14 I. Theory of Distributions
Thus
E(x, t) =
1
(4t)
n
2
β
x
2

t
δ as t +0.
5. Regularizations of nonintegrable functions
If f(x) is not a locally integrable function, the problem is how to define a
distribution naturally related to f(x).
Consider, for example, the case where f(x1) is locally integrable for
x1
R1,
|x1| δ, ∀δ 0,
|x1|δ
|f(x1)|dx1 = +∞ and |f(x1)|
C
|x1|m
, m 1.
It is natural to assume that f is a distribution that satisfies
(5.1) f(ϕ) =
R1
f(x1)ϕ(x1)dx1 if ϕ D, ϕ(x1) = 0 for |x1| δ.
The problem of defining f D such that (5.1) holds for any ϕ(x1) = 0 in a
neighborhood of x1 = 0 is called the regularization problem for f(x1). One
of the distributions f1 D that solves the problem of the regularization of
f(x1) is given by the formula:
(5.2) f1(ϕ) =
|x1|≤1
f(x1) ϕ(x1)
m−1
k=0
ϕ(k)(0)
k!
x1
k
dx1
+
|x1| 1
f(x1)ϕ(x1)dx1.
Since
ϕ(x1)
m−1
k=0
ϕ(k)(0)
k!
x1
k
=
ϕ(m)(θ)
m!
x1
m

C|x1|m,
(5.2) defines a linear continuous functional f1 on D that coincides with (5.1)
when ϕ(x) = 0 in a neighborhood of x1 = 0. However, the solution of the
regularization problem is not unique since any distribution of the form
(5.3) f2(ϕ) = f1(ϕ) +
N
k=1
ck
dkδ
dx1k
(ϕ),
where ck C are arbitrary, also coincides with (5.1) when ϕ(x1) = 0 in a
neighborhood of x1 = 0.
In applications one tries to define a regularization f D of a noninte-
grable function f(x) in such a way that f D retains some properties of
f(x) in addition to (5.1) such as symmetry, homogeneity, etc.
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