nLab convolution product of distributions

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 $u \in \mathcal{D}'(\mathbb{R}^n)$ be a distribution, and $f \in C^\infty_0(\mathbb{R}^n)$ a compactly supported smooth function?. Then the convolution of the two is the smooth function

$$u \star f \in C^\infty(\mathbb{R}^n)$$

defined by

$$(u \star f)(x) \;\coloneqq\; u\left( f(x-(-))\right) \,.$$

(Hörmander 90, top of section 4.1)

Definition

(convolution of a distribution with a compactly supported distribution)

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

$$u_1 \star u_2 \;\in \; \mathcal{D}'(\mathbb{R}^n)$$

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

$$(u_1 \star u_2) \star f = u_1 \star (u_2 \star f) \,,$$

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

$$u_1 \in \mathcal{S}'(\mathbb{R}^n)$$

be a tempered distribution and

$$u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$$

be a compactly supported distribution.

Observe here that the Paley-Wiener-Schwartz theorem implies that the Fourier transform of distributions of $u_1$ is a non-singular distribution $\widehat{u_1} \in C^\infty(\mathbb{R}^n)$ so that the product $\widehat{u_1} \cdot \widehat{u_2}$ 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:

$$\widehat{u_1 \star u_2} \;=\; \widehat{u_1} \cdot \widehat{u_2} \,.$$

(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 $u_1 \cdot u_2$ for compactly supported distributions $u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ by the formula

$$\widehat{ u_1 \cdot u_2 } \;\coloneqq\; (2\pi)^{-n} \widehat{u_1} \star \widehat{u_2} \,.$$

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 $u_1$ and $u_2$. 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 $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two compactly supported distributions. Then the wave front set of their convolution of distributions (def. ) is

$$WF(u \star v) \;\subseteq\; \left\{ (x + y, k) \;\vert\; (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u) \right\} \,.$$

(Bengel 77, prop. 3.1)

Related concepts

• product of a distribution with a smooth function

• product of distributions

• Fourier transform of distributions

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)