmonic analysisharmonic 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
- Properties of Fourier transform
- Riemann-Lebesgue lemma
- Fourier inversion theorem
- 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 TransformDFT
- 이산 푸리에 변환 행렬
- Properties of Discrete Fourier Transform
- Inverse Discrete Fourier Transform
- Fast Fourier TransformFFT
Applications
- Differential Equations: Solving the heat equation
- Signal Analysis: Sampling theorem
- Quantum Mechanics: Heisenberg uncertainty principle
Convolution
- Definition of convolution
- Support of convolution
- Convolution convergence theorem
- Norm convergence theorem of convolution
Mellin Transform
Wavelet Analysis
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)
All posts
- 이산 푸리에 변환 행렬
- 줄리아에서 이산 푸리에 변환 행렬 구현하기
- Fourier Series and Bessel's Inequality
- Derivation of Fourier Series
- Dirichlet Kernel
- Proof of the Orthogonality of the Set of Trigonometric Functions
- Sum of Trigonometric Functions Orthogonal to Each Other
- The Fourier Series of a Riemann Integrable Function Converges
- Convergence of Fourier Series at Discontinuities
- Derivative's Fourier Coefficients
- Fourier Series Integration
- The constant term of the Fourier series is equal to the average of one period of the function.
- The Limit of Fourier Coefficients is Zero
- Definition of Convolution
- Fourier Cosine Series, Sine Series, Fourier Coefficients of Even and Odd Functions
- Fourier Coefficients of Odd Functions
- The sufficient condition for the Fourier series of a function to converge absolutely and uniformly to the function
- Properties of Fourier Transform
- Fourier Transform of Gaussian Functions
- Fourier Transform of Characteristic Functions
- Riemann-Lebesgue Lemma
- Exponential Function Set and Trigonometric Function Set are Orthogonal Bases
- Fourier Inversion Theorem
- Solving Differential Equations Using Fourier Transform
- Discrete Fourier Transform
- Mellin Transformation
- Definition of Wavelets
- Multi-Resolution Analysis
- Multi-Resolution Analysis Scaling Equation
- Fourier Transform of the Dirac Delta Function
- Spline, B-Spline in Analysis
- Fourier Series in Complex Notation
- Mellin Transform Convolution
- Convolution Convergence Theorem
- Convolution Norm Convergence Theorem
- Inverse Fourier Transform Theorem for Smooth Functions
- Plancherel's Theorem
- Several Definitions and Notations of Fourier Transform
- Properties of Convolution
- Properties of B-Splines
- Fourier Transform of B-Splines
- Explicit Formulas of B-splines
- Regularity of B-Splines
- Central B-spline
- B-spline Scaling Equation
- Various Meanings of the Fourier Transform
- Convolution of Multivariable Functions
- Sampling Theorem
- Heisenberg Uncertainty Principle
- Discrete Fourier Transform Properties
- Discrete Fourier Inversion
- Convolution Support
- The Fast Fourier Transform Algorithm