Análisis de la conservación de la información en el proceso de regularización de las funciones generalizadas.

Abstract

A theoretical investigation about the regularization of generalized functions is described. In the development of the work, the mathematical process of regularization of the distributions with test functions and the regularization of real functions with generalized functions was analyzed. Conclusions were drawn about the conservation of information or integrity initially contained in the functions. This proposal was designed to achieve a better understanding of the regularization process since it is a process that is frequently used in the analysis of physics problems. The analysis of the information was made through the study of the properties of the generalized functions; the test functions and the local properties of the generalized functions were investigated, the functionalization regularization process was studied and an analysis of the characteristics of the function before and after the regularization was carried out, besides an analysis of the regularization of real functions through generalized functions. In conclusion, it was obtained that the level of conservation of information in the process of regularization of generalized functions with base functions is very high, since the regularized function changes very little in relation to the original function. In the regularization of real functions with generalized functions, the level of information conservation varies in a noticeable way since there is an important variation of the domain and range of the function. As a contribution of the research, a demonstration of the integral of Dirac is given and the possibility of evaluating a function in its singularity is presented.