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