Blending functions based on trigonometric and polynomial approximations of the Fabius function

Abstract

Most simple blending functions are polynomials, while more advanced blending functions are, for example, rational or expo-rational fractions. The Fabius function has the required properties of a blending function, but is a nowhere analytic function and cannot be calculated exactly everywhere on the required domain. We present a new set of trigonometric and polynomial blending functions with the shape and other properties similar to the Fabius function. They consist of combinations of trigonometric polynomials and piecewise polynomials. The main advanced of these are that they are easy to implement, have low processing costs and have simple derivatives. This makes them very suitable for the calculation of splines. Due to the selfdifferential property of the Fabius function, scaled versions of these functions can even be used to approximate their own derivatives.