Abstract
A compactly supported radially symmetric function \(\varPhi :\Bbb{R}^{d}\to \Bbb{R}\) is said to have Sobolev regularity k if there exist constants B≥A>0 such that the Fourier transform of Φ satisfies
Such functions are useful in radial basis function methods because the resulting native space will correspond to the Sobolev space \(W_{2}^{k}(\Bbb{R}^{d})\). For even dimensions d and integers k≥d/4, we construct piecewise polyharmonic radial functions with Sobolev regularity k. Two families are actually constructed. In the first, the functions have k nontrivial pieces, while in the second, exactly one nontrivial piece. We also explain, in terms of regularity, the effect of restricting Φ to a lower dimensional space \(\Bbb{R}^{d-2\ell}\) of the same parity.
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Communicated by Edward B. Saff.
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Johnson, M.J. Compactly Supported, Piecewise Polyharmonic Radial Functions with Prescribed Regularity. Constr Approx 35, 201–223 (2012). https://doi.org/10.1007/s00365-011-9141-z
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DOI: https://doi.org/10.1007/s00365-011-9141-z