Plancherel’s theorem

Plancherel’s theorem

0.1

Statement of theorem

Plancherel’s

Theorem

states that
the unitary

Fourier transform

of

𝐋

1

functions

(the Lebesgue-integrable functions (

http://planetmath.org/Integral3

)) on

ℝ

n

extends to a unitary

isomorphism

on

𝐋

2

(the square-integrable functions).

Thus, the following two fundamental properties hold
for the Fourier transform

ℱ

on

𝐋

2

functions

g

:

ℝ

n

→

ℂ

:

i

ℱ

-

1

⁢

(

ℱ

⁢

g

)

=

g

=

ℱ

⁢

(

ℱ

-

1

⁢

g

)

.

The equalities are as elements of

𝐋

2

; in terms of

pointwise

functions,
the equalities hold almost everywhere on

ℝ

n

.

ii

The Fourier transform preserves

𝐋

2

norms:

∫

ℝ

n

|

ℱ

⁢

g

⁢

(

ξ

)

|

2

⁢

𝑑

ξ

=

∥

ℱ

⁢

g

∥

𝐋

2

2

=

∥

g

∥

𝐋

2

2

=

∫

ℝ

n

|

g

⁢

(

x

)

|

2

⁢

𝑑

x

.

0.2

Extension of the Fourier transform to

𝐋

2

The

extension

ℱ

of the usual
Fourier transform can be described concretely as follows: given a

𝐋

2

function

g

:

ℝ

n

→

ℂ

,
take any

sequence

g

k

:

ℝ

n

→

ℂ

of

𝐋

1

functions that

converge

in

𝐋

2

to

g

.
The Fourier transforms

ℱ

⁢

g

k

⁢

(

ξ

)

=

∫

ℝ

n

g

k

⁢

(

x

)

⁢

e

-

2

⁢

π

⁢

i

⁢

ξ

⋅

x

⁢

𝑑

x

,

ξ

∈

ℝ

n

are defined as usual,
and

ℱ

⁢

g

can be obtained as the

𝐋

2

limit
of

ℱ

⁢

g

k

.

In the one-dimensional case, a
common sequence of approximating sequences to take is

g

k

=

g

⋅

𝕀

[

-

k

,

k

]

; in that case we have

ℱ

⁢

g

⁢

(

ξ

)

=

lim

T

→

∞

⁡

∫

-

T

T

g

⁢

(

t

)

⁢

e

-

2

⁢

π

⁢

i

⁢

ξ

⁢

t

⁢

𝑑

t

,

ξ

∈

ℝ

.

The inverse Fourier transform

ℱ

-

1

can be obtained in a

similar

way
to

ℱ

, using approximating functions

g

k

:

ℱ

-

1

⁢

g

k

⁢

(

x

)

=

∫

ℝ

n

g

k

⁢

(

ξ

)

⁢

e

2

⁢

π

⁢

i

⁢

ξ

⋅

x

⁢

𝑑

x

,

x

∈

ℝ

n

.

0.3

Note on different conventions

Here, we have used the convention for the Fourier transform

ℱ

that

ξ

denotes “ordinary frequency”, i.e. the exponential contains factors of

2

⁢

π

. Another common convention has

ξ

replaced by

ω

denoting the “angular frequency”, with factors

2

⁢

π

occurring not in the exponent, but as

multiplicative

constants

. In this case property (i) above still holds,
but property (ii) will not hold unless the multiplicative constants
in front of the forward and inverse Fourier transform are chosen properly.

References

Folland

Gerald B. Folland.

Real Analysis: Modern Techniques and Their Applications

, second ed. Wiley-Interscience, 1999.

Katznelson

Yitzhak Katznelson.

An

Introduction

to

Harmonic Analysis

, second ed. Dover Publications, 1976.

Wiki

“

http://en.wikipedia.org/wiki/Continuous_Fourier_transform

Fourier transform
”,

Wikipedia, The Free Encyclopedia

. Accessed 22 December, 2006.

Title

Plancherel’s theorem

Canonical name

PlancherelsTheorem

Date of creation

2013-03-22 16:29:00

Last modified on

2013-03-22 16:29:00

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

11

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 42B10

Classification

msc 42A38

Related topic

ProofOfSamplingTheorem

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