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
these properties. Hence, it should be the case that ϕf
motivates the following definition.
Definition 26.2.23 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 26.2.24 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 F−1
as it was earlier?
Theorem 26.2.25 Let f ∈G∗ and let ϕ ∈G.
Proof: Note that 26.15 follows from Definition 26.2.24 and both assertions hold for
f ∈G. Consider 26.16. Here is a simple formula involving a pair of functions in
Now for ψ ∈G
The last line follows from the following.
From 26.18 and 26.17 , since ψ was arbitrary,
which shows 26.16. ■