Truncation Dimension for Function Approximation

Abstract: We consider approximation of functions of $s$ variables, where $s$ is very large or infinite, that belong to weighted anchored spaces. We study when such functions can be approximated by algorithms designed for functions with only very small number ${\rm dim<sup>{trnc}}(\varepsilon)$ of variables. Here $\varepsilon$ is the error demand and we refer to ${\rm dim<sup>{trnc}}(\varepsilon)$ as the $\varepsilon$-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about $\varepsilon \approx 10<sup>{-5}$) the truncation dimension is surprisingly very small.