nLab
convolution product of distributions

Contents

Contents

Idea

The generalization of the convolution product on smooth function to distributions, viewing distributions as generalized functions (via this prop.).

Definition

Definition

(convolution of a distribution with a smooth function)

Let u∈𝒟′(ℝ n)u \in \mathcal{D}'(\mathbb{R}^n) be a distribution, and f∈C ∞(ℝ n)f \in C^\infty(\mathbb{R}^n) a smooth function. Then the convolution of the two is the smooth function

u⋆f∈C ∞(ℝ n) u \star f \in C^\infty(\mathbb{R}^n)

defined by

(u⋆f)(x)≔u(f(x−(−))). (u \star f)(x) \;\coloneqq\; u\left( f(x-(-))\right) \,.

(Hörmander 90, top of section 4.1)

Definition

(convolution of a distribution with a compactly supported distribution)

Let u 1,u 2∈𝒟′(ℝ n)u_1, u_2 \in \mathcal{D}'(\mathbb{R}^n) be two distributions, such that at least one of them is a compactly supported distribution in ℰ′(ℝ n)↪𝒟′(ℝ n)\mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n), then their convolution product

u 1⋆u 2∈𝒟′(ℝ n) u_1 \star u_2 \;\in \; \mathcal{D}'(\mathbb{R}^n)

is the unique distribution such that for f∈C ∞(ℝ n)f \in C^\infty(\mathbb{R}^n) a smooth function, it satisfies

(u 1⋆u 2)⋆f=u 1⋆(u 2⋆f), (u_1 \star u_2) \star f = u_1 \star (u_2 \star f) \,,

where on the right we have twice a convolution of a distribution with a smooth function according to def. .

(Hörmander 90, def. 4.2.2)

Properties

Relation to pointwise product of distributions

Proposition

(Fourier transform of distributions interchanges convolution of distributions with pointwise product)

Let

u 1∈𝒮′(ℝ n) u_1 \in \mathcal{S}'(\mathbb{R}^n)

be a tempered distribution and

u 2∈ℰ′(ℝ n)↪𝒮′(ℝ n) u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)

be a compactly supported distribution.

Observe here that the Paley-Wiener-Schwartz theorem implies that the Fourier transform of distributions of u 1u_1 is a non-singular distribution u 1^∈C ∞(ℝ n)\widehat{u_1} \in C^\infty(\mathbb{R}^n) so that the product u 1^⋅u 2^\widehat{u_1} \cdot \widehat{u_2} is always defined.

Then the Fourier transform of distributions of the convolution product of distributions (def. ) is the product of the Fourier transform of distributions:

u 1⋆u 2^=u 1^⋅u 2^. \widehat{u_1 \star u_2} \;=\; \widehat{u_1} \cdot \widehat{u_2} \,.

(e.g. Hörmander 90, theorem 7.1.15)

For the converse: Friedlander-Joshi 98, p. 102

Remark

(product of distributions via Fourier transform of distributions)

Prop. together with the Fourier inversion theorem suggests to define the product of distributions u 1⋅u 2u_1 \cdot u_2 for compactly supported distributions u 1,u 2∈ℰ′(ℝ n)↪𝒮′(ℝ n)u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n) by the formula

u 1⋅u 2^≔(2π) −nu 1^⋆u 2^. \widehat{ u_1 \cdot u_2 } \;\coloneqq\; (2\pi)^{-n} \widehat{u_1} \star \widehat{u_2} \,.

For this to make sense, the convolution product of the smooth functions on the right needs to exist, which is not guaranteed. The condition that this exists is the Hörmander-condition on the wave front set of u 1u_1 and u 2u_2. See at product of distributions for more.

Relation to singular support and wave front set

Proposition

(wave front set of convolution of compactly supported distributions)

Let u,v∈ℰ′(ℝ n)u,v \in \mathcal{E}'(\mathbb{R}^n) be two compactly supported distributions. Then the wave front set of their convolution of distributions (def. ) is

WF(u⋆v)={(x+y,k)|(x,k)∈WF(u)and(y,k)∈WF(u)}. WF(u \star v) \;=\; \left\{ (x + y, k) \;\vert\; (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u) \right\} \,.

(Bengel 77, prop. 3.1)

References

  • Lars Hörmander, section 4 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

  • Friedlander F G and Joshi M 1998 Introduction to the Theory of Distributions 2nd ed. (Cambridge: Cambridge University Press)

  • Gunter Bengel, Wave front sets and Singular Supports of Convolutions, Publ. RIMS, Kyoto Univ. 12 Suppl. (1977), 1-4 and Math. Ann. 226, 247-252 (1977) (web, pdf)

Last revised on November 28, 2017 at 05:12:13. See the history of this page for a list of all contributions to it.