Fourier Analysis

About Fourier Analysis

Fourier analysis is a mathematical technique for representing complex functions and signals as superpositions of simple sinusoidal waves (sine and cosine waves). It is an indispensable tool across all fields of science and technology, 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

High school level

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 rigorous definition and computation of Fourier series. Understand the formulas for Fourier coefficients, Parseval's equality, and various convergence theorems, and apply them to real problems.

  • Definition of Fourier series
  • Computing Fourier coefficients
  • Expansion of even and odd functions
  • Parseval's equality
  • Convergence theorems
  • Gibbs phenomenon

7 chapters

Intermediate

Undergraduate 3rd-4th year level

Introduce the Fourier transform and advance into the world of continuous spectra. Learn about convolution, the sampling theorem, the discrete Fourier transform (DFT), and the fast Fourier transform (FFT) in preparation for practical applications.

  • Definition of the Fourier transform
  • Properties of the Fourier transform
  • Convolution theorem
  • Sampling theorem
  • Discrete Fourier transform
  • Fast Fourier transform
  • Window functions and spectral leakage

8 chapters

Advanced

Graduate level

Study Fourier analysis in $L^2$ spaces, the Fourier transform of distributions, multivariable Fourier analysis, and wavelet transforms. Applications to partial differential equations and the foundations of harmonic analysis are also covered.

  • $L^2$ spaces and Hilbert spaces
  • Fourier transform of distributions
  • Multivariable Fourier transform
  • Applications to partial differential equations
  • Wavelet transforms
  • Introduction to harmonic analysis

7 chapters

Learning Roadmap

Introductory

Trigonometric functions and waves

→

Basic

Fourier series

→

Intermediate

Fourier transform / DFT

→

Advanced

Functional analysis and applications

Prerequisites

  • Introductory: Middle school math fundamentals, trigonometric ratios
  • Basic: Introductory level content, basics of calculus
  • Intermediate: Basic level content, complex numbers, improper integrals
  • Advanced: Intermediate level content, Lebesgue integration, basics of functional analysis