A Bound on the Approximation Order of Surface Splines

The functions m := j j 2m d if d is odd, and m := j j 2m d log j j if d is even, are known as surface splines, and are commonly used in the interpolation or approximation of smooth functions. We show that if one's domain is the unit ball in R, then the approximation order of the translates of m is at most m. This is in contrast to the case when the domain is all of R where it is known that the approximation order is exactly 2m.