Multivariate approximation

Definition

Let $d \ge 2$, let $\Omega \subseteq \R^d$, and let $X$ be a normed space of real functions on $\Omega$, such as $C(\Omega)$ for compact $\Omega$ or $L^p(\Omega)$. Let $V_1, V_2, \dots$ be subsets of $X$, called approximating families, usually finite-dimensional subspaces or sets described by $n$ parameters. Multivariate approximation is the approximation of functions $f \in X$ of the $d$ variables by elements of the families $V_n$. Its quality is measured by the error of best approximation $$E(f, V_n) = \inf_{v \in V_n} \norm{f - v}_X.$$ The subject studies how $E(f, V_n)$ depends on $n$, on $d$ and on the smoothness of $f$.