Blend-to-zero operators for smooth transition functions

Motivated by existing blend-to-zero techniques, a formal framework is developed for defining and constructing blend-to-zero operators on closed intervals for the generation of sufficiently smooth transitions between functions. Such transitions are first formulated as a two-point Hermite-type interpolation that is not necessarily polynomial. It is shown that, in the polynomial case, the corresponding interpolant can be explicitly represented in terms of the regularized incomplete Beta-function. This representation is then used to generate linear blend-to-zero operators. Following this, additional blend-to-zero operators are constructed by considering the algebraic and geometric properties of functions with sufficiently flat ends (e.g., smooth staircase functions and smooth step functions). Finally, explicit formulas for a family of trigonometric smooth step functions are provided, and these functions are shown to be related to certain higher-order two-point boundary value problems.