Absolutely continuous copulas obtained by regularization of the Frech\'{e}t--Hoeffding bounds

We show that the lower and upper Frech\'{e}t-Hoeffding copulas, which are singular, can be regularized to absolutely continuous copulas. The method, which is constructive and explicit, states sufficient conditions for when an absolutely continuous copula can be achieved by averaging. A higher degree of regularisation cannot be achieved with the proposed method.