Notes: Selected papers from the International Conference on Distribution Theory, Order Statistics, and Inference held in honor of the 65th birthday of Barry C.\ Arnold at the University of Cantabria, Santander, June 16â18, 2004
Abstract: In this article functional equations are used to characterize some families of surfaces. First, the most general surfaces in implicit form $f(x,y,z) = 0$, such that any arbitrary intersection with the planes $z = z0$, $y = y0$, and $x = x0$ are linear combinations of sets of functions of the other two variables, are characterized. It is shown that only linear combinations of tensor products of univariate functions are possible for $f(x,y,z)$. Second, we obtain the most general families of surfaces in explicit form such that their intersections with planes parallel to the planes $y = 0$ and $x = 0$ belong to two, not necessarily equal, parametric families of curves. Finally, functional equations are used to analyze the uniqueness of representation of Gordon-Coons surfaces. Some practical examples are used to illustrate the theoretical results.