To begin with it is necessary to discuss the meaning of ϕf where f ∈G∗ and ϕ ∈G. What should it mean?
First suppose f ∈ Lp
or measurable with polynomial growth. Then
also has these properties.
Hence, it should be the case that ϕf
This motivates the following
Definition 10.5.24 Let T ∈G∗ and let ϕ ∈G. Then ϕT ≡ Tϕ ∈G∗ will be defined by
The next topic is that of convolution. It was just shown that
whenever f ∈ L2
so the same definition is retained in the general case because it makes
perfect sense and agrees with the earlier definition.
Definition 10.5.25 Let f ∈G∗ and let ϕ ∈G. Then define the convolution of f with an element of G as
There is an obvious question. With this definition, is it true that F−1
Theorem 10.5.26 Let f ∈G∗ and let ϕ ∈G.
Proof: Note that 10.20 follows from Definition 10.5.25 and both assertions hold for f ∈G. Consider
10.21. Here is a simple formula involving a pair of functions in G.
Now for ψ ∈G
The last line follows from the following.
From 10.23 and 10.22 , since ψ was arbitrary,
which shows 10.21. ■