Complimentary information using in the task of averaging operators inversion in function space

The dual task of integral geometry — to define for a given averaging operator the function class where inversion of that operator is possible — is solved. Those classes are defined ambiguously. Full description of those classes is given in the form of minimal complimentary information necessary to know about the function. The possible to give a constructive description of the class is researched and in the case of a finite averaging system the inversion formulas are given.