convolution of functions

卷积

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

Translation evidence: Literature usage

Meaning in context

For f,g in L1(R^n), their convolution is the almost-everywhere-defined integrable function (f*g)(x)=integral f(x-y)g(y) dy.

两个 R^n 上可积函数的卷积，是几乎处处由积分 f(x-y)g(y) 定义的可积函数。

Example

For f=g=1[0,2]f=g=\mathbf1_{[0,2]}, (f∗g)(x)=max⁡(0,2−∣x−2∣)(f*g)(x)=\max(0,2-|x-2|), the length of overlap of [0,2][0,2] and [x−2,x][x-2,x].

Usage and distinctions

Convolution integrates translated products; pointwise multiplication is a different operation, and discrete sequence convolution uses a sum.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Mikko Salo — Fourier Analysis and Distribution Theory§3.8, equations (3.25)–(3.26) · 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.11.2 卷积，定义与可积函数情形 · Checked 2026-10-14
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