This chapter is concerned primarily with deriving certain theorems on convergence which will be required in subsequent chapters. It is expected that most readers will be familiar with the notions involved so that much of the material is given in a condensed manner. However, an attempt has been made to make the chapter self-contained. Some readers may find the chapter a helpful introduction to the ideas and terminology employed in other books on generalised functions. The reader who does not have a good background in analysis is strongly advised to go straight to Chapter 2 and to just refer to Chapter 1 for the theorems that are needed.

Preliminary definitions

A set is a collection of elements. A set containing no elements is called a null or empty set. There is no restriction on what an element is: it may be a number or a point or a vector and so on. Usually we shall call the elements points and take all sets to be sets of points in a fixed non-empty set Ω, which will be called a space. The empty set will be denoted by ∅ and the capitals A,B,… will denote sets. If ω is a point of A, we write ω∈A ; if ω is not a point of A, we write ω∉A.

For some years I have been offering lectures on generalised functions to undergraduate and postgraduate students. The undergraduate course was based originally on M.J. Lighthill's stimulating book An Introduction to Fourier Analysis and Generalised Functions which contains a simplified version of a theory evolved by G. Temple to make generalised functions more readily accessible and intelligible to students. It is an approach to the theory of generalised functions which permits early introduction in student courses while retaining the power and practical utility of the methods. At the same time it can be developed so as to include the more advanced aspects appropriate to postgraduate instruction. This book has grown from the courses which I have given expounding the ramifications of the Lighthill–Temple theory to various groups of students. It is arranged so that sections can be chosen relevant to any level of course.

Much of the material was originally contained in my book Generalised Functions, published by McGraw-Hill in 1966, but this book differs from the earlier version in several major respects. The treatment and definitions of the special generalised functions which are powers of the single variable x have been completely changed as well as those of the powers of the radial distance in higher dimensions.

Multiplication

It is desirable to define the product of two generalised functions g1 and g2 in such a way that it agrees with the conventional product g1g2 when g1 and g2 are conventional functions. However, it is not possible to give a meaning to the product which is applicable to all generalised functions. One reason is that, if g1∈K1 and g2∈K1, it is not necessarily true that g1g2∈K1 (e.g. g1(x) = l/|x|¼, g2(x) = l/|x|¾) and so the conventional product g1g2 may not give rise to a generalised function. Another reason is that if {γ1m} and {γ2m} are regular sequences the sequence {γ1m γ2m} need not be regular. Therefore restrictions must be placed on g1 and g2 in order that their product may be defined. We have already seen that, if g1 is limited to the class of fairly good functions, the product g1g2 can be satisfactorily specified for any generalised function g2 (Definition 3.8). If we want g1 to be less constrained then we must impose some conditions on g2.

We start by defining multiplication in such a way as to include what is customarily meant by a product in so far as this is possible.

Rotation of axes

One significant way in which calculations in Rn differ from those in R1 is that the axes can be chosen fairly freely. Furthermore one often wishes to calculate multiple integrals by means of spherical polars or cylindrical polars instead of Cartesians. In order to provide similar facilities for generalised functions it is necessary to see what effect a change of variable has.

We commence by examining the effect of choosing a different set of Cartesian axes with the same origin. (A change of origin without alteration of the directions of the axes is covered by Definition 7.9.) Regarding x as a column matrix we can obtain any other Cartesian set with the same origin by a linear transformation y = Lx where L is an orthogonal matrix, i.e. LTL = I where LT is the transpose of L. The determinant of L, det L, is either 1 or – 1. If det L = 1 the new axes are derived from the old by a proper rotation; if det L = – 1 an improper rotation, i.e. a proper rotation together with a reflection, is involved.

Definition 8.1.If {γm} is a regular sequence defining g, the sequence {γm(Lx)} is regular and defines a generalised function which is denoted by g (Lx).