Estimates of multipolar coefficients to search for cosmic ray anisotropies with non - uniform or partial sky coverage

We study the possibility to extract the multipolar moments of an underlying distribution from a set of cosmic rays observed with non-uniform or even partial sky coverage. We show that if the degree is assumed to be upper bounded by L, each multipolar moment can be recovered whatever the coverage, but with a variance increasing exponentially with the bound L if the coverage is zero somewhere. Despite this limitation, we show the possibility to test predictions of a model without any assumption on L by building an estimate of the covariance matrix seen through the exposure function.