Blend-to-zero operators for smooth transition functions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918474032349184 |
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| author | Méndez-Cruz, Ivan Amlani, Faisal |
| author_facet | Méndez-Cruz, Ivan Amlani, Faisal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26440 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blend-to-zero operators for smooth transition functions Méndez-Cruz, Ivan Amlani, Faisal Classical Analysis and ODEs 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. |
| title | Blend-to-zero operators for smooth transition functions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.26440 |