## Abstract

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_{u}f_{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.

Original language | English |
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Pages (from-to) | 351-369 |

Number of pages | 19 |

Journal | Journal of Complexity |

Volume | 29 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2013 |

## Keywords

- Tensor product problems
- Tractability

## ASJC Scopus subject areas

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Mathematics (all)
- Control and Optimization
- Applied Mathematics