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Plancherel theorem and Parseval's theorem

Let f (x) and g (x) be integrable, and let f̂ (ξ) and ĝ (ξ) be their Fourier transforms. If f (x) and g (x) are also square-integrable, then the Parseval formula follows: ⟨ f, g ⟩ L 2 = ∫ − ∞ ∞ f (x) g (x) ¯ d x = ∫ − ∞ ∞ f ^ (ξ) g ^ (ξ) ¯ d ξ, {\displaystyle \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,} where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that ‖ f ‖ L 2 2 = ∫ − ∞ ∞ | f (x) | 2 d x = ∫ − ∞ ∞ | f ^ (ξ) | 2 d ξ. {\displaystyle \|f\|_{L^{2}}^{2}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi)\right|^{2}\,d\xi.}

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on L (R). On L (R) ∩ L (R), this extension agrees with original Fourier transform defined on L (R), thus enlarging the domain of the Fourier transform to L (R) + L (R) (and consequently to L (R) for 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.