The missing Wendland functions
Advances in Computational Mathematics
Journal of Approximation Theory
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For k@?{1,2,3,...}, we construct an even compactly supported piecewise polynomial @j"k whose Fourier transform satisfies A"k(1+@w^2)^-^k@?@j@^"k(@w)@?B"k(1+@w^2)^-^k, @w@?R, for some constants B"k=A"k0. The degree of @j"k is shown to be minimal, and is strictly less than that of Wendland's function @f"1","k"-"1 when k2. This shows that, for k2, Wendland's piecewise polynomial @f"1","k"-"1 is not of minimal degree if one places no restrictions on the number of pieces.