文摘
We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of s-variate functions. Here s is large including s=∞. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions fk with only k variables, where k=k(ε) depends solely on the error demand 3689491f95f" title="Click to view the MathML source">ε and is surprisingly small when s is sufficiently large relative to 3689491f95f" title="Click to view the MathML source">ε. This holds, in particular, for s=∞ and arbitrary 3689491f95f" title="Click to view the MathML source">ε since then k(ε)<∞ for all 3689491f95f" title="Click to view the MathML source">ε. Moreover k(ε) does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.