Distributions - Part I

Distributions - Part I

Wolfgang Stefani

LMU München

Hütte am 12.-15.12.2013

Motivation

We would like

δ(

x

) : δ(

x

) =

0 forx ,⁰

∧

R

R^d

δ(

x

)

dx

=

1.

This allowsR

R^d

δ(

x

)ϕ(

x

)

dx

= ϕ(

0

)

, for

ϕ ∈

C₀^∞

(

R^d

)

. Consider now:

Z 1

−1

|

x

|ϕ

⁰

(

x

)

dx

= −

Z 1

−1

sign

(

x

)ϕ(

x

)

dx

,

Z ₁

−1

sign

(

x

)ϕ

⁰

(

x

)

dx

=

Z ₀

−1

−ϕ

⁰

(

x

)

dx

+

Z ₁

0

ϕ

⁰

(

x

)

dx

= −

2

ϕ(

0

).

Test functions - D(Ω)

Let

Ω ⊂

R^{d andC}₀^∞

(Ω) = {ϕ ∈

C^∞

(Ω)|

supp

(ϕ) ⊂⊂ Ω}

. Whereby supp

(ϕ) = {

x

∈ Ω|ϕ(

x

)

,⁰

}

.

With the usual pointwise addition and skalar multiplication this is a vector space (overC^).

We further define the following convergence:

lim

j→∞

ϕ

_j

= ϕ

inC₀^∞

(Ω),

if

•

∃

K

⊂⊂ Ω ∀

j

:

supp

(ϕ

_j

) ⊂

K,

•

∀α ∈

N^d

:

D^α

ϕ

_j

→

D^α

ϕ

uniformly, i. e.

lim_j→∞sup_x∈K

|

D^α

ϕ

_j

(

x

) −

D^α

ϕ(

x

)| =

0.

Distributions - D

⁰

(Ω)

The mapT

: D(Ω) →

Cis a distribution, if and only if it is a sequentially continuous linear form, i.e.

(ϕ

_j

→ ϕ ⇒

T

(ϕ

_j

) →

T

(ϕ)) ∧

T

(ϕ + λψ) =

T

(ϕ) + λ

T

(ψ).

Equivalently:

∀

K

⊂⊂ Ω ∃

c

,

Nconstant, such that:

∀ϕ ∈ D(

K

) : |

T

(ϕ)| ≤

c

kϕk

N,∞;K

=

cP

|α|≤Nsup_x∈K

|∂

^α

ϕ(

x

)|

.

If in the above a certainNsuffices for all compactK, then the smallest of those is called the order ofT.

Regular Distributions

Let

∅

,

Ω ⊂

R^dopen,

(

R^d

, λ

^d

)

the Lebesgue measure space and f

∈

L_loc¹

(Ω)

, then following is a Distribution of order 0:

T_f

:

C₀^∞

(Ω) →

C

, ϕ 7→

Z

Ω

f

· ϕ

d

λ

^d

.

= (

f

, ϕ)

If we additionally let

α ∈

N^{d, then}

(

D^αT_f

)(ϕ) := (−

1

)

^|α|T_f

(

D^α

ϕ) = (−

1

)

^|α|

Z

Ω

f

·

D^α

ϕ

d

λ

^d is a Distribution, since for

ϕ

_j

→ ϕ

:

|(

D^αT_f

)(ϕ

_j

− ϕ)| = |

Z

K

f

·

D^α

(ϕ

_j

− ϕ)

d

λ

^d

| ≤ k

D^α

(ϕ

_j

− ϕ)k

_∞;K

k

f

k

_1;K

→

0

.

Properties D

⁰

(Ω)

By definition,

D

⁰permits aCvectorspace structure. Denoting T

(ϕ) = (

T

, ϕ)

, we get a bilinear form on

D

⁰.

Let nowa

∈

C^∞

(Ω)

and

α, β ∈

N^d. With the following definitions we get:

(

aT

, ϕ) := (

T

,

a

ϕ) ⇒

aT

∈ D

⁰,

(

D^αT

, ϕ) := (−

1

)

^|α|

(

T

,

D^α

ϕ) ⇒

D^αT

∈ D

⁰. And further:

D_i

(

aT

) =

D_i

(

a

)

T

+

aD_i

(

T

)

, D^α+βT

=

D^α

(

D^βT

) =

D^β

(

D^αT

)

.

Localization

ForT

∈ D

⁰

(Ω)

we define its restriction on

Ω

0

⊂ Ω

, by T

_Ω

0

(ϕ) =

T

(ϕ) ∀ϕ ∈ D(Ω

₀

).

Using this we define the support ofT

∈ D

⁰

(Ω)

: supp

(

T

) = {

x

∈ Ω| ∀δ >

0 :T

_Ω∩Bδ(x),⁰

}.

For regular distributions, we have: supp

(

T_f

) =

supp

(

f

)

. We also have the general implication forT

∈ D

⁰ and

ϕ ∈ D

:

supp

(

T

) ∩

supp

(ϕ) = ∅ ⇒ (

T

, ϕ) ≡

0

.

Convolution with functions

ForT

∈ D

⁰

(Ω)

and

ψ ∈ D(Ω)

we define their convolution atx

∈ Ω

, by

(

T

∗ ψ)(

x

) :=

T

(ψ(

x

− ·)) = (

T

, ψ(

x

− ·)).

For a regular distributionT_f we have:

(

T_f

∗ ψ)(

x

) =

R

Ωf

(

y

)ψ(

x

−

y

)

dy.

This convolution has the following properties:

• T

∗ ϕ ∈

C^∞

(

R^d

)

,

• supp

(

T

∗ ϕ) ⊂

supp

(

T

) +

supp

(ϕ)

,

• D^α

(

T

∗ ϕ) = (

D^αT

) ∗ ϕ =

T

∗ (

D^α

ϕ)

.

For

η ∈ D

as well we get further:

(

T

∗ η) ∗ ϕ =

T

∗ (η ∗ ϕ)

.

Convergence in D

⁰

We define the following convergence on

D

⁰: lim

k→∞T_k

=

T

:⇔ ∀ϕ ∈ D :

lim

k→∞

(

T_k

, ϕ) = (

T

, ϕ).

This convergence makes

D

⁰a complete space.

With the mollifierJ_ε

(

x

) = ε

^−dJ

(ε

⁻¹x

) = ε

^−dcexp

(−

_1−|ε¹_−1x|2

)

1_{|ε−1x|<1}

we get the convergence:

T

∗

J_ε

→

T in

D

⁰

.

So the spaceC^∞is dense in

D

⁰and analogously for the space with compact support.

Delta Distribution

Leta

∈ Ω

and define,

δ

₀

∈ D

⁰

: δ

₀

(ϕ) = ϕ(

0

) ∀ϕ ∈ D.

Z

Ω

f

(

x

)ϕ(

x

)

dx

Approximation via Dirac sequences

(

t_k

)

_k

⊂

L¹

(

R^d

)

: 1.

∀

x

∈

R^d

∀

k

∈

N

:

t_k

(

x

) ≥

0,

2.

∀

k

∈

N

:

R

R^dt_k

(

x

)

dx

=

1, 3.

∀ε >

0

:

lim_k→∞R

R^d\Bε(0)t_k

(

x

)

dx

=

0.

For example:

t_k

(

x

) = √

¹ 2

π

k⁻¹

exp

(−

^x

2

2k⁻¹

)

or t_k

(

x

) =

J_ε=1 k

(

x

)

Radon Measures

Let

(

X

, B, µ)

withX Hausdorff, such that

•

µ

is locally finite, i.e.

∀

x

∈

X

∃

Uopen

: µ(

U

) < ∞

,

•

µ

is inner regular, i.e.

∀

A

∈ B : µ(

A

) =

sup

{µ(

K

)|

K

⊂

A

:

K compact

}

. LetM

(Ω)

be the set of Radon Measures on

Ω

and

µ ∈

M

(Ω)

, then

(µ, ϕ) :=

Z

Ω

ϕ(

x

)

d

µ(

x

)

, for

ϕ ∈ D,

defines a Distribution.

In particular, the Dirac measure δ(A) =











1, 0∈A,

0, 0<A (A ⊂R^d) gives the

Delta Distribution:

(δ

₀

, ϕ) =

Z

Ω

ϕ(

x

)

d

δ

₀

(

x

) = ϕ(

0

) = δ

₀

(ϕ)

for

ϕ ∈ D.

Distributions - Part II

Wolfgang Stefani

LMU München

Hütte am 12.-15.12.2013

Distributional Solution

LetT

∈ D

⁰,f

∈ C(

R^d

,

R

)

andP

(

D

) =

X

|α|≤k

a_α

(

x

) ∂

^α

∂

x^α. T is a distributional solution toP

(

D

)

u

=

f, if and only if

P

(

D

)

T

=

T_f

⇔ (

P

(

D

)

T

−

T_f

, ϕ) =

0

∀ϕ ∈ D.

For the Laplacian Operator

∆

onT we have:

∆

T

=

d

X

i=1

D_iD_iT

⇒ (∆

T

, ϕ) = (

T

, ∆ϕ),

for

ϕ ∈ D.

Ifg

∈

C²

(

R^d

,

R

)

andT

=

T_g we have the classical Green fromula Z

(ϕ∆

g

−

g

∆ϕ)

dx

=

0

.

Fundamental Solution

ForP

(

D

) =

X

|α|≤k

a_α

∂

^α

∂

x^α, the solution

γ

of

P

(

D

)

u

(

x

) =

0

,

i.e. P

(

D

)γ = δ

₀ is called a fundamental solution.

The inhomogeneous solution to

P

(

D

)

u

(

x

) =

f

(

x

)

is u

(

x

) = (γ ∗

f

)(

x

) =

Z

γ(

x

−

y

)

f

(

y

)

dy

.

Reminder

We have seen:

Z ₁

−1

|

x

|ϕ

⁰

(

x

)

dx

= −

Z ₁

−1

sign

(

x

)ϕ(

x

)

dx

Z 1

−1

sign

(

x

)ϕ

⁰

(

x

)

dx

= −

2

ϕ(

0

) = −

Z 1

−1

2

δ

0

(

x

)ϕ(

x

)

dx

.

So:

(

sign⁰

, ϕ) = −(

sign

, ϕ

⁰

) = −(−

2

δ

₀

, ϕ) ⇒

sign⁰

=

2

δ

₀

=

Z

2

δ

₀

·

dx

Example - I

For the Heaviside functionH

(

x

) =











1

,

x

≥

0

0

,

x

<

0 we have:

(

H⁰

, ϕ) = −(

H

, ϕ

⁰

) = −

Z ∞

0

1

ϕ

⁰

(

x

)

dx

= ϕ(

0

) = (δ

₀

, ϕ) ⇒ ∂

∂

xH

(

x

) = δ

₀

(

x

)

So for the Laplacian

∆

inR^:

∆γ = ∂

²

∂ x

²

^{γ =}

∂

∂ x

∂

∂ x ^γ

= δ

₀

⇒ ∂

∂ x ^{γ =}

^H

⁽

^x

^{) +}

^c

⇒ γ =

xH

(

x

) +

cx

+

c⁰

c=¹₂,c⁰=0

⇒ γ(

x

) =

¹ 2

|

x

|

Example - II

From classical theory inR^d:

γ(

x

) =











2π1 ln

(|

x

|)

(d−2)ω1 _d

|

x

|

^{2−d and} Z

|x|≤c

|γ(

x

)| =













 Z c

0

|

ln

(

r

)|

r dr

,

d

=

2

,

1 (d−2)

Z _c

0

r^2−dr^d−1

,

d

≥

2 Then withR sufficiently large,

(∆γ, ϕ) = (γ, ∆ϕ) =

Z

γ(

x

)∆ϕ(

x

)

dx

=

lim

ε&0

Z

ε<|x|<R

γ(

x

)∆ϕ(

x

)

dx

=

lim

x&0J_ε

.

Using Green’s second identity:

J_ε

=

Z

ε<|x|<R

ϕ∆γ

dx

+

Z

|x|=ε

γ ∂ϕ

∂ν − ϕ ∂γ

∂ν

d

σ.

Example - II cont.

Since

∆γ =

0 for

|

x

| > ε

and for

|

x

| = ε

we have

∂

∂ν = ∂

∂

r: J_ε

=

Z

|x|=ε

γ ∂ϕ

∂ν − ϕ ∂γ

∂ν

d

σ =

Z

|x|=ε

γ(

x

) ∂ϕ

∂

r

− ϕ(

x

)

¹

(2−d)

ω

_d

(

2

−

d

)|

x

|

^1−d d

σ

=

¹

ω

_d

Z

|x|=ε

ε

^2−d d

−

2

∂ϕ(

x

)

∂

r

− ϕ(

x

)ε

^1−d d

σ

Using the mean value theorem with

|

x⁰

| = |

x⁰⁰

| = ε

:

J_ε

=

¹

ω

d

ε

^2−d d

−

2

∂(

x⁰

)

∂

r

− ε

^1−d

ϕ(

x⁰⁰

)

ω

_d

ε

^d−1

= ε

d

−

2

∂ϕ(

x⁰

)

∂

r

+ ϕ(

x⁰⁰

).

As the derivative is finite for

ε &

0 we get:

(∆γ, ϕ) =

lim

ε&0J_ε

= ϕ(

0

) = (δ

₀

, ϕ).

The Schwartz Space S

The vector space of all rapidly decreasing functions

S(

R^d

) := {φ ∈

C^∞

(

R^d

) | ∀α, β ∈

N^d₀

:

sup

x∈R^d

|

x^αD^βφ(x

)| < ∞}

is called Schwartz space. We have

D

(

S

. h

e.g. exp

(−|

x

|

²

)

i Topology and convergence are induced by the seminorms

kφk

_N

=

sup

x∈R^d

|α|,|β|<Nmax

|

x^αD^β

ϕ(

x

)| ⇔

p_k,l

(

φ

) =

sup

x∈R^d

(|

x

|

^k

+

1

)

X

|β|≤l

|

D^βφ

(

x

)|

and φ_k

→

φin

S :⇔ ∀

N

∈

N0

: kφ

_k

−

φk_N

→

0

.

Fourier Transform F

For someφ

∈ S

its Fourier transform is defined as

F

φ

(ξ) = (

2

π)

^−d2

Z

R^d

e^−ix·ξφ

(

x

)

dx

, ξ ∈

R^d

.

which maps as

F : S → S

and is bijective and bicontinuous with inverse

(F

⁻¹φ)(x

) = (

2

π)

^−d2Z

R^d

e^ix·ξφ(ξ)d

ξ, ϕ ∈ S.

Forφ

∈ S

we also getx^αφ

,

D^αφ

, F

φ

,

D^α

F

φ

, F (

D^αφ

) ∈ S

and D^α

F

φ

= (−

i

)

^|α|

F (

x^αφ),

ξ

^α

F

φ

= (−

i

)

^|α|

F (

D^αφ)

So, in particular

F (

D^αφ

) = (−

i

)

^−|α|

ξ

^α

F

φ

Tempered Distributions S

⁰

The space

S

⁰contains all linear forms

S →

C, which are sequentially continous, i. e.

φ_k

→

φin

S ⇒

T

(φ

_k

) →

T

(φ).

Since φ_k

→

φin

D ⇒

φ_k

→

φin

S ⇒

T

(φ

_k

) →

T

(φ),

we haveT

∈ S

⁰

⇒

T

∈ D

⁰. And by

F

T

(ϕ) :=

T

(F ϕ)

we define the Fourier transform ofT

∈ S

⁰.

Example - III

The regular distributionT_e is not tempered. Let

ϕ ∈ D(

R

)

, then

ψ

_k

(

x

) :=

e^−k

ϕ(

x

−

k

) →

0 in

S(

R

),

with

ψ

_k

∈ D(

R

)

, but

T_e

(ψ

_k

) =

Z

R

e^x

ψ

_k

(

x

)

dx

=

Z

R

e^xe^−k

ϕ(

x

−

k

)

dx

=

Z

R

e^y

ϕ(

y

)

dy ,⁰

=

T_e

(

0

).

Example - IV.1

Consider the heat equation for distributions:

∂u∂tu

(

t

,

x

) = ∆

_xu

(

t

,

x

)

u

(

t

,

x

) =

f

(

t

,

x

)

^i.e.

(

_∂t^∂T

, ϕ) = (∆

_xT

, ϕ) (

0

, ∞) ×

R^d

(

T

, ϕ) = (

T_f

, ϕ) {

t

=

0

} ×

R^d Determine its Fourier transform forx:

F

_x

(∆

_xT

, ϕ) = F

_x

(

T

, ∆

_x

ϕ) = (

T

, F

_x

(∆

_x

ϕ)) = (

T

, (−

i

)

⁻²

ξ

^(0,2)

F

_x

ϕ)

Thus we get:

F

_x

( ∂

∂

tT

, ϕ) = F

_x

(−

T

, ∂

∂

t

ϕ) = (−

T

, ∂

∂

t

F

_x

ϕ) =

( ∂

∂

tT

, F

_x

ϕ) = ∂

∂

t

(

T

, F

_x

ϕ) = = (−|ξ|

^{! 2}T

, F

_x

ϕ)

Example - IV.2

We have now the ordinary differential equation in

F

_x

(

T

)

:

∂t∂

F

_x

(

T

, ϕ) = −|ξ|

²

F

_x

(

T

, ϕ) F

_x

(

T

, ϕ) = F

_x

(

T_f

, ϕ)

^i.e.

∂t∂

F

_x

(

T

) = −|ξ|

²

F

_x

(

T

) (

0

, ∞) ×

R^d

F

_x

(

T

) = F (

T_f

) {

t

=

0

} ×

R^d Thus we get:

F

_x

(

T

) =

e^−t|ξ|2

F (

T_f

) ⇒ (

T

, F

_x

ϕ) =

e^−t|ξ|2

(

T_f

, F ϕ)

⇒ F

_ξ⁻¹

(

T

, F ϕ) = F

_ξ⁻¹

(

T_f

,

e^−t|ξ|2

F ϕ) ⇒ (

T

, ϕ) =

T_f

, F

_ξ⁻¹

(

e^−t|ξ|2

F ϕ)

Example - IV.3

The convolution theorem forT₁

∈ S

⁰ andT₂

∈ E

⁰:

F (

T₁

∗

T₂

) = (

2

π)

^d2

F (

T₁

)F (

T₂

)

With this we get:

F

_ξ⁻¹

(

e^−t|ξ|2

F

_x

ϕ) = (

2

π)

^−d2

F

_ξ⁻¹

(

e^−t|ξ|2

) ∗ ϕ

Further

F

_ξ⁻¹

(

e^−t|ξ|2

) = (

2

π)

^−d2 Z

e^ix·ξe^−t|ξ|2 d

ξ =

Z

R^d

e^{ix·ξ−t|ξ|2}d

ξ

= (

2t

)

^−d2e^−|x|

2

4t

=

g

(

t

,

x

)

Example - IV.4

So finally:

(

T_f

,

g

∗ ϕ) =

Z

f

(

t

,

x

)(

2

π)

^−d2 Z

g

(

t

−

t⁰

,

x

−

x⁰

)ϕ(

t⁰

,

x⁰

)

d

(

t⁰

,

x⁰

)

d

(

t

,

x

) =

Z

ϕ(

t⁰

,

x⁰

)(

2

π)

^−d2 Z

f

(

t

,

x

)

g

(

t

−

t⁰

,

x

−

x⁰

)

d

(

t

,

x

)

d

(

t⁰

,

x⁰

) =

Z

(

2

π)

^−d2 Z

f

(

t

,

x

)

g

(

t

−

t⁰

,

x

−

x⁰

)

d

(

t

,

x

)

ϕ(

t⁰

,

x⁰

)

d

(

t⁰

,

x⁰

) =

Z

(

4

π(

t

−

t⁰

))

^−d2f

(

t

,

x

)

e⁻

|x−x0|2 4(t−t0) d

(

t

,

x

)

ϕ(

t⁰

,

x⁰

)

d

(

t⁰

,

x⁰

) =

Z

(

4

π(−

t⁰

))

^−d2 Z

f

(

x

)

e⁻

|x−x0|2 4(−t0) d

(

x

)

ϕ(

t⁰

,

x⁰

)

d

(

t⁰

,

x⁰

).

Example - IV.5

The solution is the regular distributionT_h with:

h

(

t

,

x

) = (

4

π

t

)

^−d2 Z

f

(

x

)

e⁻

|x−x0|2 4t d

(

x

)

^ ¨