On the information complexity for integration in subspaces of the Wiener algebra

Abstract

Recently, Goda proved the polynomial tractability of integration on the following function subspace of the Wiener algebraFd:={f∈C(Td)|‖f‖Fd:=∑k∈Zd|fˆ(k)|max⁡(1,minj∈supp(k)⁡log⁡|kj|)<∞}, where T:=R/Z=[0,1), fˆ(k) is the k-th Fourier coefficient of f and supp(k):={j∈{1,…,d}|kj≠0}. Goda raised an open question as to whether the upper bound of the information complexity for integration in Fd can be improved. In this note, we give a positive answer. By establishing a Monte Carlo sampling method and using Rademacher complexity to estimate the uniform convergence rate, the upper bound can be improved to Θ(d/ϵ3), where ϵ∈(0,1/2) is the target accuracy. We also use the same technique to estimate the information complexity for a Hölder continuous subspace of Wiener algebra. Compared to the previous upper bound Θ(max⁡(d2ϵ2,d1/qϵ1/α)), we present a new upper bound Θ((dlog⁡dq+dlog⁡(1/ϵ)α)/ϵ2), where q∈[1,∞),α∈(0,1] are the parameters of Hölder continuity. Ignoring the logarithmic factors, the order of our upper bound is superior to the previous result, especially for the case where the Hölder exponent α is small.