nLab convolution product

Idea

A convolution product is the binary operation on ring-valued (or more generally magma-valued) functions $f$ on a group $G$ or more generally on the set of morphisms $\mathcal{G}$ of a groupoid, which is given by summing or more generally integrating products of values on complementary elements of the schematic form

$$(f_1 \star f_2)(g) \;=\; \sum_{h \in G} f_1(h)\cdot f_2( g \cdot h^{-1} )$$

where $h^{-1}$ denotes the inverse element or more generally the inverse morphism of $h$.

For example if $G$ is a finite group, then this defines the group algebra, or more generally if $G$ is a suitable topological group or Lie group or more generally a suitably topological groupoid or Lie groupoid, then an integral-version of this formula defines the groupoid convolution algebra.

Often the convolution product is considered in analysis for suitably integrable functions $f \colon \mathbb{R} \to \mathbb{R}$ on the real line $\mathbb{R}^1$ regarded as a group with respect to addition of real numbers:

$$(f_1 \star f_2)(x) \;\coloneqq\; \int_{t \in\mathbb{R}} f_1(t) f_2(x-t) \, d t \,.$$

This generalizes to a convolution product of distributions.

These convolution products play a central role in Fourier analysis.

A categorification of the concept of convolution is the concept of Day convolution.

Related concepts

• group algebra

• groupoid algebra, category algebra

category: disambiguation