Plancherel theorem | Wikipedia audio article

This is an audio version of the Wikipedia Article:


00:05:16 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
Other Wikipedia audio articles at:
Upload your own Wikipedia articles through:
Speaking Rate: 0.9216210072787696
Voice name: en-US-Wavenet-F


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, the Plancherel theorem is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its 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

A more precise formulation is that if a function is in both L1(R) and L2(R), then its Fourier transform is in L2(R), and the Fourier transform map is an isometry with respect to the L2 norm. This implies that the Fourier transform map restricted to L1(R) ∩ L2(R) has a unique extension to a linear isometric map L2(R) → L2(R), sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.
Plancherel's theorem remains valid as stated on n-dimensional Euclidean space Rn. The theorem also holds more generally in locally compact abelian groups. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of 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 L2(R) inner product of two functions. That is, if



f
(
x
)


{\displaystyle f(x)}
and



g
(
x
)


{\displaystyle g(x)}
are two L2(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 L1(R) functions, then




(


P


f
)
(
ξ
)
=
...