Abstract
We obtain sufficient conditions for arrays of points, \(\mathcal {Z}=\{\mathcal {Z}(L) \}_{L\ge 1}\), on the unit sphere \(\mathcal {Z}(L)\subset \mathbb {S}^d\), to be Marcinkiewicz–Zygmund and interpolating arrays for spaces of spherical harmonics. The conditions are in terms of the mesh norm and the separation radius of \(\mathcal {Z}(L)\).
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Acknowledgments
The authors thank Joaquim Ortega-Cerdà for helpful conversations on the subject matter of this paper. We would also like to express our gratitude to the referee for all his/her comments and suggestions. Supported by Grants MTM2008-05561-C02-01 and 2009 SGR 1303.
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Communicated by Edward B. Saff.
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Marzo, J., Pridhnani, B. Sufficient Conditions for Sampling and Interpolation on the Sphere. Constr Approx 40, 241–257 (2014). https://doi.org/10.1007/s00365-014-9252-4
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DOI: https://doi.org/10.1007/s00365-014-9252-4
Keywords
- Marcinkiewicz–Zygmund inequalities
- Interpolation
- Mesh norm
- Separation radius
- Laplace–Beltrami operator
- Points on the sphere