20.500.12556/DiRROS-30698 Diagonally-reduced tensor-product polynomials: dimensions and boundary-diagonal matching 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. bivariate polynomials tensor-product polynomials diagonal curves boundary-diagonal matching Bézier-smart surfaces serendipity elements true false true Angleški jezik Ni določen Neznano 2026-07-01 10:45:01 2026-07-01 10:45:02 2026-07-02 03:59:41 0000-00-00 00:00:00 2027 0 0 11 str. article 117913 Vol. 489 Jan. 2027 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2027-01-01 519.6 0377-0427 10.1016/j.cam.2026.117913 283106819 1-s2.0-S0377042726005558-main.pdf 1-s2.0-S0377042726005558-main.pdf 1 461C89EB831659115E860DB92B097BB2 60a9119ff18ec6fb6c9e5950ef212b6444678fc45e50e8733e790e2754516090 df83fcdd-7529-11f1-b0d9-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=46829 https://www.sciencedirect.com/science/article/pii/S0377042726005558 1 280d4cef-7529-11f1-b0d9-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=46826 Inštitut za matematiko, fiziko in mehaniko 0 0 0