Distributions and Generalized Functions — Mathematics Knowledge System

Overview

The theory of distributions (generalized functions), developed by Laurent Schwartz in the 1940s, extends the notion of a function to objects that may not be defined pointwise but act on smooth test functions through integration. Every locally integrable function defines a distribution, but distributions also include objects like the Dirac delta $\delta_0$—which assigns to each test function its value at the origin—and derivatives of any order of any distribution. This framework resolves the problem that many natural 'functions' arising in physics and PDE theory (impulse forces, point charges, fundamental solutions) are not classical functions. Every distribution is infinitely differentiable in the distributional sense.

Intuition

Rather than defining $f$ pointwise, think of $f$ by how it 'weighs' test functions: $\phi \mapsto \int f \phi$. A distribution is any continuous linear functional on the space of smooth compactly supported test functions. The Dirac delta is then perfectly natural: $\delta_0(\phi) = \phi(0)$. Differentiation of distributions is defined by duality—the distributional derivative of $T$ is the distribution $-T \circ (d/dx)$—which means every distribution can be differentiated arbitrarily many times.

Formal Definition

Definition

Let $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ be the space of smooth compactly supported test functions on an open set $\Omega \subset \mathbb{R}^n$, equipped with the topology of uniform convergence of all derivatives on compact sets (a locally convex space). A distribution is a continuous linear functional $T: \mathcal{D}(\Omega) \to \mathbb{R}$. The space of all distributions is $\mathcal{D}'(\Omega)$. For a locally integrable function $f$, the associated distribution is $T_f(\phi) = \int_\Omega f\phi\,dx$. The distributional derivative is defined by $(D^\alpha T)(\phi) = (-1)^{|\alpha|} T(D^\alpha \phi)$.

T: C_c^\infty(\Omega) \to \mathbb{R} \text{ continuous and linear}

Distribution

T_f(\phi) = \int_\Omega f(x)\,\phi(x)\,dx, \quad f \in L^1_{\mathrm{loc}}(\Omega)

Regular distribution

(D^\alpha T)(\phi) = (-1)^{|\alpha|} T(D^\alpha \phi), \quad \phi \in C_c^\infty(\Omega)

Distributional derivative

\delta_0(\phi) = \phi(0)

Dirac delta distribution

Properties

Every distribution is infinitely differentiable

D^\alpha T \in \mathcal{D}'(\Omega) for every T \in \mathcal{D}'(\Omega) and every multi-index \alpha.

Distributional differentiation is sequentially continuous

If T_j \to T in \mathcal{D}' (i.e., T_j(\phi) \to T(\phi) for all \phi), then D^\alpha T_j \to D^\alpha T in \mathcal{D}'.

Support of a distribution

The support of T \in \mathcal{D}' is the smallest closed set S such that T(\phi) = 0 whenever \phi is supported outside S.

Theorems

Theorem 7.1: Every distribution is locally a finite-order derivative of a continuous function

For every T \in \mathcal{D}'(\Omega) and every relatively compact open set \omega \subset\subset \Omega, there exist a continuous function f and a multi-index \alpha such that T|_\omega = D^\alpha T_f.

Theorem 7.2: Fundamental solution of the Laplacian

In \mathbb{R}^n (n \ge 3), the function \Phi(x) = c_n |x|^{2-n} satisfies -\Delta \Phi = \delta_0 in the sense of distributions, where c_n = 1/(n(n-2)\omega_n) and \omega_n is the volume of the unit ball.

Theorem 7.3: Schwartz kernel theorem

Every continuous linear operator A: C_c^\infty(\Omega) \to \mathcal{D}'(\Omega) is given by a distribution K \in \mathcal{D}'(\Omega \times \Omega) (the Schwartz kernel): A\phi(x) = \int K(x,y)\phi(y)\,dy in the distributional sense.

Worked Examples

Compute $T_{H}'(\phi) = -T_H(\phi') = -\int_{-\infty}^\infty H(x) \phi'(x)\,dx = -\int_0^\infty \phi'(x)\,dx$.
$T_H'(\phi) = -\int_0^\infty \phi'(x)\,dx$
Evaluate: $-\int_0^\infty \phi'(x)\,dx = -[\phi(x)]_0^\infty = -(\phi(\infty) - \phi(0)) = \phi(0)$ since $\phi$ is compactly supported.
$-\int_0^\infty \phi'(x)\,dx = \phi(0)$
So $T_H'(\phi) = \phi(0) = \delta_0(\phi)$ for all test functions $\phi \in C_c^\infty(\mathbb{R})$.

Answer: $H' = \delta_0$ in the sense of distributions.

Practice Problems

Difficulty 7/10

Show that the map $\phi \mapsto \phi(0)$ is a continuous linear functional on $\mathcal{D}(\mathbb{R})$ (i.e., $\delta_0$ is indeed a distribution).

Difficulty 8/10

Compute the distributional derivative of $f(x) = |x|$ on $\mathbb{R}$ and explain why it equals the sign function $\text{sgn}(x)$.

Difficulty 9/10

Prove that the Dirac delta $\delta_0$ is not a regular distribution, i.e., it cannot be represented as $T_f$ for any $f \in L^1_{\text{loc}}(\mathbb{R})$.

Common Mistakes

Common Mistake

$\delta_0$ is a function that equals $\infty$ at 0 and 0 elsewhere

$\delta_0$ is not a function at all—it is a distribution (continuous linear functional on test functions). The informal description is a mnemonic for its action $\delta_0(\phi) = \phi(0)$, not a pointwise definition.

Common Mistake

Distributional derivatives agree with classical derivatives whenever the latter exist

If $f$ is a $C^1$ function, its classical derivative equals its distributional derivative. But for less regular $f$, they may differ: the distributional derivative of $|x|$ is $\text{sgn}(x)$, which is the pointwise derivative wherever it exists (all $x\neq 0$)—here they agree a.e., which is the general rule for absolutely continuous functions.

Quiz

Summary

A distribution is a continuous linear functional on $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$; every locally integrable function defines a regular distribution.
The Dirac delta $\delta_0(\phi) = \phi(0)$ is a canonical example of a singular (non-regular) distribution.
Every distribution has distributional derivatives of all orders, defined by $(D^\alpha T)(\phi) = (-1)^{|\alpha|}T(D^\alpha\phi)$.
Fundamental solutions satisfy $P(D)E = \delta_0$ distributionally; e.g., $-\Delta(1/(4\pi|x|)) = \delta_0$ in $\mathbb{R}^3$.
Sobolev spaces embed into $\mathcal{D}'$: every $W^{k,p}$ function is a distribution, and the Sobolev derivative equals the distributional derivative.

References

BookSchwartz, L. Théorie des distributions. Hermann, 1950–1951.
BookEvans, L. C. Partial Differential Equations. 2nd ed., AMS, 2010. Appendix D.
WebsiteWikipedia — Distribution (mathematics)