On a mixed Monge-Amp\`ere operator for quasiplurisubharmonic functions with analytic singularities

We consider mixed Monge-Amp`ere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Amp`ere products of smooth functions, generalizing results of Andersson, B{\l}ocki and the last author in the case of non-mixed Monge-Amp`ere products. Connections to the theory of residue currents, going back to Coleff-Herrera, Passare and others, play an important role in the proof. As a consequence we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.