Strong tractability for multivariate integration in a subspace of the Wiener algebra
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Building upon recent work by the author, we prove that multivariate integration in the following subspace of the Wiener algebra over [0,1)^d is strongly polynomially tractable: F_d:={ f∈ C([0,1)^d) | f:=∑_k∈ℤ^d|f̂(k)|max(width(supp(k)),min_j∈supp(k)log |k_j|)<∞}, with f̂(k) being the k-th Fourier coefficient of f, supp(k):={j∈{1,…,d}| k_j≠ 0}, and width: 2^{1,…,d}→{1,…,d} being defined by width(u):=max_j∈ uj-min_j∈ uj+1, for non-empty subset u⊆{1,…,d} and width(∅):=1. Strong polynomial tractability is achieved by an explicit quasi-Monte Carlo rule using a multiset union of Korobov's p-sets. We also show that, if we replace width(supp(k)) with 1 for all k∈ℤ^d in the above definition of norm, multivariate integration is polynomially tractable but not strongly polynomially tractable.
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