The generalization of the convolution product on smooth function to distributions, viewing distributions as generalized functions (via this prop.).
(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
defined by
(Hörmander 90, top of section 4.1)
(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
is the unique distribution such that for $f \in C^\infty(\mathbb{R}^n)$ a smooth function, it satisfies
where on the right we have twice a convolution of a distribution with a smooth function according to def. .
(Fourier transform of distributions interchanges convolution of distributions with pointwise product)
Let
be a tempered distribution and
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:
(e.g. Hörmander 90, theorem 7.1.15)
For the converse: Friedlander-Joshi 98, p. 102
(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
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.
(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
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 20, 2021 at 09:46:56. See the history of this page for a list of all contributions to it.