Parseval identity for Fourier series

Parseval等式

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

Translation evidence: Literature usage

Meaning in context

For a 2 pi-periodic function f in L2 and coefficients c_k=(2 pi)^(-1) integral f(x)e^{-ikx} dx, Parseval's identity states (2 pi)^(-1) integral_{-pi}^pi |f|^2=sum_{k in Z}|c_k|^2.

对平方可积的 2π 周期函数，以通常平均积分定义 Fourier 系数时，Parseval 等式把周期内平方模的平均积分等同于所有系数平方模的和。

Example

For $f=2+\cos x$, the coefficient sum is $4+1/4+1/4=9/2$, matching the mean-square integral.

Usage and distinctions

Bessel's inequality gives only an upper bound for a general orthonormal system; equality requires completeness.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Mikko Salo — Fourier Analysis and Distribution Theory§2.1, Theorem 2.1.3, Parseval identity · 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，均方收敛与Parseval等式 · Checked 2026-10-14

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