Distributions and Generalised Functions — Mathematics Knowledge System

Overview

Distributions (or generalised functions) extend the notion of function to include objects like the Dirac delta that behave like functions under integration but are not classical functions. A distribution is a continuous linear functional on a space of test functions. This theory, developed by Laurent Schwartz, provides a rigorous framework for the differentiation of non-smooth functions, impulse forces in physics, and the analysis of PDEs. The derivative of any distribution is defined and is again a distribution.

Intuition

Instead of asking 'what is the value of f at a point x?' we ask 'what is the integral of f against a test function phi?' A test function is smooth and compactly supported — it probes f in an averaged, smeared-out way. A distribution is a consistent rule assigning a number to each such test function. The Dirac delta delta_0 assigns to phi the value phi(0): it 'probes' the value at 0. Every locally integrable function defines a distribution, but distributions also include derivatives of such functions and much more exotic objects.

Formal Definition

Definition

The space of test functions D(Omega) = C_c^inf(Omega) consists of all smooth, compactly supported functions on an open set Omega in R^n. A distribution on Omega is a continuous linear functional T: D(Omega) -> R (or C). Continuity means: if phi_j -> phi in D(Omega) (all supported in a fixed compact set and all derivatives converge uniformly), then T(phi_j) -> T(phi). The derivative of a distribution T is defined by (dT/dx_j)(phi) = -T(dphi/dx_j). The space of tempered distributions S'(R^n) is the dual of the Schwartz space S(R^n) = {f in C^inf : x^alpha d^beta f -> 0 for all alpha,beta}.

\langle T, \varphi \rangle \in \mathbb{R} \quad \forall\, \varphi \in \mathcal{D}(\Omega)

Distribution as a functional

\langle \delta_0, \varphi \rangle = \varphi(0)

Dirac delta distribution

\langle T_f, \varphi \rangle = \int_\Omega f(x)\varphi(x)\,dx

Regular distribution from L^1_{loc}

\left\langle \frac{\partial T}{\partial x_j}, \varphi \right\rangle = -\left\langle T, \frac{\partial \varphi}{\partial x_j}\right\rangle

Distributional derivative

Notation

Notation	Meaning
$\mathcal{D}(\Omega)$	Test function space C_c^inf(Omega)
$\mathcal{D}'(\Omega)$	Space of distributions on Omega
$\delta_x$	Dirac delta at x: <delta_x, phi> = phi(x)
$\mathcal{S}(\mathbb{R}^n)$	Schwartz space (rapidly decreasing smooth functions)
$\mathcal{S}'(\mathbb{R}^n)$	Tempered distributions (dual of Schwartz space)

Theorems

Theorem 12.1: Every Locally Integrable Function Defines a Distribution

If f in L^1_{loc}(Omega), then the map phi |-> integral f phi dx defines a distribution T_f in D'(Omega). Moreover, T_f = T_g (as distributions) if and only if f = g almost everywhere. Thus L^1_{loc} embeds injectively into D'.

Theorem 12.2: Distributional Derivatives Always Exist

Every distribution T in D'(Omega) has distributional derivatives of all orders. The k-th derivative is defined by <(d/dx)^k T, phi> = (-1)^k <T, (d/dx)^k phi>.

Theorem 12.3: Fundamental Theorem for Distributions on R

Every distribution T in D'(R) with T' = 0 (distributionally) is a constant. For any distribution S, T' = S has a solution T.

Theorem 12.4: Structure Theorem

Every distribution T in D'(Omega) is locally a finite-order derivative of a continuous function: for compact K in Omega, there exists continuous F and multi-index alpha with T|_K = d^alpha F.

Worked Examples

1
H defines a regular distribution T_H via <T_H, phi> = integral_0^inf phi(x) dx.
2
Compute the distributional derivative <T_H', phi> = -<T_H, phi'> = -integral_0^inf phi'(x) dx.
$\langle T_H', \varphi \rangle = -\int_0^\infty \varphi'(x)\,dx$
3
Evaluate: -integral_0^inf phi'(x) dx = -[phi(x)]_0^inf = -phi(inf) + phi(0) = phi(0) since phi has compact support.
$-\int_0^\infty \varphi'\,dx = \varphi(0) = \langle \delta_0, \varphi \rangle$
4
Therefore H' = delta_0 in the distributional sense.

✓ Answer

The distributional derivative of the Heaviside function is the Dirac delta: H' = delta_0.

Practice Problems

Mediumproof writing

Prove that the distributional derivative is well-defined: if phi_j -> phi in D(Omega), then <T', phi_j> -> <T', phi>.

Mediumfree response

Explain why distributions are useful for PDEs. Give a specific example where a classical solution does not exist but a distributional solution does.

MediumMultiple choice

The Schwartz space S(R^n) consists of smooth functions that, along with all their derivatives, decay faster than any polynomial. Its dual S'(R^n) is the space of:

Compactly supported distributionsTempered distributionsAll distributions on R^nL^2 functions

Common Mistakes

Common Mistake

Thinking of the Dirac delta as a function that is zero everywhere except at 0.

The Dirac delta is NOT a function. It is a distribution: a linear functional on test functions. The informal statement 'delta(0) = infinity' is not rigorous. The correct statement is <delta_0, phi> = phi(0).

Common Mistake

Believing distributions can be multiplied freely.

Distributions cannot in general be multiplied. The product of two distributions is not always defined. For example, delta_0^2 is not defined as a distribution. Multiplication by smooth functions is well-defined, but multiplication of two arbitrary distributions requires extra structure (e.g., Colombeau algebras).

Quiz

Historical Background

Paul Dirac introduced his delta 'function' in 1930 in quantum mechanics. Sergei Sobolev introduced generalised functions in the context of PDEs in 1936. Laurent Schwartz gave the first comprehensive and rigorous theory of distributions in 1945-1950, for which he received the Fields Medal in 1950. Schwartz's formulation using test functions and duality has become the standard approach. The theory of tempered distributions (dual of the Schwartz space) was developed simultaneously to handle Fourier transforms.

1930
Dirac introduces the delta function in quantum mechanics
Paul Dirac
1936
Sobolev studies generalised derivatives in PDEs
Sergei Sobolev
1945
Schwartz develops the rigorous theory of distributions
Laurent Schwartz
1950
Schwartz receives the Fields Medal for the theory of distributions
Laurent Schwartz

Summary

Distributions are continuous linear functionals on the test function space C_c^inf(Omega), extending the notion of function.
Every locally integrable function defines a regular distribution; singular distributions like the Dirac delta are not regular.
Every distribution has distributional derivatives of all orders: <T', phi> = -<T, phi'>.
Tempered distributions (dual of the Schwartz space) are the natural setting for the Fourier transform.
Distributions provide rigorous foundations for the differentiation of non-smooth functions and the analysis of PDEs.

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 6
BookBrezis, H. — Functional Analysis, Sobolev Spaces and PDEs, Chapter 1
WebsiteWikipedia: Distribution (mathematics)