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Methods are studied to compute the boosting effects produced by the observermotion that modifies and transfers to higher l the isotropic monopole frequencyspectrum of the cosmic background. Explicit analytical solutions for sphericalharmonic coefficients are presented and applied to various background spectra,alleviating computational effort. High l frequency spectra are led by higherorder derivatives of the spectrum. Tabulated frequency spectra are computedwith a relatively poor frequency resolution in comparison with the Dopplershift, calling for interpolation. They are affected by uncertainties due tointrinsic inaccuracies in modelling, observational data or limited computationaccuracy, propagate and increase with the derivative order, possibly preventinga trustworthy computation to higher l and of the observed monopole. We filterthe original function and the multipole spectra to derive reliable predictionsof the harmonic coefficients. For spectra expressed in Taylor series, we deriveexplicit solutions for the harmonic coefficients up to l=6 in terms of spectraderivatives. We consider filters and study the quality of these methods onsuitable analytical approximations, polluted with simulated noise. We considerthe extragalactic sources microwave background from radio loud AGN and the 21cmline superimposed to the CMB. Gaussian pre-filtering coupled to a real spacefiltering of derivatives allows accurate predictions up to l=6, while log-logpolynomial representation gives accurate solutions at any l. Describing the 21cm model variety is difficult, so it is relevant to relax assumptions.Pre-filtering gives accurate predictions up to l=3-4, while further filteringor boosting amplification/deamplification method improves the results allowingreasonable estimations. The methods can extend the range of realisticbackground models manageable with a fast computation.

On the fast computation of the observer motion effects induced on monopole frequency spectra for tabulated functions