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: 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 $k=k(\varepsilon)$ such that the corresponding error is bounded by a given error demand $\varepsilon$? Surprisingly, $k(\varepsilon)$ could be very small even for weights with a modest speed of convergence to zero.