Abstract
Approximations from the L 2-closure S of the finite linear combinations of the shifts of a radial basis function are considered, and a thorough analysis of the least-squares approximation orders from such spaces is provided. The results apply to polyharmonic splines, multiquadrics, the Gaussian kernel and other functions, and include the derivation of spectral orders. For stationary refinements it is shown that the saturation class is trivial, i.e., no non-zero function in the underlying Sobolev space can be approximated to a better rate. The approach makes essential use of recenl results of de Boor, DeVore and the author.
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Dedicated to the memory of Lothar Collatz
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Ron, A. (1992). The L2-Approximation Orders of Principal Shift-Invariant Spaces Generated by a Radial Basis Function. In: Braess, D., Schumaker, L.L. (eds) Numerical Methods in Approximation Theory, Vol. 9. ISNM 105: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 105. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8619-2_14
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DOI: https://doi.org/10.1007/978-3-0348-8619-2_14
Publisher Name: Birkhäuser, Basel
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