Blend-to-zero operators for smooth transition functions

TL;DR

This paper develops a formal framework for creating smooth transition functions called blend-to-zero operators, using Hermite interpolation, Beta functions, and boundary value problems, applicable to various function types.

Contribution

It introduces a novel formal framework for blend-to-zero operators, including explicit formulas and constructions for polynomial, staircase, step, and trigonometric functions.

Findings

Explicit representation of polynomial blend-to-zero operators using Beta functions

Construction of smooth staircase and step functions with flat ends

Formulas for trigonometric smooth step functions related to boundary value problems

Abstract

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.