distribution

distribution

Motivation

The main motivation behind

distribution

theory is to
extend the common

linear operators

on

functions

,
such as the

derivative

,

convolution

, and the

Fourier transform

,
so that they also apply to the

singular

, non-smooth, or non-integrable
functions that regularly appear in both theoretical and applied

analysis

.

Distribution theory also seeks to define suitable

structures

on the

spaces of functions

involved
to ensure the

convergence

of suitable approximating functions,
and the

continuity

of certain

operators

.
For example, the limit of derivatives should be equal
to the derivative of the limit, with some definition of the limiting

operation

.

When this program is carried out,
inevitably we find that we have to enlarge the space of objects that we
would consider as “functions”. For example, the derivative of a

step
function

is the Dirac delta function with a spike at the

discontinuous

step;
the Fourier transform of a

constant function

is also a Dirac delta
function, with the spike representing

infinite

spectral magnitude
at one single frequency. (These facts, of course, had long been
used in engineering mathematics.)

Remark:

Dirac’s notion of

delta distributions

was introduced to facilitate computations in Quantum Mechanics,
however without having at the beginning a proper mathematical definition. In part as
a (negative) reaction to such a state of affairs, von Neumann produced a mathematically
well-defined

foundation

of Quantum Mechanics (

http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras

) based on actions of

self-adjoint operators

on

Hilbert spaces

which is still currently in use, with several significant

additions

such as Frechét

nuclear spaces

and

quantum groups

.

There are several theories of such

‘

generalized functions

’

.
In this entry, we describe Schwartz’ theory of

distributions

,
which is probably the most widely used.

Essentially, a distribution on

ℝ

is a linear mapping that takes a

smooth function

(with compact support) on

ℝ

into a

real number

.
For example, the delta distribution is the map,

f

↦

f

⁢

(

0

)

while any smooth function

g

on

ℝ

induces a distribution

f

↦

∫

ℝ

f

⁢

g

.

Distributions are also well behaved under coordinate changes, and
can be defined onto a manifold. Differential forms with
distribution valued

coefficients

are called

currents

.
However, it is not possible to define a

product

of two
distributions generalizing the product of usual functions.

Formal definition

A note on notation.

In distribution theory, the

topological vector space

of

smooth functions with compact support

on
an

open set

U

⊆

ℝ

n

is traditionally denoted by

𝒟

⁢

(

U

)

. Let us also denote by

𝒟

K

⁢

(

U

)

the subset of

𝒟

⁢

(

U

)

of functions with

support

in a

compact set

K

⊂

U

.

Definition 1

(Distribution).

A

distribution

is a linear continuous functional on

D

⁢

(

U

)

,
i.e., a linear

continuous mapping

D

⁢

(

U

)

→

C

.
The set of all distributions on

U

is denoted by

D

′

⁢

(

U

)

.

Suppose

T

is a

linear functional

on

𝒟

⁢

(

U

)

.
Then

T

is

continuous

if and only if

T

is continuous
in the origin (see this page (

http://planetmath.org/ContinuousLinearMapping

)).
This condition can be rewritten in various ways, and
the below

theorem

gives two convenient conditions that can be used to prove
that a linear mapping is a distribution.

Theorem 1

.

Let

U

be an open set in

R

n

,
and let

T

be a linear functional on

D

⁢

(

U

)

. Then the
following are

equivalent

:

1.

T

is a distribution.

2.

If

K

is a compact set in

U

, and

{

u

i

}

i

=

1

∞

be a

sequence

in

𝒟

K

⁢

(

U

)

, such that
for any multi-index

α

, we have

D

α

⁢

u

i

→

0

in the

supremum norm

as

i

→

∞

,
then

T

⁢

(

u

i

)

→

0

in

ℂ

.

3.

For any compact set

K

in

U

, there are

constants

C

>

0

and

k

∈

{

1

,

2

,

…

}

such that for all

u

∈

𝒟

K

⁢

(

U

)

, we have

|

T

⁢

(

u

)

|

≤

C

⁢

∑

|

α

|

≤

k

||

D

α

⁢

u

||

∞

,

(1)

where

α

is a multi-index, and

|

|

⋅

|

|

∞

is the supremum norm.

Proof

The equivalence of (2) and (3) can be
found on this page (

http://planetmath.org/EquivalenceOfConditions2And3

),
and the equivalence of (1) and (3) is shown in

[

1

]

.

Distributions of order

k

If

T

is a distribution on an open set

U

,
and the same

k

can be used for any

K

in the above

inequality

, then

T

is a

distribution of order

k

.
The set of all such distributions is denoted by

D

′

⁣

k

⁢

(

U

)

.

Both usual functions and the delta distribution are of order

0

.
One can also show that by differentiating a distribution its order increases
by at most one. Thus, in some sense, the order is a measure of how
”smooth” a distribution is.

Topology for

𝒟

′

⁢

(

U

)

The

standard topology

for

𝒟

′

⁢

(

U

)

is the weak

∗

topology

.
In this topology, a sequence

{

T

i

}

i

=

1

∞

of distributions
(in

𝒟

′

⁢

(

U

)

)

converges

to a distribution

T

∈

𝒟

′

⁢

(

U

)

if and only if

T

i

⁢

(

u

)

→

T

⁢

(

u

)

⁢

(in

ℂ

) as

i

→

∞

for every

u

∈

𝒟

⁢

(

U

)

.

Notes

A common notation for the action of a distribution

T

onto a test function

u

∈

𝒟

⁢

(

U

)

(i.e.,

T

⁢

(

u

)

with above notation) is

⟨

T

,

u

⟩

.
The motivation for this
comes from this example (

http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution

).

References

1

W. Rudin,

Functional Analysis

,
McGraw-Hill Book Company, 1973.

2

L. H

ö

rmander,

The Analysis of Linear Partial Differential Operators I,
(Distribution theory and

Fourier Analysis

)

, 2nd ed, Springer-Verlag, 1990.

Title

distribution

Canonical name

Distribution

Date of creation

2013-03-22 13:44:08

Last modified on

2013-03-22 13:44:08

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

23

Author

matte (1858)

Entry type

Definition

Classification

msc 46-00

Classification

msc 46F05

Synonym

‘generalized function’

Related topic

ExampleOfDiracSequence

Related topic

DiracDeltaFunction

Related topic

DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom

Related topic

QuantumGroups

Related topic

FourierStieltjesAlgebraOfAGroupoid

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Related topic

QFTOrQuantumFieldTheories

Related topic

QuantumGroup

Defines

distribution of finite order

Generated on Fri Feb 9 21:34:29 2018 by

LaTeXML