Confusion between Wendland RBF functions - missing Wendland functions

Question

I need to compute Wendland functions for a project, and got confused between the formula to construct Wendland functions $\phi(d,k)$ where d is the dimension

So in the original paper introducing these functions (Wendland, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in computational Mathematics, 4(1), 389-396.), the author seems to only compute the functions for odd dimension d and integer k.

Schaback added missing Wendland functions later on (Schaback, R. (2011). The missing Wendland functions. Advances in Computational Mathematics, 34(1), 67-81.), with a different formula, and stated that now $\phi(d,k)$ can be constructed for even d and half-integer k.

In my problem, I don't really care about the value of k, but I need RBFs that work for even dimensions. I am not sure if I should follow the formula in the original paper, or the one in the later paper (and does it matter if I use the method in the original paper for even dimension d?)

Thanks a lot.

Answer 1

Wendland's original RBFs are reproducing kernels for the Sobolev spaces of smoothness $d/2 + k + 1/2$. If $d$ is even, we can't get integer smoothness. The missing Wenland functions fix that.