What is a multivariate spline?

The various concepts and ideas that have contributed to univariate spline theory are considered with a view to finding a suitable definition of a multivariate spline. In this way, an overview of the existing more or less complete univariate spline theory is given along with a survey of some of the high points of the current research in multivariate splines. My very first paper dealt with multivariate (well, bivariate) splines and I was then quite certain of what a multivariate spline, i.e., a spline function of many variables, might be. Now, many years and several answers later, I am not so sure any more and therefore consider the question worth a forty-minute talk. It is a worthwhile question since univariate splines have been phenomenally successful and one would wish to have available a similarly useful tool for the approximation of functions of several variables. This raises the question of just which features of the univariate spline to generalize. My talk will therefore be in part a survey of the more or less complete univariate spline theory with the aim of deciding which parts to take along into the multivariate context. But before embarking on that discussion, I want to point out that there is available one way of generalization that is specifically designed to require no thought, no new idea (if this construction is satisfactory for you, I have nothing further to tell you). This is the tensor product construct. Here one takes one’s favorite univariate spline class $ and fashion from it splines IR −→ IR : (x, y, . . . , z) 7−→ f(x)g(y) · · ·h(z) This work was sponsored by the U.S. Army under Contract DAAG29-80-C-0041. *CMS, University of Wisconsin, 610 Walnut St., Madison WI 53705