---
tag: 4PZ35C0GV3
kind: concept
name: Wavelet
msc: [42C40, 65T60]
sources:
  - url: https://en.wikipedia.org/wiki/Wavelet
same_as:
  wikidata: Q831390
  wikipedia: Wavelet
---

## Summary

A wavelet is a function $\psi$ on $\R$ whose dilations by powers of $2$ and integer translations form an [[orthonormal basis|0S9GGTR0EX]] of $L^2(\R)$.
Each basis function is localized near a position and at a scale, so a wavelet expansion describes a function at all levels of detail.
In a wider sense, any oscillating function with integral zero that is used in a [[continuous wavelet transform|57FR21NDGV]] is called a wavelet.
Wavelets are used in signal and image compression, numerical analysis and harmonic analysis.

## Explanations

### L1 human

A wavelet is a small wave: it wiggles up and down and then dies out.
Stretching it gives slow wiggles for coarse features, and squeezing it gives fast wiggles for fine detail.
Sliding these copies along the line, one can match the features of a signal wherever they occur.
The simplest wavelet is the Haar wavelet, which is $+1$ on $[0, \tfrac12)$, $-1$ on $[\tfrac12, 1)$ and $0$ elsewhere.