Regularizations of products of residue and principal value currents

We prove that the Coleff-Herrera residue current, corresponding to a pair of holomorphic functions defining a complete intersection, can be obtained as the unrestricted weak limit of a natural smooth $(0,2)$-form depending on two parameters. Moreover, we prove that the rate of convergence is H\"older. This result is in contrast to the fact, first discovered by Passare and Tsikh, that the residue integral in general is discontinuous at the origin. We also generalize our regularization results to pairs of so called Bochner-Martinelli, or more generally, Cauchy-Fantappie-Leray blocks in the case of a complete intersection.