| Title: | Diagonally-reduced tensor-product polynomials: dimensions and boundary-diagonal matching |
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| Authors: | ID Gradišek, Domen (Author)
ID Grošelj, Jan (Author) |
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| Files: |  PDF - Presentation file, download (3,42 MB)
MD5: 461C89EB831659115E860DB92B097BB2
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0377042726005558
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | We study subspaces of tensor-product polynomials of bi-degree $(n, n)$ on the unit square whose restrictions to the domain diagonals have degree reduced by an integer $k \le n$. For maximal reduction $k = n$, these diagonally-reduced spaces coincide with the Bézier-Smart surface spaces and have dimension $n^2 + 2$. We derive exact dimension formulas and solve the boundary-diagonal matching problem: when can such a polynomial be replaced by a total-degree polynomial that agrees on the boundary and both diagonals? The answer is a sharp threshold on the degree parameters, with a complete characterisation of existence, uniqueness, and non-uniqueness. For the first non-trivial reduction parameters we provide explicit matching formulas and closed-form error estimates; when the solution is non-unique, the remaining freedom is resolved by $L^2$ optimality. At the endpoint $k = n$, where matching always fails, we show for $n = 3$ that the $L^2$-best approximation from the total-degree subspace is also the $L^\infty$-best approximation. |
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| Keywords: | bivariate polynomials, tensor-product polynomials, diagonal curves, boundary-diagonal matching, Bézier-smart surfaces, serendipity elements |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2027 |
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| Year of publishing: | 2027 |
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| Number of pages: | 11 str. |
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| Numbering: | Vol. 489, article 117913 |
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| PID: | 20.500.12556/DiRROS-30698  |
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| UDC: | 519.6 |
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| ISSN on article: | 0377-0427 |
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| DOI: | 10.1016/j.cam.2026.117913 |
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| COBISS.SI-ID: | 283106819 |
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| Note: |
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| Publication date in DiRROS: | 01.07.2026 |
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| Views: | 108 |
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| Downloads: | 100 |
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