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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 year COBE data, yielding a power spectrum that is consistent with a standard n=1, Q=20 micro-Kelvin model. By using weight functions that fall off smoothly near the galactic cut, it is found that the spectral resolution \Delta l 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 CDM Doppler peaks would be enhanced by a factor of about 100, down to \Delta l\approx 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.