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.