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=Σ_uf_u with f _u∈H_u, where u are finite subsets of ^N+ and H_u are Hilbert spaces of functions with variables listed in u. The norm of f is defined by the ^ℓq norm of {γ_u ^-1-f_u-H_u: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 L ₂-approximation.