Multivariate Polynomials: A Spanning Question

There are many different methods of approximating multivariate functions. We have, for example, the classic methods of polynomials, Fourier series, or tensor products, and more modern methods using wavelets, radial basis functions, multivariate splines, or ridge functions. Many of these are natural generalizations of methods developed for approximating univariate functions. However, functions of many variables are fundamentally different from functions of one variable, and approximation techniques for such problems are much less developed and understood. We discuss in these pages one approach to this problem which is truly multivariate in character. To motivate the approach considered, we recall Hilbert 's 13th problem. Although not formulated in the following terms, Hilbert 's 13th problem was interpreted by some as conjecturing that not all functions of three variables could be represented as superposi t ions (compositions) and sums of functions of two variables. Surprisingly it transpired that all functions could be so represented, and even more was true. Kolmogorov and his student Arnold proved in a series of papers in the late 1950s that fixed continuous one-variable functions hii exist such that every continuous function f of d variables on [0, 1] d could be represented in the form