Sobolev-type L p -approximation orders of radial basis function interpolation to functions in fractional Sobolev spaces

Sobolev-type error analysis has recently been intensively studied for radial basis function interpolation. Although the results have been very successful, some limitations have been found. First, the spaces of target functions are not large enough for thecase $1\leqslant p\leqslant \infty$ to be used practically in some applications. Second, error estimates are confined to the case of finitely smooth radial basis functions. Thus, the primary goal of this paper is to provide Sobolev-type $L_{p}$-error bounds ( $1\leqslant p\leqslant \infty$) to functions in fractional Sobolev spaces for a wide class of radial functions including some infinitely smooth radial functions. Here an infinitely smooth radial function is required to be conditionally positive definite of a certain order $m>0$. In addition we provide numerical results that illustrate our theoretical error bounds.