convolution product

A convolution product is the binary operation on ring-valued (or more generally magma-valued) functions ff on a group GG or more generally on the set of morphisms 𝒢\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 1f 2)(g)= hGf 1(h)f 2(gh 1) (f_1 \star f_2)(g) \;=\; \sum_{h \in G} f_1(h)\cdot f_2( g \cdot h^{-1} )

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

For example if GG is a finite group, then this defines the group algebra, or more generally if GG 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:f \colon \mathbb{R} \to \mathbb{R} on the real line 1\mathbb{R}^1 regarded as a group with respect to addition of real numbers:

(f 1f 2)(x) tf 1(t)f 2(xt)dx. (f_1 \star f_2)(x) \;\coloneqq\; \int_{t \in\mathbb{R}} f_1(t) f_2(x-t) \, d x \,.

This generalizes to a convolution product of distributions. Thes convolution products plays a central role in Fourier analysis

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

category: disambiguation

Revised on November 2, 2017 18:17:19 by Urs Schreiber (