Plancherel's theorem

The Plancherel theorem (after Michel Plancherel which he proved in 1910) is a statement from the mathematical branch of Fourier analysis , which for functional analysis belongs. He states that the Fourier transform on the space of square integrable functions an isometric view is, so that a function and its Fourier transform the same - standard have. ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$

statement

There is an isometry that is unitary and uniquely determined by ${\displaystyle \Psi \colon L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}$

${\displaystyle \Psi (f)={\mathcal {F}}(f)}$

for everyone , being ${\displaystyle f\in {\mathcal {S}}}$

• ${\displaystyle {\mathcal {F}}}$the Fourier transform and
• ${\displaystyle {\mathcal {S}}}$called the Schwartz room .

Remarks

1. The equality applies not only to , but also to , since both in and in are dense . Since on and the Fourier transformation on is defined, one can understand as a continuation of the Fourier transformation on . This continuation is again called a Fourier transformation or, more rarely, the Fourier-Plancherel transformation. ${\displaystyle \Psi (f)={\mathcal {F}}(f)}$ ${\displaystyle f\in {\mathcal {S}}}$ ${\displaystyle f\in L^{1}(\mathbb {R} ^{n})\cap L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle L^{1}(\mathbb {R} ^{n})}$ ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle \Psi }$ ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle L^{1}(\mathbb {R} ^{n})}$ ${\displaystyle \Psi }$ ${\displaystyle L^{2}(\mathbb {R} ^{n})}$
2. The Parseval's theorem is the analogue of the set of Plancherel for Fourier series . However, the sentences are not directly related, since the continuous Fourier transformation is not based on an orthogonal system.

See also

• Harmonic analysis

literature

• Walter Rudin: Functional Analysis . McGraw-Hill, New York 1991, pp. 188-189, ISBN 0070542368

Web links

• Plancherel's Theorem by Mathworld