(Sometimes called energy spectrum.) The square of the amplitude of the (complex) Fourier transform of an aperiodic function.
Thus, if f(t) is the given function, its Fourier transform is 
. It is assumed that the total energy 
is finite.
and the energy density spectrum is
. It is assumed that the total energy is finite.
An analytical transformation of a function f(x) obtained (if it exists) by multiplying the function by e-iux and integrating over all x,
If the Fourier transform of a function is known, the function itself may be recovered by use of the inversion formula:
See also Fourier integral, spectral function.
where u is the new variable of the transform F(u) and i2 = -1.
If the Fourier transform of a function is known, the function itself may be recovered by use of the inversion formula:
The Fourier transform has the same uses as the Fourier series: For example, the integrand F(u) exp (iux) is a solution of a given linear equation, so that the integral sum of these solutions is the most general solution of the equation. When the variable u is complex, the Fourier transform is equivalent to the Laplace transform.
See also Fourier integral, spectral function.
