nLab product of distributions with a smooth function

Definition

Let $n \in \mathbb{N}$ and consider $\mathbb{R}^n$ the Cartesian space of dimension $n$.

Definition

(product of a distribution with a smooth function)

For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution, and $f \in C^\infty(\mathbb{R}^n)$ a smooth function, their product

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

is the distribution given on a compactly supported smooth function $g \in C^\infty_{cp}(\mathbb{R}^n) = \mathcal{D}(\mathbb{R}^n)$ by

$$(f\cdot u)(g) \;\coloneqq\; u\left( f \cdot g\right) \,,$$

where on the right we have the application of $u$ regarded as a continuous linear functional $u \colon \mathcal{D}(\mathbb{R}^n) \to \mathbb{C}$ to the ordinary pointwise product of smooth functions $f \cdot g$.

Properties

Definition

(product of a distribution with a non-singular distributions is product of distribution with a smooth function)

The wave front set of a non-singular distribution $u_f$ corresponding to a smooth function $f \in C^\infty(\mathbb{R}^n)$, is empty (this prop.). Therefore the product of distributions (def. ) of a non-singular distribution with any distribution $u$ is defined, and given by the product of distributions with smooth functions according to def. :

$$u_f \cdot u = f \cdot u = u(f\cdot (-)) \,.$$

Related concepts

• convolution product

• convolution product of distributions

• derivative of a distribution

References

See also

• Wikipedia, Distribution (mathematics) – Multiplication by a smooth function