Theorem in harmonic analysis
In
frequency spectrum
. That is, if
f
(
x
)
{\displaystyle f(x)}
is a function on the real line, and
f
^
(
ξ
)
{\displaystyle {\widehat {f}}(\xi )}
is its frequency spectrum, then
∫
−
∞
∞
|
f
(
x
)
|
2
d
x
=
∫
−
∞
∞
|
f
^
(
ξ
)
|
2
d
ξ
{\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=\int _{-\infty }^{\infty }|{\widehat {f}}(\xi )|^{2}\,d\xi }
A more precise formulation is that if a function is in both L p
L
1
(
R
)
{\displaystyle L^{1}(\mathbb {R} )}
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
Fourier transform is in
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
L 2 norm. This implies that the Fourier transform map restricted to
L
1
(
R
)
∩
L
2
(
R
)
{\displaystyle L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} )}
linear isometric map
L
2
(
R
)
↦
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )\mapsto L^{2}(\mathbb {R} )}
, sometimes called the Plancherel transform. This isometry is actually a
quadratically integrable functions
.
Plancherel's theorem remains valid as stated on n -dimensional Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
non-commutative harmonic analysis
.
The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series .
Due to the polarization identity , one can also apply Plancherel's theorem to the
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
inner product
of two functions. That is, if
f
(
x
)
{\displaystyle f(x)}
and
g
(
x
)
{\displaystyle g(x)}
are two
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
functions, and
P
{\displaystyle {\mathcal {P}}}
denotes the Plancherel transform, then
∫
−
∞
∞
f
(
x
)
g
(
x
)
¯
d
x
=
∫
−
∞
∞
(
P
f
)
(
ξ
)
(
P
g
)
(
ξ
)
¯
d
ξ
,
{\displaystyle \int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }({\mathcal {P}}f)(\xi ){\overline {({\mathcal {P}}g)(\xi )}}\,d\xi ,}
and if
f
(
x
)
{\displaystyle f(x)}
and
g
(
x
)
{\displaystyle g(x)}
are furthermore
L
1
(
R
)
{\displaystyle L^{1}(\mathbb {R} )}
functions, then
(
P
f
)
(
ξ
)
=
f
^
(
ξ
)
=
∫
−
∞
∞
f
(
x
)
e
−
2
π
i
ξ
x
d
x
,
{\displaystyle ({\mathcal {P}}f)(\xi )={\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi i\xi x}\,dx,}
and
(
P
g
)
(
ξ
)
=
g
^
(
ξ
)
=
∫
−
∞
∞
g
(
x
)
e
−
2
π
i
ξ
x
d
x
,
{\displaystyle ({\mathcal {P}}g)(\xi )={\widehat {g}}(\xi )=\int _{-\infty }^{\infty }g(x)e^{-2\pi i\xi x}\,dx,}
so
∫
−
∞
∞
f
(
x
)
g
(
x
)
¯
d
x
=
∫
−
∞
∞
f
^
(
ξ
)
g
^
(
ξ
)
¯
d
ξ
.
{\displaystyle \int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\widehat {f}}(\xi ){\overline {{\widehat {g}}(\xi )}}\,d\xi .}
See also References Dixmier, J. (1969), Les C*-algèbres et leurs Représentations , Gauthier VillarsYosida, K. (1968), Functional Analysis , Springer VerlagExternal links
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