Rigorous luminosity function determination in presence of a background: theory and application to two intermediate redshift clusters

In this paper we present a rigorous derivation of the luminosity function (LF) in presence of a background. Our approach is free from the logical contradictions of assigning negative values to positively defined quantities and avoid the use of incorrect estimates for the 68 % confidence interval (error bar). It accounts for Poisson fluctuations ignored in previous approaches and does not requires binning of the data. The method is extensible to more complex situations, does not require the existence of an environment--independent LF, and clarifies issues common to field LF derivations. We apply the method to two clusters of galaxies at intermediate redshift (z~0.3) with among the deepest and widest K_s observations ever taken. Finally, we point out short-comings of flip--flopping magnitudes.