Gabor Transforms on the Sphere with Applications to CMB Power Spectrum Estimation

The Fourier transform of a data set apodized with a window function is known as the Gabor transform. In this paper we extend the Gabor transform formalism to the sphere with the intention of applying it to cosmic microwave background (CMB) data analysis. The Gabor coefficients on the sphere known as the pseudo power spectrum is studied for windows of different size. By assuming that the pseudo power spectrum coefficients are Gaussian distributed, we formulate a likelihood ansatz using these as input parameters to estimate the full-sky power spectrum from a patch on the sky. As this likelihood can be calculated quickly without having to invert huge matrices, this allows for fast power spectrum estimation. By using the pseudo power spectrum from several patches on the sky together, the full-sky power spectrum can be estimated from full-sky or nearly full-sky observations.