Abstract
In this paper, we present a generalization of Lehmann’s approach for solving approximation problems on hypersurfaces to situations with arbitrary codimension. We show that as in the case of hypersurfaces, the method is able to transfer approximation orders from the ambient space to the submanifold. In particular, the resulting approximant is \({\mathrm {C}}^{m-2}\) and the error decays at an optimal \(h^m\) for tensor product B-splines of order m. Additionally, the method is easily implemented and comes with an optimal computational expense when applying quasi interpolation techniques. Applications include, in particular, surfaces embedded into \({{\mathbb {R}}}^4\) but not into \({{\mathbb {R}}}^3\).
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Communicated by Larry L. Schumaker.
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Maier, LB. Ambient Approximation on Embedded Submanifolds. Constr Approx 52, 183–211 (2020). https://doi.org/10.1007/s00365-020-09502-5
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DOI: https://doi.org/10.1007/s00365-020-09502-5
Keywords
- Spline approximation
- Manifold
- Closest point representation
- Multivariate spline
- Quasi-interpolation
- Data compression
- Local approximation order