A Generalized Doppler and Aberration Kernel for Frequency-Dependent
  Cosmological Observables
  We introduce a $\textit{frequency-dependent}$ Doppler and aberration
transformation kernel for the harmonic multipoles of a general cosmological
observable with spin weight $s$, Doppler weight $d$ and arbitrary frequency
spectrum. In the context of Cosmic Microwave Background (CMB) studies, the
frequency-dependent formalism allows to correct for the motion-induced
aberration and Doppler effects on individual frequency maps with different
masks. It also permits to deboost background radiations with non-blackbody
frequency spectra, like extragalactic foregrounds and CMB spectra with
primordial spectral distortions. The formalism can also be used to correct
individual E and B polarization modes and account for motion-induced E/B mixing
of polarized observables with $d\neq1$ at different frequencies.
  We apply the generalized aberration kernel on polarized and unpolarized CMB
specific intensity at 100 and 217 GHz and show that the motion-induced effects
typically increase with the frequency of observation. In all-sky CMB
experiments, the frequency-dependence of the motion-induced effects for a
blackbody spectrum are overall negligible. However in a cut-sky analysis,
ignoring the frequency dependence can lead to percent level error in the
polarized and unpolarized power spectra over all angular scales. In the
specific cut-sky used in our analysis ($b > 45^\circ, f_\text{sky}\simeq14\%$),
and for the dipole-inferred velocity $\beta=0.00123$ typically attributed to
our peculiar motion, the Doppler and aberration effects can change polarized
and unpolarized power spectra of specific intensity in the CMB rest frame by
$1-2\%$, but we find the polarization cross-leakage between E and B modes to be
negligible.
