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.