Fourier transform

傅立叶变换

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

Translation evidence: Literature usage

Meaning in context

For f in L1(R^n), the Fourier transform in the convention used here is the function f-hat(xi)=integral f(x)e^{-i x dot xi} dx.

对 R^n 上可积函数，此处把积分 f(x)e^{-ix·ξ} 定义为傅立叶变换；采用这一约定时，逆变换需乘 (2π)^{-n}。

Example

For f(x)=e−2x1[0,∞)(x)f(x)=e^{-2x}\mathbf1_{[0,\infty)}(x), f^(ξ)=1/(2+iξ)\widehat f(\xi)=1/(2+i\xi) by integrating the exponential.

Usage and distinctions

The transform has a continuous frequency variable; a periodic Fourier series has integer-indexed coefficients. Normalization constants vary by source.

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Definition sources

  1. Mikko Salo — Fourier Analysis and Distribution TheoryChapter 3 opening and §3.3, equation (3.5) · 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，开头定义；Chinese source uses a different normalization · Checked 2026-10-14
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