Littlewood–Paley decomposition

Littlewood-Paley分解

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Translation evidence: Literature usage

Meaning in context

An inhomogeneous Littlewood–Paley decomposition splits a tempered distribution into smooth Fourier multiplier pieces using a low-frequency cutoff and dyadic annular cutoffs whose sum is one; the piece sum reconstructs the distribution in S′.

非齐次 Littlewood-Paley 分解，用一个低频截断和一列二进制尺度圆环上的光滑频率截断分割单位，将缓增分布拆成频率局部化片段，并在缓增分布意义下重构。

Example

If $\psi_0(0)=1$ and all annular $\psi_j(0)=0$, the constant distribution $1$ has low-frequency piece $1$ and every higher piece $0$.

Usage and distinctions

The homogeneous decomposition omits a low-frequency block and requires attention to the zero frequency and polynomial ambiguity.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Mikko Salo — Function Spaces§3.3, Theorem 2 and dyadic pieces; §3.4, general partitions · Checked 2026-10-14

Translation sources

1. 北京国际数学研究中心 — Enhanced Program for Graduate Study（2024）五、课程简介，4.微分方程基础选讲，（2）课程内容：Littlewood-Paley分解 · Checked 2026-10-14

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