  Gravitational-wave memory is characterized by a signal component that
persists after a transient signal has decayed. Treating such signals in the
frequency domain is non-trivial, since discrete Fourier transforms assume
periodic signals on finite time intervals. In order to reduce artifacts in the
Fourier transform, it is common to use recipes that involve windowing and
padding with constant values. Here we discuss how to regularize the Fourier
transform in a straightforward way by splitting the signal into a given sigmoid
function that can be Fourier transformed in closed form, and a residual which
does depend on the details of the gravitational-wave signal and has to be
Fourier transformed numerically, but does not contain a persistent component.
We provide a detailed discussion of how to map between continuous and discrete
Fourier transforms of signals that contain a persistent component. We apply
this approach to discuss the frequency-domain phenomenology of the $(\ell=2,
m=0)$ spherical harmonic mode, which contains both a memory and an oscillatory
ringdown component.
