Applicability of Smolyak's Algorithms to Certain Banach Spaces of Multivariate Functions

Abstract

We investigate the applicability of Smolyak's algorithm to tensor product problems on certain Banach spaces of multivariate functions. First, we show that the algorithm can be efficiently used for the integration problem on these function classes. For approximation problems on the Sobolev space W 1, γ r 1,…, r d , we prove that the algorithm is applicable as well; the range spaces can be any Banach spaces of functions, provided that the tensor product of these spaces is natural. On the other hand, if the range spaces are the univariate smooth function classes C γ k r k , the same conclusion can be drawn for approximation problems on any natural tensor products of Banach spaces of functions. Applications are illustrated for the integration problem on W p r 1,…, r d ([0,1] d ).