  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.
