The solutions of the lineal ill-posed problems are frequently obtained by approximation using the regularization functions.
In the present work a new class of the regularization functions is presented. Convergence is obtained under smoothness condition
of the Penrose solution and qualification property of the regularization functions. The presented class of functions is constructed
through conjugation technique by using Julia's functional equation. This procedure allows to incorporate the properties of the operator
into the regularization class. Some results about the solutions of Julia's equation are also obtained. As examples of an application,
classical Tikhonov-Phillips regularization is generated.A numerical result is presented showing the robustness of a regularization by conjugation.