I shall introduce a natural and simple to implement renormalization scheme of the Hartree-Fock-Bogoliubov (HFB) equations for the case of zero range pairing interaction. This renormalization scheme proves to be equivalent to a simple energy cut-off with a position dependent running coupling constant, which qualitatively emulates a density dependent zero range pairing interaction suggested by phenomenological analyses. I shall describe also fully self--consistent HFB calculations with a full consideration of the continuum of long chains of spherical nuclei (essentially from one drip line to another) using the SLy interaction. where we used a zero range pairing interaction, which for the first time was treated without an arbitrary cutoff, using the new regularization procedure recently developed by us. In the end I shall formulate a local density approximation for fermion systems with pairing correlations based on a rapidly converging renormalization scheme for the pairing gap in the case of zero range pairing interaction.