Abstract: The behavior of a variety of phenomena measurable on the boundary of 3D shapes is studied by modeling the set of known measurements as a scalar function f :P $rightarrow $ R, defined on a surface P. Furthermore, the large amount of scientific data calls for efficient techniques to correlate, describe, and analyze this data. In this context, we focus on the problem of extending the measures captured by a scalar function f, defined on the boundary surface P of a 3D shape, to its surrounding volume. This goal is achieved by computing a sequence of volumetric functions that approximate f up to a specified accuracy and preserve its critical points. More precisely, we compute a smooth map g : R³ $rightarrow $ R such that the piecewise linear function h :=gP : P $rightarrow $ R, which interpolates the values of g at the vertices of the triangulated surface P, is an approximation of f with the same critical points. In this way, we overcome the limitation of traditional approaches to function approximation, which are mainly based on a numerical error estimation and do not provide measurements of the topological and geometric features of f. The proposed approximation scheme builds on the properties of f related to its global structure, that is, its critical points, and ignores the local details of f, which can be successively introduced according to the target approximation accuracy.