Knowlpedia

Definition

Convolution of distributions

Distributional convolution extends ordinary convolution under support hypotheses and includes convolution with test functions.

Definition

Let S,T∈D′(Rn)S,T\in\mathcal D'(\mathbb R^n) be , with at least one of them . Their convolution S∗TS*T is the distribution defined by

⟨S∗T,φ⟩=⟨Sx,⟨Ty,φ(x+y)⟩⟩,φ∈Cc∞(Rn).\langle S*T,\varphi\rangle =\big\langle S_x,\langle T_y,\varphi(x+y)\rangle\big\rangle, \qquad \varphi\in C_c^\infty(\mathbb R^n).

The support hypothesis makes the iterated pairing well defined. For arbitrary TT and a ψ\psi, convolution is always defined by

(T∗ψ)(x)=⟨Ty,ψ(x−y)⟩,(T*\psi)(x)=\langle T_y,\psi(x-y)\rangle,

and produces a smooth function. Both constructions extend classical convolution of functions and are independent of which factor is evaluated first whenever the distributional convolution exists.

Smoothing and differentiation

Convolution with a test function regularizes a distribution: T∗ψ∈C∞(Rn)T*\psi\in C^\infty(\mathbb R^n), and derivatives may be moved between the factors,

∂α(T∗ψ)=(∂αT)∗ψ=T∗(∂αψ).\partial^\alpha(T*\psi) =(\partial^\alpha T)*\psi =T*(\partial^\alpha\psi).

This identity underlies mollification and the use of fundamental solutions. When both distributions have compact support, convolution is associative and commutative and remains compactly supported Hörmander, §4.2.

Support and standard examples

Whenever the convolution is defined,

supp⁡(S∗T)⊆supp⁡S+supp⁡T,\operatorname{supp}(S*T) \subseteq\operatorname{supp}S+\operatorname{supp}T,

where the right side is the Minkowski sum. The Dirac distribution is the identity: δ0∗T=T\delta_0*T=T, and δa∗T\delta_a*T translates TT. If ff and gg are locally integrable functions and their classical convolution is meaningful, their regular distributions have the same distributional convolution.

Broader support conditions
References
  1. Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: §4.2, convolution of distributions and support conditions.
  2. François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Publisher record. Relevant: Chapter 27, convolution of distributions.