inverse Fourier transform

逆傅立叶变换

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For g in Schwartz space on R^n, the inverse Fourier transform in the nonunitary convention is (2 pi)^(-n) integral e^{i x dot xi}g(xi) dxi; it inverts the forward transform with the negative exponential.

对速降函数 g，非酉归一化约定下的逆傅立叶变换是正指数积分乘 (2π)^{-n}，它与使用负指数的正变换互为逆。

Example

If g(ξ)=πe−ξ2/4g(\xi)=\sqrt\pi e^{-\xi^2/4}, then (2π)−1∫eixξg(ξ) dξ=e−x2(2\pi)^{-1}\int e^{ix\xi}g(\xi)\,d\xi=e^{-x^2}.

Usage and distinctions

The sign and constant are paired with the forward convention; this integral need not converge absolutely for arbitrary L2 data.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Mikko Salo — Fourier Analysis and Distribution Theory§3.3, equation (3.6) and Theorem 3.3.1 · Checked 2026-10-14

Translation sources

  1. 李东风（北京大学）— 金融时间序列分析备课笔记，附录EE.11.3 傅立叶变换，速降函数的逆傅立叶变换定理 · Checked 2026-10-14
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