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.