
1270 ELEMENTS OF THEORY OF GENERALIZED FUNCTIONS
21.1.6 Tempered Distributions and Fo u rier Transform
◮ Tempered distributions.
The space J of rapidly decaying test functions contains all infinitely differentiable func-
tions ϕ(x) decaying, together with all of their derivatives, more rapidly than any power of
|x|
−1
as |x| → ∞. The convergence in J is defined as follows: a sequence of functions
ϕ
n
∈ J converges to a function ϕ ∈ J if and only if x
β
d
m
dx
m
ϕ
n
(x) ⇒ x
β
d
m
dx
m
ϕ(x) as
n → ∞ for any β and m = 0, 1, . . . .
We have the following consequences of these definitions: (i) J is a linear space; (ii)
K ⊂ J; (iii) the convergence in K implies the convergence ...