Fast cosmic microwave background power spectrum estimation of temperature and polarization with Gabor transforms

We extend the analysis of Gabor transforms on a cosmic microwave background temperature map to polarization. We study the temperature and polarization power spectra on the cut sky, the so-called pseudo-power spectra. The transformation kernels relating the full-sky polarization power spectra and the polarization pseudo-power spectra are found to be similar to the kernel for the temperature power spectrum. This fact is used to construct a fast power spectrum estimation algorithm using the pseudo-power spectrum of temperature and polarization as data vectors in a maximum-likelihood approach. Using the pseudo-power spectra as input to the likelihood analysis solves the problem of having to invert huge matrices, which makes the standard likelihood approach infeasible.