Summary
For spaces of polynomial functions on the sphere which are invariant against rotation, the square-sums of the Lagrangians can be estimated by means of the smallest eigenvalue of a positive definite system matrix defined by the reproducing kernel and the nodes used. As a consequence, bounds for the corresponding Lebesgueconstants are obtained. There are examples where the method leads to an estimation of the square-sum by one, which cannot be improved. In this case the Lagrangians perform an extremal basis.
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Reimer, M. Interpolation on the sphere and bounds for the Lagrangian square sums. Results. Math. 11, 144–164 (1987). https://doi.org/10.1007/BF03323265
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DOI: https://doi.org/10.1007/BF03323265