The mathematical theories of multivariate polynomial interpolation and multivariate spline approximation differ in content and goals, yet share a common source. In addition, many of the mathematical tools used to analyze basic questions are similar. Underlying much of this work has been the problem of developing a strategy for developing a theory of splines in several dimensions which is both computationally effective and as accurate as the one dimensional basis from which the subject arose. Work on multivariate polynomial interpolation derives from problems in what became known as box spline theory. A surprisingly simple and general method for choosing, for any given finite set of points in a space of several variables, a good polynomial space for interpolation at those points, has been discovered. Work will now be done exploring the theoretical and practical ramifications of this discovery. A long-term objective is to construct a coherent theory of interpolation, one which will play a more important role in multivariate numerical analysis. There are many approaches to spline approximation in higher dimensions currently in use. Each is associated with a certain type of mesh along which the elements are joined. The focus of this work will be that of approximation order. A better understanding of what makes for a good approximation order is expected to lead to the construction of better approximation methods. Ultimately, one would like to develop a unification of the various theories and techniques now extant. A particularly important area of application of this work is in providing mathematical models of surfaces, often from a given set of bounding curves. The mainstay of industrial work at this time is a method which only works for relatively flat surfaces. One immediate goal is to obtain a better understanding of how one can tell whether a given surface can be well represented by a small number of patches.
