K-functionals on L1 (Rn) related to the Laplacian

The K-functional with respect to L-1 (R-n) and the domain of the Laplacian Delta in the weak sense in L-1(R-n) (as well as of (-Delta)(alpha/2), alpha > 0,) is characterized in different ways. These characterizations make evident that the K-functional is a perfect interface between the smoothness of a function and the rate of convergence arising when the function is approximated by appropriate linear means. In particular, a short and straightforward proof of a result of Ditzian 1991 is given.