A Bernstein‐Like Trigonometric Basis: Properties, Curve Design, and Operator Construction
We introduce a novel family of trigonometric basis functions equipped with a shape parameter, analogous to Bernstein functions. These basis functions are employed to construct Bézier‐like curves, termed “trigo‐curves”, which retain the fundamental properties of classical Bézier curves while offering...
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| Published in | Journal of applied mathematics Vol. 2025; no. 1 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
John Wiley & Sons, Inc
01.01.2025
Wiley |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1110-757X 1687-0042 |
| DOI | 10.1155/jama/5676548 |
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| Abstract | We introduce a novel family of trigonometric basis functions equipped with a shape parameter, analogous to Bernstein functions. These basis functions are employed to construct Bézier‐like curves, termed “trigo‐curves”, which retain the fundamental properties of classical Bézier curves while offering enhanced shape control through parameter adjustment, independent of control point positions. We establish the monotonicity‐preserving nature of these curves. Additionally, we develop a new class of linear positive operators based on these trigonometric basis functions. The operators, incorporating an auxiliary parameter, are thoroughly analyzed for their fundamental properties. We establish their convergence rate, derive a modified Voronovskaja theorem, and obtain error bounds in terms of the modulus of continuity. Furthermore, the monotonicity‐preserving properties of these operators are also investigated. |
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| AbstractList | We introduce a novel family of trigonometric basis functions equipped with a shape parameter, analogous to Bernstein functions. These basis functions are employed to construct Bézier-like curves, termed “trigo-curves”, which retain the fundamental properties of classical Bézier curves while offering enhanced shape control through parameter adjustment, independent of control point positions. We establish the monotonicity-preserving nature of these curves. Additionally, we develop a new class of linear positive operators based on these trigonometric basis functions. The operators, incorporating an auxiliary parameter, are thoroughly analyzed for their fundamental properties. We establish their convergence rate, derive a modified Voronovskaja theorem, and obtain error bounds in terms of the modulus of continuity. Furthermore, the monotonicity-preserving properties of these operators are also investigated. |
| Audience | Academic |
| Author | Nouri, Bahareh Saeidian, Jamshid Azizi, Aram |
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| Cites_doi | 10.1016/j.cam.2014.06.014 10.33401/fujma.903140 10.1016/j.matcom.2021.12.011 10.1016/j.cam.2021.113626 10.1016/j.jmaa.2016.12.075 10.4208/jcm.1509-m4414 10.3934/math.2021410 10.1007/BF02591381 10.1080/00207160701821715 10.1016/j.amc.2013.07.073 10.1016/j.advengsoft.2018.09.002 10.37193/CJM.2022.01.01 10.3934/mfc.2021024 10.1155/2023/5516510 10.3390/mca21040044 10.1016/0377-0427(95)00240-5 10.3390/math10030376 10.1016/j.cad.2017.05.008 10.1371/journal.pone.0303340 10.2298/FIL2402705K 10.1016/j.amc.2011.08.030 10.31801/cfsuasmas.510382 10.1186/s13662-020-03001-4 10.18514/MMN.2018.2216 10.3390/sym13101919 10.1002/mma.4559 10.3390/math10071045 10.1007/s00025-018-0922-6 10.1007/s10851-021-01046-y 10.1016/S0097-8493(03)00084-0 10.1007/s10543-014-0468-2 10.22067/ijnao.2022.72818.1064 10.1186/s13660-022-02832-x 10.1155/2020/4036434 10.1016/S0167-8396(98)00031-4 10.1631/jzus.2005.AS0116 10.1109/ACCESS.2019.2953496 10.1007/s13398-020-00903-6 10.3390/math10071149 10.1016/j.cagd.2018.03.020 10.1155/2014/174716 |
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| Copyright | COPYRIGHT 2025 John Wiley & Sons, Inc. Copyright © 2025 Jamshid Saeidian et al. Journal of Applied Mathematics published by John Wiley & Sons Ltd. This is an open access article under the terms of the Creative Commons Attribution License (the “License”), which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. https://creativecommons.org/licenses/by/4.0 |
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| Title | A Bernstein‐Like Trigonometric Basis: Properties, Curve Design, and Operator Construction |
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