# Parseval's theorem

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## Parseval's theorem

[′pär·sə·vəlz ′thir·əm]
(mathematics)
A theorem that gives the integral of a product of two functions, ƒ(x) and F (x), in terms of their respective Fourier coefficients; if the coefficients are defined by and similarly for F (x), the relationship is
References in periodicals archive ?
By Parseval's theorem, since u, [[partial derivative].sup.2.sub.t]u [member of] [L.sub.2]([OMEGA]), for every [epsilon] > 0, there exists an N [greater than or equal to] N([epsilon]) such that
The desired result follows immediately from Parseval's theorem, noting that [u.sub.[omega]] and V[u.sub.[omega]] are [L.sup.2]([OMEGA]) functions.
The above series converges in [L.sub.2] ([I.sub.a]) and by Parseval's theorem, this implies the convergence of the sampling series (47) in [L.sub.2] ([R.sub.+])
Therefore on account of Parseval's theorem and the orthogonality relation (33),
From (2.1) and Parseval's theorem, it follows by easily that
and by Parseval's theorem again we have the orthogonal decomposition:
We expand the two terms in the inner product in a series of eigenfunctions (5) and use Parseval's theorem.
Applying the Parseval's theorem to the unit function f(x) [equivalent to] with respect to the latter basis, we conclude the well-known identity
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