Distributions and Generalized Functions | Malone Math

The Central Idea

A distribution is not primarily defined by assigning a numerical value to each point. Instead, it assigns a number to each smooth test function.

The central shift is

\[ \boxed{ \text{pointwise values} \longrightarrow \text{action on test functions} }. \]

This change allows differentiation to be extended far beyond the class of classically differentiable functions.

Why Generalized Functions Are Needed

Several familiar mathematical objects resist a purely classical treatment.

A point mass should be concentrated at one point but have nonzero total mass.
A step function has a jump and no classical derivative at the jump.
A differential equation may contain an impulse or point source.
A limit of smooth functions may become discontinuous or singular.
A weak solution may possess derivatives only in an integrated sense.

Distribution theory gives all of these objects a consistent analytical framework.

Test Functions

Let \(\Omega\subseteq\mathbb{R}^n\) be open. A test function on \(\Omega\) is a smooth function

\[ \varphi:\Omega\to\mathbb{R} \]

or \(\mathbb{C}\) that has compact support inside \(\Omega\).

The standard space of test functions is denoted

\[ \boxed{ C_c^\infty(\Omega) } \]

or, commonly,

\[ \boxed{ \mathcal{D}(\Omega) }. \]

Smoothness

The symbol

\[ C^\infty \]

means that derivatives of every order exist and are continuous.

Thus a test function can be differentiated repeatedly without leaving the class of smooth functions.

Compact Support

The support of a function \(\varphi\) is the closure of the set where it is nonzero:

\[ \operatorname{supp}\varphi = \overline{\{x:\varphi(x)\ne0\}}. \]

A test function has compact support, meaning that all of its nonzero behavior occurs inside some compact subset of \(\Omega\).

Why Test Functions Vanish Away from a Compact Region

Compact support removes boundary and infinity complications from many integration-by-parts arguments.

If \(\varphi\in C_c^\infty(\mathbb{R})\), then \(\varphi\) and all of its derivatives vanish outside a bounded interval.

This makes test functions ideal probes for generalized derivatives.

What Is a Distribution?

A distribution on \(\Omega\) is a continuous linear functional

\[ T:\mathcal{D}(\Omega)\to\mathbb{R} \]

or \(\mathbb{C}\).

We write the action of \(T\) on a test function \(\varphi\) as

\[ \boxed{ \langle T,\varphi\rangle }. \]

Linearity of a Distribution

For test functions \(\varphi,\psi\) and scalars \(a,b\),

\[ \boxed{ \langle T,a\varphi+b\psi\rangle = a\langle T,\varphi\rangle + b\langle T,\psi\rangle }. \]

Distribution theory therefore fits naturally into the operator and functional framework developed in the previous lesson.

Continuity of a Distribution

The continuity condition on distributions is adapted to the topology of the test-function space.

Informally, if a sequence of test functions and all of their derivatives approach zero uniformly while their supports remain inside one fixed compact set, then

\[ \langle T,\varphi_n\rangle\to0. \]

This condition rules out arbitrary discontinuous linear functionals.

Regular Distributions

Ordinary locally integrable functions define distributions.

\[ f\in L_{\mathrm{loc}}^1(\Omega), \]

define

\[ \boxed{ \langle T_f,\varphi\rangle = \int_\Omega f(x)\varphi(x)\,dx }. \]

The resulting \(T_f\) is called the regular distribution associated with \(f\).

Local Integrability

A function belongs to

\[ L_{\mathrm{loc}}^1(\Omega) \]

if it is integrable on every compact subset of \(\Omega\).

This condition is weaker than requiring

\[ \int_\Omega|f|<\infty. \]

Test functions have compact support, so only local integrability is needed.

Example: The Constant Function as a Distribution

Let

\[ f(x)=1. \]

Then

\[ \boxed{ \langle T_1,\varphi\rangle = \int_{\mathbb{R}}\varphi(x)\,dx }. \]

This is an ordinary function viewed through the distributional framework.

Example: A Polynomial as a Distribution

Let

\[ f(x)=x^2. \]

Then

\[ \langle T_f,\varphi\rangle = \int_{\mathbb{R}}x^2\varphi(x)\,dx. \]

Since test functions have compact support, the integral is finite even though \(x^2\notin L^1(\mathbb{R})\).

Distributions Are Larger Than Ordinary Functions

Every locally integrable function determines a distribution, but not every distribution arises from a locally integrable function.

The most important example is the Dirac delta.

The Dirac Delta

Fix \(a\in\Omega\). The Dirac delta at \(a\) is the distribution defined by

\[ \boxed{ \langle\delta_a,\varphi\rangle = \varphi(a) }. \]

It extracts the value of the test function at one point.

Why the Dirac Delta Is Not an Ordinary Function

Informally, the delta is often described as being zero away from \(a\) and infinitely concentrated at \(a\), with total mass \(1\).

No ordinary locally integrable function has exactly that behavior.

The rigorous object is the functional

\[ \varphi\mapsto\varphi(a). \]

Delta and Point Mass

The Dirac delta is closely related to the Dirac measure.

For a test function \(\varphi\),

\[ \int\varphi\,d\delta_a = \varphi(a). \]

Thus the measure-theoretic point mass and distributional delta encode the same evaluation behavior in compatible frameworks.

Distributional Derivatives

Let \(T\) be a distribution on an open interval. Its derivative \(T'\) is defined by

\[ \boxed{ \langle T',\varphi\rangle = -\langle T,\varphi'\rangle }. \]

The minus sign comes from integration by parts.

Why the Definition Matches Classical Differentiation

Suppose \(f\) is classically differentiable and sufficiently regular. Then

\[ \langle T_{f'},\varphi\rangle = \int f'(x)\varphi(x)\,dx. \]

Integration by parts gives

\[ \int f'(x)\varphi(x)\,dx = -\int f(x)\varphi'(x)\,dx, \]

because the compact support of \(\varphi\) removes boundary terms.

Therefore,

\[ \boxed{ (T_f)'=T_{f'} }. \]

Every Distribution Can Be Differentiated

One of the major advantages of distributions is that differentiation is always defined.

Even if \(T\) does not come from a classically differentiable function, the formula

\[ \langle T',\varphi\rangle = -\langle T,\varphi'\rangle \]

still makes sense.

Higher Distributional Derivatives

Repeating the definition gives

\[ \boxed{ \langle T^{(k)},\varphi\rangle = (-1)^k \langle T,\varphi^{(k)}\rangle }. \]

Thus every distribution has derivatives of every order.

The Heaviside Step Function

Define

\[ H(x) = \begin{cases} 0, & x<0,\\ 1, & x>0. \end{cases} \]

The value assigned at \(x=0\) does not affect the associated regular distribution.

Derivative of the Heaviside Function

For a test function \(\varphi\),

\[ \langle H',\varphi\rangle = -\int_0^\infty\varphi'(x)\,dx. \]

Since \(\varphi\) vanishes for sufficiently large \(x\),

\[ -\int_0^\infty\varphi'(x)\,dx = \varphi(0). \]

Therefore,

\[ \boxed{ H'=\delta_0 }. \]

What the Heaviside Example Means

Classically, \(H'\) is zero away from the jump and undefined at the jump.

Distributionally, the entire change is concentrated at the discontinuity as a Dirac delta.

This is a precise mathematical form of the idea that an instantaneous jump produces an impulse.

Derivative of the Dirac Delta

Using the definition,

\[ \langle\delta_0',\varphi\rangle = -\langle\delta_0,\varphi'\rangle. \]

Therefore,

\[ \boxed{ \langle\delta_0',\varphi\rangle = -\varphi'(0) }. \]

Higher Derivatives of the Delta

More generally,

\[ \boxed{ \langle\delta_0^{(k)},\varphi\rangle = (-1)^k\varphi^{(k)}(0) }. \]

These derivatives are legitimate distributions even though they are not ordinary functions.

Multiplication by a Smooth Function

Let \(T\) be a distribution and let \(a\in C^\infty(\Omega)\). Define

\[ \boxed{ \langle aT,\varphi\rangle = \langle T,a\varphi\rangle }. \]

Multiplication by smooth functions is always well defined in distribution theory.

Example: Multiplying the Delta

Let \(a\) be smooth. Then

\[ \langle a\delta_0,\varphi\rangle = \langle\delta_0,a\varphi\rangle = a(0)\varphi(0). \]

Hence

\[ \boxed{ a\delta_0 = a(0)\delta_0 }. \]

The Product Rule

If \(a\) is smooth and \(T\) is a distribution, then

\[ \boxed{ (aT)' = a'T+aT' }. \]

The familiar product rule survives in the distributional setting when the multiplier is smooth.

Support of a Distribution

A distribution \(T\) is said to vanish on an open set \(U\) if

\[ \langle T,\varphi\rangle=0 \]

for every test function \(\varphi\) supported inside \(U\).

The support of \(T\) is the complement of the largest open set on which \(T\) vanishes.

Support of the Dirac Delta

The Dirac delta at \(a\) responds only to the value of a test function at \(a\).

Therefore,

\[ \boxed{ \operatorname{supp}\delta_a=\{a\} }. \]

Weak Derivatives

Let \(u\in L_{\mathrm{loc}}^1(\Omega)\). A locally integrable function \(v\) is called a weak derivative of \(u\) in one dimension if

\[ \boxed{ \int_\Omega u(x)\varphi'(x)\,dx = -\int_\Omega v(x)\varphi(x)\,dx } \]

for every test function \(\varphi\).

Distributional versus Weak Derivatives

Every locally integrable function defines a distribution.

Its distributional derivative always exists as a distribution.

If that derivative happens to be represented by another locally integrable function \(v\), then \(v\) is the weak derivative of \(u\).

Classical Derivatives Are Weak Derivatives

If \(u\) is classically differentiable with suitable local regularity, then its weak derivative agrees almost everywhere with the classical derivative.

Thus weak differentiation extends classical differentiation rather than replacing it.

Example: Absolute Value

Consider

\[ u(x)=|x|. \]

The classical derivative fails at \(0\), but away from \(0\),

\[ u'(x) = \begin{cases} -1, & x<0,\\ 1, & x>0. \end{cases} \]

This sign function is the weak derivative of \(|x|\).

A Second Derivative of \(|x|\)

Since

\[ \frac{d}{dx}|x| = \operatorname{sgn}(x) \]

weakly, and the sign function jumps by \(2\) at \(0\),

\[ \boxed{ \frac{d^2}{dx^2}|x| = 2\delta_0 } \]

in the distributional sense.

Weak Solutions of Differential Equations

A classical solution satisfies a differential equation pointwise.

A weak solution satisfies an equivalent integrated identity after derivatives have been transferred onto test functions.

This makes it possible to solve equations when classical derivatives do not exist everywhere.

Example: Weak Form of a First Derivative Equation

Suppose formally

\[ u'=f. \]

Multiplying by a test function and integrating gives

\[ \int u'\varphi = \int f\varphi. \]

Integration by parts yields

\[ \boxed{ -\int u\varphi' = \int f\varphi }. \]

This identity can remain meaningful even when \(u'\) does not exist classically.

Point Sources in Differential Equations

A point source at \(a\) can be modeled by

\[ \delta_a. \]

For example, an equation of the form

\[ Lu=\delta_a \]

describes a response generated by a unit source concentrated at one point.

Fundamental solutions of differential operators are often defined through such equations.

Convolution with a Distribution

When one factor has suitable support or regularity, convolution can be extended from functions to distributions.

The Dirac delta behaves as an identity:

\[ \boxed{ \delta_0*f=f }. \]

More generally,

\[ \delta_a*f \]

produces a translated version of \(f\).

Convolution and Differentiation

Under appropriate conditions,

\[ \boxed{ (T*S)' = T'*S = T*S' }. \]

This makes convolution especially useful for differential equations and smoothing arguments.

Mollifiers

A mollifier is a smooth, compactly supported function used to smooth rough functions or distributions.

A typical family has the form

\[ \rho_\varepsilon(x) = \frac1{\varepsilon^n} \rho\left(\frac{x}{\varepsilon}\right), \]

where

\[ \rho\ge0, \qquad \int_{\mathbb{R}^n}\rho=1. \]

Approximation by Mollification

If \(T\) is a distribution, convolution with a mollifier produces a smooth approximation

\[ T*\rho_\varepsilon. \]

\[ \varepsilon\to0, \]

these smooth objects approach \(T\) in the distributional sense under standard conditions.

Distributional Convergence

A sequence of distributions \(T_n\) converges to \(T\) if

\[ \boxed{ \langle T_n,\varphi\rangle \to \langle T,\varphi\rangle } \]

for every test function \(\varphi\).

Convergence is therefore tested through all smooth compactly supported probes.

Example: Approximate Identities Converging to Delta

Suppose \(\rho\) is smooth, compactly supported, and

\[ \int\rho=1. \]

Define

\[ \rho_\varepsilon(x) = \frac1\varepsilon \rho\left(\frac{x}{\varepsilon}\right). \]

Then, in the distributional sense,

\[ \boxed{ \rho_\varepsilon \to \delta_0 }. \]

Why the Approximate Delta Works

For a test function \(\varphi\),

\[ \int \rho_\varepsilon(x)\varphi(x)\,dx = \int \rho(y)\varphi(\varepsilon y)\,dy. \]

As \(\varepsilon\to0\),

\[ \varphi(\varepsilon y)\to\varphi(0). \]

Since \(\int\rho=1\), the limit is

\[ \varphi(0) = \langle\delta_0,\varphi\rangle. \]

The Space of Distributions

The space of all distributions on \(\Omega\) is denoted

\[ \boxed{ \mathcal{D}'(\Omega) }. \]

The prime indicates a continuous dual space:

\[ \mathcal{D}'(\Omega) = \bigl(\mathcal{D}(\Omega)\bigr)'. \]

Schwartz Functions

For Fourier analysis on \(\mathbb{R}^n\), another test-function space is particularly useful.

The Schwartz space

\[ \mathcal{S}(\mathbb{R}^n) \]

consists of smooth functions whose derivatives decay faster than every inverse polynomial.

Rapid Decay

A function \(\varphi\) belongs to \(\mathcal{S}(\mathbb{R}^n)\) when for every pair of multi-indices \(\alpha,\beta\),

\[ \sup_{x\in\mathbb{R}^n} |x^\alpha D^\beta\varphi(x)| < \infty. \]

This condition controls both smoothness and decay at infinity.

Tempered Distributions

A tempered distribution is a continuous linear functional on the Schwartz space:

\[ \boxed{ \mathcal{S}'(\mathbb{R}^n) }. \]

Tempered distributions are especially important because the Fourier transform extends naturally to them.

Examples of Tempered Distributions

Important tempered distributions include:

all rapidly decaying integrable functions;
all polynomially growing functions that are locally integrable;
the Dirac delta;
derivatives of the Dirac delta;
many principal-value distributions.

Fourier Transform of the Delta

Under a common Fourier-transform convention,

\[ \widehat{\delta_0}=1. \]

Conversely, the Fourier transform of the constant function is a multiple of the delta, with the exact constant depending on the transform convention.

Distribution theory makes these identities rigorous.

Differentiation and Fourier Transform

For tempered distributions, differentiation in physical space corresponds to multiplication by frequency in Fourier space.

Under a standard convention,

\[ \boxed{ \widehat{T'} (\xi) = i\xi\,\widehat{T}(\xi) }. \]

This identity is a cornerstone of transform methods for differential equations.

Principal-Value Distributions

Some singular expressions that are not ordinary integrable functions can still define distributions through symmetric limiting procedures.

A standard example is the principal value of

\[ \frac1x. \]

It is defined by

\[ \left\langle \operatorname{p.v.}\frac1x, \varphi \right\rangle = \lim_{\varepsilon\to0^+} \int_{|x|>\varepsilon} \frac{\varphi(x)}{x}\,dx, \]

when interpreted in the appropriate symmetric sense.

Distributions Are Not Arbitrary Symbolic Objects

Although distribution notation may resemble formal algebra, every valid operation must be justified through the action on test functions.

Some operations extend cleanly, such as:

addition;
scalar multiplication;
differentiation;
multiplication by smooth functions;
convolution under suitable hypotheses.

Products of Distributions Are Delicate

There is no general multiplication operation

\[ T\cdot S \]

defined for arbitrary distributions.

Expressions such as

\[ \delta_0^2 \]

are not defined in ordinary distribution theory without additional structure.

Common Error: Treating Delta as an Ordinary Function

The Dirac delta is not a classical function with an infinite value at one point.

Its rigorous definition is

\[ \boxed{ \langle\delta_a,\varphi\rangle=\varphi(a) }. \]

Common Error: Forgetting the Minus Sign in Distributional Differentiation

The derivative is defined by

\[ \boxed{ \langle T',\varphi\rangle = -\langle T,\varphi'\rangle }. \]

The minus sign comes from integration by parts and is essential.

Common Error: Assuming Every Distribution Is a Function

Locally integrable functions produce regular distributions, but singular distributions such as \(\delta_0\) cannot be represented by ordinary locally integrable functions.

Common Error: Confusing Weak and Classical Derivatives

A classical derivative is defined pointwise.

A weak derivative is defined through an integral identity against every test function.

They agree when sufficient regularity is present, but weak derivatives exist in many cases where classical derivatives do not.

Common Error: Multiplying Arbitrary Distributions

Distribution theory supports multiplication by smooth functions, but not a general product of two arbitrary distributions.

A formal expression is not automatically a mathematically defined distribution.

Common Error: Ignoring the Test-Function Space

A distribution is defined relative to a specific class of test functions.

Ordinary distributions act on

\[ \mathcal{D}(\Omega), \]

while tempered distributions act on

\[ \mathcal{S}(\mathbb{R}^n). \]

The choice of test-function space determines which generalized objects are admitted and which operations are natural.

A Reliable Procedure for Turning a Function into a Distribution

Verify local integrability of \(f\).
Choose a test function \(\varphi\in C_c^\infty(\Omega)\).
Define \[ \langle T_f,\varphi\rangle = \int_\Omega f\varphi. \]
Use linearity of the integral to verify linearity.
Use local integrability and compact support to justify finiteness.

A Reliable Procedure for Finding a Distributional Derivative

Begin with \[ \langle T',\varphi\rangle = -\langle T,\varphi'\rangle. \]
Replace \(T\) by its defining action.
Use integration by parts or interval decomposition when appropriate.
Collect boundary or jump contributions.
Identify the resulting functional as a regular distribution, delta term, or combination.

A Reliable Procedure for Checking a Weak Derivative

Choose an arbitrary test function \(\varphi\).
Compute \[ \int u\varphi'. \]
Integrate by parts on regions where \(u\) is classically regular.
Track any jump or boundary contributions carefully.
Compare with \[ -\int v\varphi. \]
If the identity holds for every test function, \(v\) is the weak derivative.

A Reliable Procedure for Proving Distributional Convergence

Fix an arbitrary test function \(\varphi\).
Compute or estimate \(\langle T_n,\varphi\rangle\).
Take the limit as \(n\to\infty\).
Show the limit equals \(\langle T,\varphi\rangle\).
Because the argument holds for every test function, conclude \(T_n\to T\) distributionally.

Example: Derivative of a Jump

Let

\[ f(x) = \begin{cases} 0, & x<0,\\ 3, & x>0. \end{cases} \]

Then \(f=3H\) as a distribution.

Therefore,

\[ f' = 3H' = \boxed{3\delta_0}. \]

Example: A Piecewise Smooth Function

Suppose \(f\) is smooth except for a jump of size

\[ J=f(a^+)-f(a^-) \]

at \(x=a\).

Then the distributional derivative contains the ordinary derivative away from \(a\), together with the singular term

\[ \boxed{ J\delta_a }. \]

Example: Delta as a Limit of Narrow Peaks

Let \(\rho\) be smooth with

\[ \int_{\mathbb{R}}\rho(x)\,dx=1, \]

and define

\[ \rho_n(x) = n\rho(nx). \]

The functions become taller and narrower, but preserve total mass \(1\).

Distributionally,

\[ \boxed{ \rho_n\to\delta_0 }. \]

Practice 1: Regular Distribution

Let

\[ f(x)=x. \]

Write the action of the regular distribution \(T_f\) on a test function \(\varphi\).

Practice 2: Dirac Delta

Compute

\[ \langle\delta_2,\varphi\rangle \]

in terms of \(\varphi\).

Practice 3: Derivative of Delta

Compute

\[ \langle\delta_0',\varphi\rangle. \]

Practice 4: Heaviside Derivative

Show that

\[ H'=\delta_0 \]

in the distributional sense.

Practice 5: Jump of Size Five

Let

\[ f(x) = \begin{cases} 1, & x<0,\\ 6, & x>0. \end{cases} \]

Find the singular contribution to the distributional derivative.

Practice 6: Weak Derivative of \(|x|\)

Identify the weak derivative of

\[ u(x)=|x|. \]

Practice 7: Second Distributional Derivative

Find the second distributional derivative of

\[ |x|. \]

Practice 8: Multiplication by a Smooth Function

Let \(a\in C^\infty(\mathbb{R})\). Simplify

\[ a\delta_0. \]

Practice 9: Distributional Convergence

Suppose

\[ \rho_n(x)=n\rho(nx), \qquad \int\rho=1. \]

State what must be shown for

\[ \rho_n\to\delta_0 \]

in the distributional sense.

Practice 10: Identify the Error

A student says:

The Dirac delta is an ordinary function that equals zero everywhere except at \(0\), where it equals infinity.

Explain the error.

Practice Answers

Practice 1

The regular distribution associated with \(f(x)=x\) is

\[ \boxed{ \langle T_f,\varphi\rangle = \int_{\mathbb{R}}x\varphi(x)\,dx }. \]

Practice 2

By definition,

\[ \boxed{ \langle\delta_2,\varphi\rangle = \varphi(2) }. \]

Practice 3

By distributional differentiation,

\[ \langle\delta_0',\varphi\rangle = -\langle\delta_0,\varphi'\rangle. \]

Therefore,

\[ \boxed{ \langle\delta_0',\varphi\rangle = -\varphi'(0) }. \]

Practice 4

We compute

\[ \langle H',\varphi\rangle = -\int_0^\infty\varphi'(x)\,dx. \]

Since \(\varphi\) has compact support,

\[ -\int_0^\infty\varphi'(x)\,dx = \varphi(0). \]

Hence

\[ \boxed{H'=\delta_0}. \]

Practice 5

The jump size is

\[ 6-1=5. \]

Therefore the singular contribution is

\[ \boxed{5\delta_0}. \]

Practice 6

The weak derivative is

\[ \boxed{ \operatorname{sgn}(x) = \begin{cases} -1, & x<0,\\ 1, & x>0. \end{cases} }. \]

The value assigned at \(0\) is irrelevant almost everywhere.

Practice 7

The first weak derivative is \(\operatorname{sgn}(x)\), which jumps by \(2\) at the origin. Therefore,

\[ \boxed{ \frac{d^2}{dx^2}|x| = 2\delta_0 }. \]

Practice 8

For every test function,

\[ \langle a\delta_0,\varphi\rangle = a(0)\varphi(0). \]

Therefore,

\[ \boxed{ a\delta_0 = a(0)\delta_0 }. \]

Practice 9

One must show that for every test function \(\varphi\),

\[ \boxed{ \int_{\mathbb{R}} \rho_n(x)\varphi(x)\,dx \to \varphi(0) }. \]

The right side is exactly

\[ \langle\delta_0,\varphi\rangle. \]

Practice 10

The Dirac delta is not an ordinary pointwise function. The phrase “infinite at one point” is only informal intuition.

Its rigorous definition is the linear functional

\[ \boxed{ \langle\delta_0,\varphi\rangle = \varphi(0) }. \]

What to Remember

Test functions belong to

\[ \boxed{ \mathcal{D}(\Omega) = C_c^\infty(\Omega) }. \]

A distribution is a continuous linear functional on the test-function space:

\[ \boxed{ T:\mathcal{D}(\Omega)\to\mathbb{R} }. \]

Locally integrable functions define regular distributions through

\[ \boxed{ \langle T_f,\varphi\rangle = \int_\Omega f\varphi }. \]

The Dirac delta is defined by

\[ \boxed{ \langle\delta_a,\varphi\rangle = \varphi(a) }. \]

Distributional differentiation is

\[ \boxed{ \langle T',\varphi\rangle = -\langle T,\varphi'\rangle }. \]

This single definition extends differentiation to discontinuous functions, point sources, impulses, and objects that are not classical functions at all.

The governing habit is: define every generalized object through its action on test functions and justify every operation at that level.

Where We Go Next

Distributions reveal how operator theory, duality, integration, and weak differentiation fit together. They also show why infinite-dimensional spaces require structural theorems rather than only computational techniques.

The next lesson, Introductory Functional Analysis, consolidates these ideas around Banach spaces, Hilbert spaces, bounded operators, duality, weak convergence, compactness, and the major principles that govern infinite-dimensional analysis.