Introduction to the RAFU method on approximation

The RAFU method on approximation uses radical functions (RAFU functions) of odd index to approximate an arbitrary function from the knowledge of its values, or its approximate values, at a set of points. The main goal of this book is to show to the scientific community all the results published until now about this approximation procedure [Co1, Co2, Co3] in a complete work so that they could be applied in the cases that the researchers consider appropriate to do it. As far as we know, the work is original. It does not impress with the difficulties it overcomes. It does not contain complicated calculations or reasonings, but we think that the importance of this technique to solve approximation problems will balance the deficiency of difficulties. We will show some ideas about the beginning of the RAFU method. We will study the approximation of continuous functions defined in [a,b] by means of the RAFU method and its topological foundation. Moreover, we will use this approximation procedure to approach functions belonging to different smoothness spaces.