1212 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

Proof: For h a function defined on [0,T ], let h1 be even, 2T periodic, and h1 (t) = h(t)for all t ∈ [0,T ]. Let C (·) ∈C∞

0 (−T,2T ) ,C (t) ∈ [0,1],C (t) = 1 on [0,T ].

0 T 2T−T

h1(t)

0 T

C(t)

Let B̃(t) =C (t)B1 (t) for all t ∈ R and define

ũ(t) ={

u1 (t) , t ∈ [−T,2T ]0, t /∈ [−T,2T ] .

Now let u ∈ X . Then

(i∗B̃ũ

)′(t) =

0, t <−TC′ (t)(i∗Bu)(−t)−C (t)(i∗Bu)′ (−t) , t ∈ [−T,0](i∗Bu)′ (t) , t ∈ [0,T ]C′ (t)(i∗Bu)(2T − t)−C (t)(i∗Bu)′ (2T − t) , t ∈ [T,2T ]0, t > 2T

(34.6.46)

Thus, if I ⊇ [−T,2T ], then(i∗B̃ũ

)′ ∈ V ′I . Defining un ≡ ũ∗φ n, then for a.e. t,

Re⟨(i∗B̃un

)′(t) ,un (t)⟩=

12[⟨B̃un,un⟩′ (t)+ ⟨B̃′ (t)un (t) ,un (t)⟩

]. (34.6.47)

From 34.6.46 and Proposition 34.6.1, the following holds in V ′[−T,2T ].

limn→∞

(i∗B̃un

)′= lim

n→∞

(i∗B̃(ũ∗φ n)

)′ (34.6.48)

= limn→∞

((i∗B̃ũ

)∗φ n

)′= lim

n→∞

(i∗B̃ũ

)′ ∗φ n

=(i∗B̃ũ

)′Where the second equality follows from Corollary 34.6.2, the third follows from the point-wise a.e. equality of

((i∗B̃ũ

)∗φ n

)′ and(i∗B̃ũ

)′ ∗φ n, while the fourth follows from 34.6.46and standard properties of convolutions.