Transfinite interpolation over implicitly defined sets

In a general setting, the transfinite interpolation problem requires constructing a single function f(x) that takes on the prescribed values and/or derivatives on some collection of point sets. The sets of points may contain isolated points, bounded or unbounded curves, as well as surfaces and regions of arbitrary topology. All such closed semi-analytic sets may be represented implicitly by real valued functions with guaranteed differential properties. Furthermore, such functions may be constructed automatically using the theory of R-functions. We show that such implicit representations may be used to solve the general transfinite interpolation problem using a generalization of the classical inverse distance weighting interpolation for scattered data. The constructed interpolants may be used to approximate boundary value and smoothing problems in a meshfree manner.