Distributions and Fundamental Solutions - Key Properties

Distributions possess algebraic and topological properties that make them powerful for PDE analysis, including operations like convolution, multiplication, and Fourier transform.

DefinitionSupport of a Distribution

The support of a distribution $T \in \mathcal{D}'(\Omega)$ is the complement of the largest open set $U$ where $T = 0$ : $\text{supp}(T) = \Omega \setminus \bigcup\{U \subset \Omega : \langle T, \phi \rangle = 0 \text{ for all } \phi \in \mathcal{D}(U)\}$

A distribution with compact support belongs to $\mathcal{E}'(\Omega)$ , the dual of $C^\infty(\Omega)$ .

Convolution: For $T \in \mathcal{E}'(\mathbb{R}^n)$ and $f \in C^\infty(\mathbb{R}^n)$ : $\langle T * f, \phi \rangle = \langle T, \tilde{f} * \phi \rangle$ where $\tilde{f}(x) = f(-x)$ .

Convolution with distributions extends the classical operation and satisfies $\partial(T * f) = (\partial T) * f = T * (\partial f)$ .

DefinitionFourier Transform of Distributions

For tempered distributions $T \in \mathcal{S}'(\mathbb{R}^n)$ , the Fourier transform is defined by: $\langle \hat{T}, \phi \rangle = \langle T, \hat{\phi} \rangle$ for all $\phi \in \mathcal{S}$ (Schwartz space).

This extends the Fourier transform to generalized functions while preserving key properties like $\mathcal{F}[\partial_x T] = 2\pi i\xi\hat{T}$ .

ExampleFourier Transform of Delta

$\hat{\delta}_0 = 1 \quad \text{(constant distribution)}$ $\mathcal{F}[1] = \delta_0$

These reciprocal relations show that perfect localization in position corresponds to complete delocalization in frequency.

Remark

Structure Theorem: Every distribution $T \in \mathcal{D}'(\Omega)$ is locally a finite-order derivative of a continuous function. More precisely, for any compact $K \subset \Omega$ , there exists $m \in \mathbb{N}$ and continuous functions $f_\alpha$ such that: $T = \sum_{|\alpha| \leq m} \partial^\alpha f_\alpha \text{ on } K$

Multiplication: Distributions can be multiplied by smooth functions: $(aT)(\phi) = T(a\phi)$ for $a \in C^\infty$ . However, multiplication of distributions is generally undefined (e.g., $\delta^2$ has no consistent meaning).

These properties make distributions an algebra under differentiation and convolution, ideal for studying linear PDEs with singular data or coefficients.