Truncation dimension for linear problems on multivariate function spaces

The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are the following: when is it possible to approximate the solution for the original function of very many variables by the solution for the same function, however with all but the first k variables set to zero, so that the corresponding error is small? What is the truncation dimension , i.e., the smallest number k = k ( ε ) such that the corresponding error is bounded by a given error demand ε ? Surprisingly, k ( ε ) could be very small even for weights with a modest speed of convergence to zero.