Parseval's identity

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Parseval's identity

[′pär·sə·vəlz ī′den·əd·ē]
(mathematics)
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It is necessary to describe all distribution functions [sigma](*) such that instead of (2) the following Parseval identity holds:
A matrix-valued distribution function [sigma] : R [right arrow] [C.sup.nxn] will be called a spectral function for the vector-valued Fourier transform (1) (with respect to [??]) if the following Parseval identity holds:
A distribution function [??](*) : R [right arrow] B([C.sup.n]) is called a spectral function of system (20) if the following Parseval identity holds:
Assume that [sigma](*) : R [right arrow] B([C.sup.n]) is an operator-valued distribution function such that for each [phi] [member of] [[??].sup.2] (R, [C.sup.n]; [??]) the following Parseval identity holds:
If in addition [sigma](*) is absolutely continuous and [summation](s) = [sigma]'(s) is the density of [sigma](*), then Parseval identity (38) and the inverse transform (39) can be written as
The equation (4) follows from the Parseval identity or the Funibi theorem.
By the Parseval identity, the mean square error of [[psi].sup.(0)] satisfies