On tractability of linear tensor product problems for ∞∞-variate classes of functions

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=∑ufuf=∑ufu with fu∈Hufu∈Hu, where uu are finite subsets of N+N+ and HuHu are Hilbert spaces of functions with variables listed in uu. The norm of ff is defined by the ℓqℓq norm of {γu−1‖fu‖Hu:u⊂N}, where γuγu’s are given weights and q∈[1,∞]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 γuγu, the value of  qq, and the complexity of the corresponding problem for univariate functions. The previous results were obtained only for the Hilbert case of q=2q=2 and dealt with weighted integration and weighted L2L2-approximation.