---
tag: FAYWXJ68SB
kind: concept
name: Fourier analysis
msc: [42A16, 42A38, 42B10]
sources:
  - url: https://en.wikipedia.org/wiki/Fourier_analysis
same_as:
  wikidata: Q1365258
  wikipedia: Fourier analysis
---

## Summary

Fourier analysis is the study of how functions are represented by sums or integrals of sines, cosines and complex exponentials.
Its main tools are the [[Fourier series|G12KXDHPPS]] of periodic functions, the [[Fourier transform|DPVM34ADZV]] on $\R^n$ and the [[discrete Fourier transform|TST0GH66A5]] on $\Z / N\Z$.
It is the classical core of [[harmonic analysis|7QNRC9SJVV]].
It turns differentiation into multiplication and [[convolution|2Z29JN3VES]] into pointwise product, which makes it central for differential equations and signal processing.

## Explanations

### L1 human

Fourier analysis splits a signal into its frequencies, much as a prism splits white light into colours.
A periodic signal is a sum of waves whose frequencies are whole multiples of one base frequency.
A non-periodic signal needs all frequencies, and the sum becomes an integral.
For example, the function $\cos^2 x$ equals $\frac{1}{2} + \frac{1}{2} \cos 2x$, so it contains only the frequencies $0$ and $2$.
Joseph Fourier used such decompositions to solve the [[heat equation|VP2HN86VAG]].