Fourier series

Fourier级数

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

Translation evidence: Literature usage

Meaning in context

For a 2 pi-periodic square-integrable function f, its Fourier series is sum_{k in Z} c_k e^{ikx}, where c_k=(2 pi)^(-1) integral_{-pi}^pi f(x)e^{-ikx} dx. The series converges in L2, and stronger modes of convergence need extra justification.

对 2π 周期平方可积函数，以其与各整数频率复指数的积分配对所得系数构成的级数称为 Fourier 级数；它在 L2 意义下收敛，更强的收敛方式须另加论证。

Example

For f(x)=4+cos⁡(2x)f(x)=4+\cos(2x), c0=4c_0=4 and c2=c−2=1/2c_2=c_{-2}=1/2; all other coefficients vanish.

Usage and distinctions

L2 convergence, pointwise convergence, and uniform convergence are different assertions about the same coefficient series.

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Definition sources

  1. Mikko Salo — Fourier Analysis and Distribution Theory§2.1, Theorem 2.1.3 and Fourier coefficient formula · Checked 2026-10-14

Translation sources

  1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第三部分，级数理论：四、Fourier级数 · Checked 2026-10-14
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