smooth wavefront set

波前集

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Translation evidence: Literature usage

Meaning in context

For a distribution u, its smooth wavefront set consists of pairs (x,xi) with xi nonzero for which no smooth compactly supported cutoff equal to one near x has a Fourier transform of its product with u decaying faster than every inverse power in a conic neighborhood of xi.

分布的光滑波前集由点与非零余向量方向的配对组成；若任何在该点附近为一的光滑局部截断，都不能使其傅立叶变换在此方向的锥邻域中快速衰减，则该配对属于波前集。

Example

For the Dirac distribution at aa in one dimension, the localized transform is e−iaξe^{-ia\xi}, so WF⁡(δa)={(a,ξ):ξ≠0}\operatorname{WF}(\delta_a)=\{(a,\xi):\xi\ne0\}.

Usage and distinctions

Singular support records positions only; the wavefront set also records covector directions and excludes the zero covector.

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Definition sources

  1. Mikko Salo — Applications of Microlocal Analysis to Inverse Problems§2.1, Definition (Wave front set), printed p.8 · Checked 2026-10-14

Translation sources

  1. 国立台湾大学科学推展中心 — 表現理論波前集之新突破开篇引言“波前集(wavefront set) WF(f)”；图1说明 · Checked 2026-10-14
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