A BOUND ON THE APPROXIMATION ORDER OF SURFACE SPLINES

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