Distributions_Generalized_Functions-Ivan_Wilde.pdf

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 Dierential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 1

Introduction

−∞ (x) dx = 1. (The function on R d is similarly

described.) Consequently,

∞

f(x) (x) dx =

(f(x)−f(0)) (x) dx + f(0)

(x) dx = f(0)

−∞

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

step-function

(

0, x < 0

1, x≥0 ,

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

then integration by parts gives

H(x) =

∞

−∞

f(x) H ′ (x) dx =

f(x) H(x)

−

f ′ (x) H(x) dx

−∞

∞

=−

f ′ (x) H(x) dx

−∞

∞

=−

f ′ (x) dx

f(x)

∞

=−

= f(0)

∞

f(x) (x) dx.

−∞

∞

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

is thought of as a generalized function.

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

Clearly , f + g= ,f+ ,gfor functions f, g and constants

and . In other words, the Dirac delta-function can be dened 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.

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

is supposed to satisfy

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

interpret

Chapter 1

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

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

Figure 1.1: Approximation to the -function.

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

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

orthogonality properties of non square-integrable wave functions.

Distributions play a crucial role in the study of (partial) dierential equa-

tions. As an introductory remark, consider the equations

@ 2 u

@ x @ y = 0

and

@ 2 u

@ y @ x = 0 .

These “ought” to be equivalent. However, the rst 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

' @ 2 u

u @ 2 '

@ x @ y dxdy =

@ x @ y dxdy.

So we might interpret @ 2 u

@ x @ y = 0 as

u @ 2 '

@ x @ y dxdy = 0

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

this makes sense for non-dierentiable u and, since ' is supposed smooth,

u @ 2 '

@ x @ y dxdy =

@ y @ x dxdy,

@ y @ x in a certain weak sense. These then are weak or

distributional derivatives.

Finally, we note that distributions also play a central role in quantum eld

theory, where quantum elds are dened as operator-valued distributions.

@ 2 u

@ x @ y

= @ 2 u

ifwilde

Notes

that is,

Introduction

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, Theorie 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

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