This operation is extended to locally compact Abelian groups by the Pontryagin duality, and it is unitary by Parseval's theorem.
This operation is related to the autocorrelation function by the Wiener- Khinchin theorem, and its "fractional" extension is used as a noise filter and allows this operation to be applied a non-integer number of times. When p is between one and two inclusive, this operation goes from Lp to Lq [ell-pee to ell-queue] where q is the reciprocal of one minus one over p by the Hausdorff- Young inequality. This operation is extended to locally compact Abelian groups by the Pontryagin duality, and it is unitary by Parseval's theorem. The recursive Cooley-Tukey algorithm may be used to calculate a "discrete" one. This operation is applied to produce readable NMR spectra, and is defined such that g of zeta is equal to the integral over all x of the function of x times the exponential of negative two pi i x zeta. FTP, name this operation that moves from a time domain to a frequency domain.
This operation is related to the autocorrelation function by the Wiener- Khinchin theorem, and its "fractional" extension is used as a noise filter and allows this operation to be applied a non-integer number of times.
{Fourier transform}