Conditionally Lower Riesz Bounds for Scattered Data Interpolation K. J e tte r

Abstract. A general approach for deriving lower Riesz bounds for interpolation with conditionally positive definite radial basis functions is presented. The method refers to the representation of the basis function in terms of a Laplace - Stieltjes integral. Therefore, one can apply the wellknown fact that (scaled) exponentials are positive definite functions. The method is appropriate for getting upper Riesz bounds as well (and hence condition numbers for the collocation matrix ) only for basis functions which are (order 0 conditionally) positive definite. In other cases one is forced to use a different method; e.g., the estimates of [4] can be applied to preconditioned basis functions.