Truncation Dimension for Function Approximation

We consider the 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 dimtrnc(e, s) of variables. Here e is the error demand and we refer to dimtrnc(e, s) as the e-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about e ≈ 10−5) the truncation dimension is surprisingly very small.