Editing Fourier transform (section)

=== Plancherel theorem and Parseval's theorem ===
{{main|Plancherel theorem|Parseval's theorem}}
Let {{math|''f''(''x'')}} and {{math|''g''(''x'')}} be integrable, and let {{math|''f̂''(''ξ'')}} and {{math|''ĝ''(''ξ'')}} be their Fourier transforms. If {{math|''f''(''x'')}} and {{math|''g''(''x'')}} are also [[square-integrable]], then the Parseval formula follows:<ref>{{harvnb|Rudin|1987|p=187}}</ref>
<math display="block">\langle f, g\rangle_{L^{2}} = \int_{-\infty}^{\infty} f(x) \overline{g(x)} \,dx = \int_{-\infty}^\infty \hat{f}(\xi) \overline{\hat{g}(\xi)} \,d\xi,</math>
where the bar denotes [[complex conjugation]].

The [[Plancherel theorem]], which follows from the above, states that<ref>{{harvnb|Rudin|1987|p=186}}</ref>
<math display="block">\|f\|^2_{L^{2}} = \int_{-\infty}^\infty \left| f(x) \right|^2\,dx = \int_{-\infty}^\infty \left| \hat{f}(\xi) \right|^2\,d\xi. </math>

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a [[unitary operator]] on {{math|''L''<sup>2</sup>('''R''')}}. On {{math|''L''<sup>1</sup>('''R''') ∩ ''L''<sup>2</sup>('''R''')}}, this extension agrees with original Fourier transform defined on {{math|''L''<sup>1</sup>('''R''')}}, thus enlarging the domain of the Fourier transform to {{math|''L''<sup>1</sup>('''R''') + ''L''<sup>2</sup>('''R''')}} (and consequently to {{math|{{math|''L''{{i sup|''p''}}}}('''R''')}} for {{math|1 ≤ ''p'' ≤ 2}}). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the [[energy]] of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem.

See [[Pontryagin duality]] for a general formulation of this concept in the context of locally compact abelian groups.