**Keywords:**Linear ill-posed problems | regularization | functional equation | Julia's equation.

The solutions of the linear ill-posed problems are frequently obtained by approximation using the regularization functions. In the present work, bounds of the noise-free part of the regularization error of a certain class of functions are obtained through the norm bound functions that decreasing to zero along the regularization parameter, the so called profile functions. Using the properties of such class and assuming apriori smoothness of the true solution, in terms of sourcewise representations, the profile functions are obtained. We verify that qualification property of the regularization class is a consequence of the properties of such functions. We present a method of construction of a regularization class through conjugation technique by using Julia's functional equation. The conjugation procedure allows incorporating the properties of the operator into the regularization class. Some properties of Julia's equation solution are obtained which are useful to construct filters for the discrete regularization. Other examples of application consist of generating the classical Tikhonov-Phillips regularization and the non-smooth regularization methods as Landweber and spectral cut-off are embedding in a regularization class by using a mollification process. Numerical examples are presented showing the robustness of the regularization by conjugation developed here.