4. Convergence of distributions 13
Indeed, changing the variables y =
x
ε
, dy =
1
εn
dx, we get
βε(ϕ) ϕ(0) =
1
εn
Rn
β
x
ε
ϕ(x)dx ϕ(0) =
Rn
β(y)(ϕ(εy) ϕ(0))dy.
Since |ϕ(εy) ϕ(0)| 2C and ϕ(εy) ϕ(0) for ∀y
Rn,
we obtain, by the
Lebesgue convergence theorem, that
Rn
β(y)(ϕ(εy) ϕ(0))dy
Rn
β(y)|ϕ(εy) ϕ(0)|dy 0
as ε 0. The sequence of regular functionals βε is called a delta-like
sequence.
Consider some particular cases of delta-like sequences.
b) The Poisson kernel for the Laplace equation:
P (x1,x2) =
x2
π(x1 2 + x2)2
satisfies the Laplace equation
∂2P
∂x1
2
+
∂2P
∂x22
= 0
in the half-plane x2 0. Here β(x1) =
1
π(1+x1)
2
, ε = x2. Note that

−∞
dx1
π(1 +
x1)2
= 1.
Thus
P (x1,x2) =
1
x2π(1 +
x2
1
x2
2
)
=
1
x2
β
x1
x2
and
P (x1,x2) δ as x2 +0.
c) The heat kernel:
E(x, t) =
1
(4πt)
n
2
e−
|x|
2
4t
satisfies the heat equation

∂t
E
n
k=1
∂2
∂xk2
E = 0
for t 0, ∀x
Rn.
Here β(x) =
1
π
n
2
e−|x|2
, ε =

4t. Note that
1
π
n
2 Rn
e−|x|2
dx =
1

π


e−x1 2
dx1
n
= 1.
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