  Methods are studied to compute the boosting effects produced by the observer
motion that modifies and transfers to higher l the isotropic monopole frequency
spectrum of the cosmic background. Explicit analytical solutions for spherical
harmonic coefficients are presented and applied to various background spectra,
alleviating computational effort. High l frequency spectra are led by higher
order derivatives of the spectrum. Tabulated frequency spectra are computed
with a relatively poor frequency resolution in comparison with the Doppler
shift, calling for interpolation. They are affected by uncertainties due to
intrinsic inaccuracies in modelling, observational data or limited computation
accuracy, propagate and increase with the derivative order, possibly preventing
a trustworthy computation to higher l and of the observed monopole. We filter
the original function and the multipole spectra to derive reliable predictions
of the harmonic coefficients. For spectra expressed in Taylor series, we derive
explicit solutions for the harmonic coefficients up to l=6 in terms of spectra
derivatives. We consider filters and study the quality of these methods on
suitable analytical approximations, polluted with simulated noise. We consider
the extragalactic sources microwave background from radio loud AGN and the 21cm
line superimposed to the CMB. Gaussian pre-filtering coupled to a real space
filtering of derivatives allows accurate predictions up to l=6, while log-log
polynomial representation gives accurate solutions at any l. Describing the 21
cm model variety is difficult, so it is relevant to relax assumptions.
Pre-filtering gives accurate predictions up to l=3-4, while further filtering
or boosting amplification/deamplification method improves the results allowing
reasonable estimations. The methods can extend the range of realistic
background models manageable with a fast computation.
