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

• Authors: G. W. Wasilkowski
• Venue: Journal of Complexity
• Year: 2013

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=@?"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 @?"q norm of {@c"u^-^1@?f"u@?"H"""u:u@?N}, where @c"u'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 @c"u, 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.