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 30.3.23 Let T ∈G∗ and let ϕ ∈G. Then ϕT ≡ Tϕ ∈G∗ will be defined
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 30.3.24 Let f ∈G∗ and let ϕ ∈G. Then define the convolution of f with an
element of G as follows.
There is an obvious question. With this definition, is it true that
as it was earlier?
Theorem 30.3.25 Let f ∈G∗ and let ϕ ∈G.
Proof: Note that 30.3.16 follows from Definition 30.3.24 and both assertions hold for
f ∈G. Consider 30.3.17. Here is a simple formula involving a pair of functions in
Now for ψ ∈G
The last line follows from the following.
From 30.3.19 and 30.3.18 , since ψ was arbitrary,
which shows 30.3.17. ■