---
tag: D1PAT0K446
kind: concept
name: Littlewood–Paley decomposition
msc: [42B25]
sources:
  - url: https://en.wikipedia.org/wiki/Littlewood%E2%80%93Paley_theory
---

## Summary

The Littlewood–Paley decomposition writes a function $f$ on $\R^n$ as a sum $f = \sum_{j \in \Z} \Delta_j f$ of pieces whose [[Fourier transforms|DPVM34ADZV]] are supported in dyadic annuli $\set{2^{j-1} < \abs{\xi} < 2^{j+1}}$.
The pieces are obtained by smooth cut-offs in frequency, so each $\Delta_j f$ is a convolution of $f$ with a rescaled Schwartz function.
It is the basic construction of [[Littlewood–Paley theory|AA29VBWBFH]], and it is used to define Besov and Triebel–Lizorkin spaces and paraproducts.

## Explanations

### L1 human

The decomposition separates a function into frequency bands, each one octave wide.
The piece $\Delta_j f$ contains the oscillations of $f$ at wavelengths about $2^{-j}$, while coarser and finer details are filtered out.
Smooth filters are used instead of sharp cut-offs, so that each piece is still nicely localized in space.
For instance, $\cos(2\pi \cdot 5x)$ on $\R$ has frequencies $\pm 5$, and only the pieces with $j = 2$ and $j = 3$ can be nonzero, since $2^{j-1} < 5 < 2^{j+1}$ holds only for these $j$.