A method for extracting maximum resolution power spectra from microwave sky maps

A method for extracting maximal resolution power spectra from microwave sky maps is presented and applied to the 2-yr COBE data, yielding a power spectrum that is consistent with a standard n=1, Q_rms-ps=20 muK model. By using weight functions that fall off smoothly near the Galactic cut, it is found that the spectral resolution DeltaL can be more than doubled at L=15 and more than tripled at L=20 compared to simply using Galaxy-cut spherical harmonics. For a future high-resolution experiment with reasonable sky coverage, the resolution around the cold dark matter (CDM) Doppler peaks would be enhanced by a factor of about 100, down to DeltaL~1, thus allowing spectral features such as the locations of the peaks to be determined with great accuracy. The reason that the improvement is so large is basically that functions with a sharp edge at the Galaxy cut exhibit considerable `ringing' in the Fourier domain, whereas smooth functions do not. The method presented here is applicable to any survey geometry, chopping strategy and exposure pattern whatsoever. The so-called signal-to-noise eigenfunction technique is found to be a special case, corresponding to ignoring the width of the window functions.