Informational entropy of non-random non-differentiable functions. An approach via fractional calculus

Abstract

By combining the explicit formula of the Shannon informational entropy ) , ( Y X H for two random variables X and Y , with the entropy )) ( ( X f H of ) (X f where (.) f is a real-valued differentiable function, we have shown that the density of the amount of information in Shannon sense involved in a non-random differentiable function is defined by the logarithm of the absolute value of its derivative (Maximum entropy, information without probability and complex fractals, Springer, 2009). Here, we shall show that in the case when the function ) (x f is non-differentiable, then it is possible to extend this result, but now, by using fractional derivative. The article contains three main parts. In the first one, one gives the essential of the fractional differential calculus as we re-visited it (Fractional differential calculus for non-differentiable function, LAP-LAMBERT, 2013), then we shall bear in mind some basic results on the Shannon informational entropy of non-random functions, and then we shall arrive at the entropy of functions which are non-differential but have fractional derivative of order lower than the unity. It is shown that the density of entropy of a given function ) (x f is c x f ) ( ) ( where c denotes a positive constant. As a result, it will be possible to meaningfully replace )) ( ( X f H by (.)) ( ) ( f H X H  and to consider some problems like Jaynes’ maximum entropy principle for instance with new points of view.