Abstract
This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given in Hangelbroek et al. (Math Comput, 2017, in press) for restricted surface splines on \({\mathbb {R}}^d\). The kernels for which the theory applies includes the Sobolev-Matérn kernels for closed, compact, connected, C ∞ Riemannian manifolds.
Dedicated to Ian H. Sloan on the occasion of his 80th Birthday
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Notes
- 1.
The integral can be done exactly. However, we don’t need to do that here.
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Acknowledgements
Thomas Hangelbroek is supported by grant DMS-1413726 from the National Science Foundation. Francis Narcowich and Joseph Ward are supported by grant DMS-1514789 from the National Science Foundation. Christian Rieger is supported by (SFB) 1060 of the Deutsche Forschungsgemeinschaft.
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Hangelbroek, T., Narcowich, F.J., Rieger, C., Ward, J.D. (2018). Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_24
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