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Interpolation properties of \(C^1\) quadratic splines on hexagonal cells. (English) Zbl 1418.41004

Summary: Let \(\Delta_n\) be a cell with a single interior vertex and \(n\) boundary vertices \(v_1,\dots,v_n\). Say that \(\Delta_n\) has the interpolation property if for every \(z_1,\dots,z_n\in\mathbb R\) there is a spline \(s\in\mathcal S_2^1(\Delta_n)\) such that \(s(v_i)=z_i\) for all \(i\). We investigate under what conditions does a cell fail the interpolation property. The question is related to an open problem posed by P. Alfeld et al. [Approximation Theory Appl. 3, No. 4, 1–10 (1987; Zbl 0687.41017)] about characterization of unconfinable vertices.
For hexagonal cells, we obtain a geometric criterion characterizing the failure of the interpolation property. As a corollary, we conclude that a hexagonal cell such that its six interior edges lie on three lines fails the interpolation property if and only if the cell is projectively equivalent to a regular hexagonal cell. Along the way, we obtain an explicit basis for the vector space \(\mathcal S_2^1(\Delta_n)\) for \(n\geq 5\).

MSC:

41A15 Spline approximation
41A05 Interpolation in approximation theory
41A63 Multidimensional problems
65D07 Numerical computation using splines

Citations:

Zbl 0687.41017
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References:

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