The concept of pseudodifference operator is introduced. The properties of a class of pseudodifference operators in spaces of fractional quotients are studied. A local theorem on the uniform convergence of divided differences of arbitrary order for an approximate solution is established. In particular, the local infinite differentiability of a precise solution of operator equations of elliptic type with locally infinitely differentiable right-hand side is proved on the basis of a numerical method. Examples related to applications are presented.