Approximation theory in multiple dimensions (reference request)

Question

I understand the following from approximation theory: if $f(x)$ is a well-behaved function on some interval $[a,b]\subset\mathbb{R}$ then for any tolerance $\varepsilon$, there exists an $N$th-degree approximating polynomial $P(x)$ whose worst-case error is $\varepsilon$.

I assume that this is also true in multiple dimensions, ie. the case where $f:I\rightarrow\mathbb{R}$ for some compact simply connected $I\subset\mathbb{R}^n$ (or even just a product of intervals). Please could someone give a reference for this?

Answer 1

The paper "Tchebycheff Approximation in Several Variables" by John R. Rice appeared in the Trans. A.M.S. 109 p.444-466 (1963) and is available online at ams.org.

Some difficulties, by comparison with the elegant theory in one variable, were noticed. There is in general "lack of uniqueness of best approximations to functions of more than one variable." However existence and sufficient conditions for a best approximation were developed in terms of a suitably modified concept of "critical set".

There are two basic features of the one variable theory which do not generalize readily. The first of these is the idea of a Tchebycheff set. No counterpart of these sets exists in this paper. A heuristic argument is given that shows that there is no possibility of defining Tchebycheff sets in several variables with the important properties present in the case of one variable. The second feature which does not generalize readily is that of alternation or oscillation. This is an intrinsic feature of Tchebycheff approximations, but it does not appear to be possible to give a simple geometric interpretation of it for functions of several variables.

This doesn't mean that a Remez-type exchange algorithm is impossible. "A Multiple Exchange Algorithm for Multivariate Chebyshev Approximation" by G. A. Watson (SIAM Journal on Numerical Analysis, Vol. 12, No. 1 (Mar., 1975), pp. 46-52) described a linear programming approach with some numerical examples in two dimensions.