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arXiv · 2604.26440

Blend-to-zero operators for smooth transition functions

Ivan Méndez-Cruz·Faisal Amlani
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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.

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Ivan Méndez-Cruz, Faisal Amlani. 2026-04-29. Blend-to-zero operators for smooth transition functions. https://arxiv.org/abs/2604.26440

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