Distribution theory

DISTRIBUTION THEORY



GENERALIZED FUNCTIONS

NOTES

Ivan F Wilde

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. The spaces S and S ′

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3. The spaces D and D′

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4. The Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5. Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6. Fourier-Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7. Structure Theorem for Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8. Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 1

Introduction

The so-called Dirac delta function (on R) obeys δ(x) = 0 for all x 6= 0 but

is supposed to satisfy R ∞

−∞ δ(x) dx = 1. (The δ function on R

d

is similarly

described.) Consequently,

Z ∞

−∞

f(x) δ(x) dx =

Z ∞

−∞

(f(x) − f(0)) δ(x) dx + f(0) Z ∞

−∞

δ(x) dx = f(0)

because (f(x)−f(0)) δ(x) ≡ 0 on R. Moreover, if H(x) denotes the Heaviside

step-function

H(x) = (

0, x < 0

1, x ≥ 0 ,

then we see that H′ = δ, in the following sense. If f vanishes at infinity,

then integration by parts gives

Z ∞

−∞

f(x) H′

(x) dx =

f(x) H(x)

∞

−∞

−

Z ∞

−∞

f

′

(x) H(x) dx

= −

Z ∞

−∞

f

′

(x) H(x) dx

= −

Z ∞

0

f

′

(x) dx

= −

f(x)

∞

0

= f(0)

=

Z ∞

−∞

f(x) δ(x) dx .

Of course, there is no such function δ with these properties and we cannot

interpret R ∞

−∞ f(x) δ(x) dx as an integral in the usual sense. The δ function

is thought of as a generalized function.

However, what does make sense is the assignment f 7→ f(0) = hδ, fi, say.

Clearly hδ, αf + βgi = αhδ, fi + βhδ, gi for functions f, g and constants α

and β. In other words, the Dirac delta-function can be defined not as a

function but as a functional on a suitable linear space of functions. The

development of this is the theory of distributions of Laurent Schwartz.

1

2 Chapter 1

One might think of δ(x) as a kind of limit of some sequence of functions

whose graphs become very tall and thin, as indicated in the figure.

Figure 1.1: Approximation to the δ-function.

The Dirac δ function can be thought of as a kind of continuous version of

the discrete Kronecker δ and is used in quantum mechanics to express the

orthogonality properties of non square-integrable wave functions.

Distributions play a crucial rˆole in the study of (partial) differential equa￾tions. As an introductory remark, consider the equations

∂

2u

∂x∂y

= 0 and ∂

2u

∂y∂x

= 0 .

These “ought” to be equivalent. However, the first holds for any function u

independent of y, whereas the second may not make any sense. By (formally)

integrating by parts twice and discarding the surface terms, we get

Z

ϕ

∂

2u

∂x∂y

dx dy =

Z

u

∂

2ϕ

∂x∂y

dx dy .

So we might interpret ∂

2u

∂x∂y

= 0 as

Z

u

∂

2ϕ

∂x∂y

dx dy = 0

for all ϕ in some suitably chosen set of smooth functions. The point is that

this makes sense for non-differentiable u and, since ϕ is supposed smooth,

Z

u

∂

2ϕ

∂x∂y

dx dy =

Z

u

∂

2ϕ

∂y∂x

dx dy ,

that is, ∂

2u

∂x∂y

=

∂

2u

∂y∂x

in a certain weak sense. These then are weak or

distributional derivatives.

Finally, we note that distributions also play a central rˆole in quantum field

theory, where quantum fields are defined as operator-valued distributions.

ifwilde Notes

Introduction 3

Bibliography

I. M. Gelfand and G. E. Shilov, Generalized Functions, Academic Press,

Inc., 1964.

J. Lighthill, Introduction to Fourier Analysis and Generalized Functions,

Cambridge University Press, 1958.

M. Reed and B. Simon, Methods of Mathematical Physics, Volume II,

Academic Press, Inc., 1975.

W. Rudin, Functional Analysis, McGraw-Hill, Inc,. 1973.

L. Schwartz, Th´eorie des distributions, Hermann & Cie, Paris, 1966.

The proper approach to the theory is via topological vector spaces—see

Rudin’s excellent book for the development along these lines, as well as

much background material. The approach via approximating sequences of

functions is to be found in Lighthill’s book.

For the preparation of these lecture notes, extensive use was made of the

books of Rudin and Reed and Simon.

November 9, 2005

4 Chapter 1

ifwilde Notes