On tractability of linear tensor product problems for [formula omitted]-variate classes of functions

Abstract

We study tractability of linear tensor product problems defined on special Banach spaces of ∞-variate functions. In these spaces, functions have a unique decomposition f=∑ufu with fu∈Hu, where u are finite subsets of N+ and Hu are Hilbert spaces of functions with variables listed in u. The norm of f is defined by the ℓq norm of {γu−1‖fu‖Hu:u⊂N}, where γu’s are given weights and q∈[1,∞]. We derive sufficient and necessary conditions for the problem to be tractable. These conditions are expressed in terms of the properties of the weights γu, the value of q, and the complexity of the corresponding problem for univariate functions. The previous results were obtained only for the Hilbert case of q=2 and dealt with weighted integration and weighted L2-approximation.