Chapter 1

Distributions

The concept of distribution generalises and extends the concept of function.

A distribution is basically defined by its action on a set of test functions.

It is indeed a linear functional T

T : F → C φ(x) → T (φ) . (1.1)

from a space of functions F of real variable, called the test functions, to C. For example, to any integrable real function f(x) we can associate the linear

functional Tf : F → R defined as

φ(x)→ Tf (φ) = ∫∞ −∞ dx f(x)φ(x) . (1.2)

The idea is that, knowing the value of the integral Tf (φ) on a large enough

number of test functions φ, we can reconstruct the original function f . Al-

lowing for general linear functionals we extend the concept of function.

We can define operations on distributions even in cases where they are

not well defined on functions. For example, we will be able to define the

derivative of a discontinuous function when it is seen as a distribution. In

the world of distributions, everything is infinitely differentiable and limits

always commute with other operations. The distributions are also called

generalized functions.

1

2 CHAPTER 1. DISTRIBUTIONS

1.1 Test functions

In this section we discuss the space F of test functions and its properties. A distribution will be a linear functional on F . For simplicity, we consider test functions of real variable. We want to choose test functions φ ∈ F that are as smooth as possible, in particular that are infinitely differentiable,

φ ∈ C∞(R). Moreover, in order to guarantee the existence of the integral in (1.2) we require that φ vanishes sufficiently rapidly at infinity. There are two

canonical choices for F , the space of bump functions D(R) and the space of rapidly decreasing functions S(R), which we now discuss.

1.1.1 The space D(R)

The support of a function of real variable is the closure of the set of points

where the function is non-zero

supp(f) = {x ∈ R | f(x) 6= 0} (1.3)

A function with compact support is a function whose support is compact (a

closed and bounded subset of R). We call D(R) the space of real functions of real variable that are infinitely

differentiable and with compact support,

D(R) = {φ(x) |φ ∈ C∞(R) with compact support} (1.4)

D(R) is obviously a vector space. It is also a topological space, but the notion of limit that is useful for the theory of distribution is not induced by

any norm or metric. For our purposes, it is enough to define the notion of

convergence of a sequence in the topology of D(R). We say that the sequence {φn ∈ D(R)} converges to the function φ ∈ D(R) if the following conditions are satisfied,

• it exists a bounded interval Î ⊂ R which contains the supports of all the functions φn and φ. In other words, it exists a real number R such

that

φn(x) = 0 and φ(x) = 0 ∀ |x| ≥ R ; (1.5)

1.1. TEST FUNCTIONS 3

• the sequence of the p-th derivatives of φn(x) converges uniformly to the p-th derivative of φ(x)

sup x∈R

∣∣∣∣ dpdxpφn(x)− dpdxpφ(x) ∣∣∣∣ n−→∞−→ 0 ∀ p = 0, 1, . . . . (1.6)

In particular, φn(x) converges uniformly to φ(x).

The generalization of the space D(R) to the case of complex-valued func- tions or of functions defined in Rn is straightforward.

Example 1.1. The function

f(x) =

{ e − 1

1−x2 |x| < 1 0 |x| ≥ 1

(1.7)

belongs to D(R) since it is zero outside the interval [−1, 1], infinitely differ- entiable in (−1, 1) and with vanishing derivatives of all orders in x = ±1. Notice that the function is C∞(R) but not analytic. The Taylor series in x = ±1 is identically zero since all derivatives are zero, but the function is not zero in the neighborhood of x = ±1. All the functions in D(R) are necessarily non analytic.

1.1.2 The space S(R)

A function f of real variable is called a rapidly decreasing function, or a

Schwarz function, if it is infinitely differentiable on R, f ∈ C∞(R), and

xp dq

dxq f(x)

x→±∞−→ 0 ∀ p, q = 0, 1, . . . . (1.8)

In other words, f is infinitely differentiable and f and all its derivatives

vanish at infinity more rapidly than any polynomial.

The vector space of rapidly decreasing functions is denoted

S(R) = {φ(x) |φ ∈ C∞(R) and rapidly decreasing} . (1.9)

We define a notion of convergence in S(R) as follows. We say that the sequence of functions {φn ∈ S(R)} converges to the function φ ∈ S(R) if xp d

q

dxq φn(x) converges uniformly to x

p dq

dxq φ(x) for all p and q

sup x∈R

∣∣∣∣xp dqdxqφn(x)− xp dqdxqφ(x) ∣∣∣∣ n−→∞−→ 0 ∀ p, q = 0, 1, . . . . (1.10)

4 CHAPTER 1. DISTRIBUTIONS

Example 1.2. The functions

φ(x) = P (x)e−x 2

, φ(x) = sinx e−x 2

, (1.11)

where P (x) is an arbitrary polynomial, are elements of S(R). On the other hand, the functions

φ(x) = 1

1 + x2 , φ(x) = e−|x| (1.12)

are not, for the first vanishes as an inverse polynomial at infinity and the

second, while vanishing faster than any polynomial at infinity, is not C∞(R).

Example 1.3. Since every function in S(R) is square integrable, S(R) is a vector subspace of L2(R). S(R) is dense in L2(R). In fact, any element of L2(R) is the limit of a sequence of elements of S(R). To see this, it is enough to expand f ∈ L2(R) in the Hermite basis, f =

∑∞ n=0 cnun, where

un(x) ∼ Hn(x)e−x 2

are obviously rapidly decreasing functions.

1.2 Distributions

We call distribution a functional T : F → C

φ(x) → T (φ) . (1.13)

where F = D(R) or F = S(R), with the properties

• T is linear: T (λ1φ1 + λ2φ2) = λ1T (φ1) + λ2T (φ2) , (1.14)

for all λ1, λ2 ∈ C and φ1, φ2 ∈ F ;

• T is continuous in the following sense: if the sequence φn(x) converges to φ(x) in F , limn→∞ φn(x) = φ(x), then

lim n→∞

T (φn) = T (φ) . (1.15)

1.2. DISTRIBUTIONS 5

It is also customary to use the notation

(T, φ) ≡ T (φ) (1.16)

for the action of a distribution on a test function.

In the theory of distributions, the functions φ in D(R) and S(R) are called test functions. The functionals T acting on F = D(R) are simply called distributions and form a space denoted D′(R). The functionals T acting on F = S(R) are called tempered distributions and form a space denoted S ′(R).

We say that two distributions, T1 et T2, are equal if they act in the same

way on all test functions

T1(φ) = T2(φ) ∀φ ∈ D(R) (S(R)) . (1.17)

The spaces of distributions D′(R) et S ′(R) are vector spaces if we define sum and multiplication by scalars by

(T1 + T2)(φ) = T1(φ) + T2(φ) , (1.18)

(λT )(φ) = λT (φ) , (1.19)

for all φ ∈ D′(R) (S ′(R)) and all complex numbers λ. The functions with compact support are a subset of the rapidly decreasing

functions

D(R) ⊂ S(R) . (1.20)

This implies the opposite inclusion for the space of distributions

S ′(R) ⊂ D′(R) . (1.21)

since every functional that is defined on all the functions in S(R) is a fortiori defined on the functions in D(R).

It is straightforward to extend the previous definitions to the case of

multi-variable distributions.

In the following we discuss the case of distributions in D′(R), since all properties we will introduce in this chapter are also valid for tempered dis-

tributions. More sophisticated operations, like the Fourier transform, can be

only defined for tempered distributions.

6 CHAPTER 1. DISTRIBUTIONS

1.2.1 Regular distributions

We say that a function f(x) is locally integrable, and we write f ∈ L1loc(R), if its integral on any compact subset K of R is finite∫

K

|f(x)|dx

1.2. DISTRIBUTIONS 7

The presence of the complex conjugate of f in the definition (1.23) can

be startling but it is dictated by convenience. In particular, with this choice,

Tf (φ) can be formally written as the scalar product (f, φ) of f with φ. This

definition also explains the notation (1.16) which is often used for the action

of a distribution on a test function.

Example 1.4. The function f(x) = sign(x) is not continuous, it is not

differentiable and it is not integrable on R. It is however integrable on any finite interval [a, b] and thus defines the distribution

Tf (φ) =

∫ ∞ −∞

dx f(x)φ(x) = − ∫ 0 −∞

dxφ(x) +

∫ ∞ 0

dxφ(x) .

Since f has an algebraic growth, Tf is a tempered distribution. The previous

expression is indeed finite for all φ ∈ S(R).

Example 1.5. The Heaviside function

θ(x) =

{ 1 x ≥ 0 0 x < 0 ,

(1.24)

is similarly associated to the tempered distribution

Tθ(φ) =

∫ ∞ −∞

dx θ(x)φ(x) =

∫ ∞ 0

dxφ(x) .

Example 1.6. By identifying locally integrable functions with the associ-

ated distribution, we have the following inclusion: C∞(R) ⊂ D′(R). The elementary functions sin(x), cos(x), log(x), P (x), where P (x) is a polynomial

are tempered distr