Fourier Analysis
From trigonometric functions to wavelet transforms
About Fourier Analysis
Fourier analysis studies functions and signals through their frequency components. At an elementary level it begins by representing complex functions and signals as superpositions of simple sinusoidal waves (sine and cosine waves). It is an important tool across a wide range of mathematics, physics and engineering, including audio processing, image processing, quantum mechanics, and heat conduction.
This series covers Fourier analysis systematically across four levels, from a high-school-level introduction to graduate-level advanced topics.
Level-by-Level Study Guide
Introductory
Accessible with high-school-level mathematics
Starting with a review of trigonometric functions, you will learn about periodic functions and gain an intuitive understanding of Fourier series. With an emphasis on the physical image of wave superposition, you will be able to perform basic calculations.
- Review of trigonometric functions
- What are periodic functions?
- Superposition of waves
- Introduction to Fourier series
- Expanding simple functions
- Intuitive understanding of convergence
6 chapters
Basic
Undergraduate 1st-2nd year level
Learn the definition and computation of Fourier series systematically. Covers both real and complex Fourier forms, Parseval's identity, convergence theory, kernel-based analysis (Dirichlet/Fejér), the Gibbs phenomenon, and the generalization to non-periodic functions (the Fourier transform), in 11 chapters.
- Definition of Fourier series
- Computing Fourier coefficients
- Expansion of even and odd functions
- Complex Fourier series
- Parseval's identity & Bessel's inequality
- Convergence theorems
- Dirichlet and Fejér kernels
- Gibbs phenomenon
- Fourier transform
11 topics
Intermediate
Undergraduate 3rd-4th year level
Develop the Fourier transform introduced at the Basic level in greater depth, advancing into the world of continuous spectra. Beyond the central theorems (convolution, sampling, Plancherel), the level also covers applied topics such as Fourier cosine/sine transforms, integral transforms, power spectra, harmonic analysis, and window functions. The discrete Fourier transform (DFT) and its fast algorithms (FFT) are consolidated in the Numerical Analysis Advanced series.
- Definition & properties of the Fourier transform
- Important Fourier transform pairs
- Convolution and convolution theorem
- Sampling theorem & Nyquist frequency
- Plancherel theorem & Poisson summation formula
- Fourier cosine and sine transforms, and related integral transforms
- Power spectrum & harmonic analysis (overview)
- Window functions and spectral leakage
16 topics
Advanced
Graduate level
Study Fourier analysis in $L^2$ spaces, orthogonal polynomials, the Fourier transform of distributions, multivariable Fourier analysis, spectral theory, wavelet transforms, harmonic analysis, and Bohr's theory of almost periodic functions. Also covers a range of integral transforms (Abel, Hankel, Hilbert, Mellin) and the Wiener filter / Wiener-Khinchin theorem.
- $L^2$ spaces and Hilbert spaces
- Orthogonal systems, orthogonal polynomials and generalized Fourier expansions
- Fourier transform of distributions
- Multivariable Fourier transform
- Applications to partial differential equations
- Spectral theory
- Wavelet transforms
- Harmonic analysis (maximal functions, singular integrals, BMO)
- Almost periodic functions & Bohr-Fourier expansion
- Integral transforms (Abel / Hankel / Hilbert / Mellin)
- Wiener filter; the Wiener-Khinchin theorem linking autocorrelation and the power spectrum
16 topics
Learning Roadmap
Introductory
Trigonometric functions and waves
Basic
Fourier series
Intermediate
Fourier transform / spectral analysis
Advanced
Functional analysis and applications
Prerequisites
- Introductory: Basics of trigonometric functions, elementary definite integration
- Basic: Introductory level content, basics of calculus, an introduction to complex numbers and the complex exponential (used for complex Fourier series)
- Intermediate: Basic level content, improper integrals, basics of series and sequences of functions
- Advanced: Intermediate level content, Lebesgue integration, $L^p$ spaces, basics of functional analysis
Frequently Asked Questions
What is Fourier analysis?
Fourier analysis is the branch of mathematics concerned with decomposing functions and signals into superpositions of sinusoidal waves. Its core tools — Fourier series (for periodic functions) and the Fourier transform (for general functions) — are fundamental in signal processing, PDEs, image compression, and many areas of physics and engineering.
What is the difference between Fourier series and the Fourier transform?
Fourier series represent periodic functions as discrete sums of sinusoids with integer frequencies $e^{2\pi inx/T}$. The Fourier transform represents functions and signals in terms of a continuous frequency variable $e^{2\pi i\xi x}$, and is particularly suited to non-periodic signals (extended to distributions it also applies to periodic functions, where the result is a train of delta functions). Heuristically, as $T$ grows the spacing between frequencies tends to zero and the Fourier-series sum leads to the Fourier-transform integral.
What prerequisites are needed to study Fourier analysis?
At the introductory level, basic trigonometry and elementary definite integration are enough. The Basic level covers complex Fourier series, so it needs calculus together with an introduction to complex numbers and the complex exponential $e^{i\theta}$. The Intermediate level needs improper integrals and the basics of series and sequences of functions; the Advanced level needs Lebesgue integration, $L^p$ spaces and Hilbert space theory.