Littlewood–Paley decomposition

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 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, and it is used to define Besov and Triebel–Lizorkin spaces and paraproducts.