The missing Wendland functions

Abstract

The Wendland radial basis functions (Wendland, Adv Comput Math 4: 389-396, 1995) are piecewise polynomial compactly supported reproducing kernels in Hilbert spaces which are norm-equivalent to Sobolev spaces. But they only cover the Sobolev spaces H(d/2+ k+ 1/2)(R(d)), k is an element of N (1) and leave out the integer order spaces in even dimensions. We derive the missing Wendland functions working for half-integer k and even dimensions, reproducing integer-order Sobolev spaces in even dimensions, but they turn out to have two additional non-polynomial terms: a logarithm and a square root. To give these functions a solid mathematical foundation, a generalized version of the "dimension walk" is applied. While the classical dimension walk proceeds in steps of two space dimensions taking single derivatives, the new one proceeds in steps of single dimensions and uses "halved" derivatives of fractional calculus.