Distributions and Fundamental Solutions - Core Definitions

Distribution theory provides a rigorous framework for treating generalized functions like the Dirac delta, enabling precise formulation of weak solutions to PDEs and fundamental solutions with singularities.

DefinitionDistributions (Generalized Functions)

Let $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ be the space of smooth, compactly supported test functions on $\Omega \subset \mathbb{R}^n$ .

A distribution (or generalized function) is a continuous linear functional $T: \mathcal{D}(\Omega) \to \mathbb{R}$ : $\phi \mapsto \langle T, \phi \rangle$

The space of distributions is denoted $\mathcal{D}'(\Omega)$ .

Regular distributions come from locally integrable functions: if $f \in L^1_{\text{loc}}(\Omega)$ , define: $\langle T_f, \phi \rangle = \int_\Omega f(x)\phi(x)\,dx$

ExampleDirac Delta Distribution

The Dirac delta $\delta_a$ concentrated at $a \in \Omega$ is defined by: $\langle \delta_a, \phi \rangle = \phi(a)$

This is not a function, but a distribution measuring the value at a point. It satisfies $\delta * f = f$ (convolution identity).

DefinitionDerivatives of Distributions

The derivative of a distribution $T$ is defined by: $\langle \frac{\partial T}{\partial x_i}, \phi \rangle = -\langle T, \frac{\partial \phi}{\partial x_i} \rangle$

This extends differentiation to non-differentiable and even discontinuous functions. Every distribution has derivatives of all orders.

ExampleDerivative of Heaviside Function

The Heaviside step function $H(x) = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$ has distributional derivative: $\frac{dH}{dx} = \delta_0$

This rigorously captures the intuition that the "derivative" of a jump is a spike.

Remark

Distribution theory resolves paradoxes like "what is the derivative of $|x|$ at zero?" The answer: the distributional derivative is the function $\text{sgn}(x)$ , which has further distributional derivative $2\delta_0$ .

This framework is essential for weak formulations of PDEs, where classical derivatives may not exist but distributional derivatives are well-defined.