On tractability of linear tensor product problems for infinity-variate classes of functions
We study tractability of linear tensor product problems defined on special Banach spaces of infinity-variate functions. In these spaces, functions have a unique decomposition f = Sigma(u)f(u) with f(u) is an element of H-u, where u are finite subsets of N+ and H-u are Hilbert spaces of functions with variables listed in u. The norm off is defined by the l(q) norm of {gamma(-1)(u) parallel to f(u)parallel to H-u : u subset of N}, where gamma(u)'s are given weights and q is an element of [1, infinity]. We derive sufficient and necessary conditions for the problem to be tractable. These conditions are expressed in terms of the properties of the weights gamma(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. (C) 2013 Elsevier Inc. All rights reserved.
tractability tensor product problems