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convolution product of distributions

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.