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 algebra$F_d:=\{f\in C(T^d\left)\right|{\Vert f\Vert }_{F_d}:=\sum _{k\in Z^d}\left|\hat{f}\right(k\left)\right|\max (1,\underset{j\in \mathrm{supp}\left(k\right)}{\min }\log |k_j\left|\right)<\infty \},$ where $T:=R/Z=[0,1)$, $\hat{f}\left(k\right)$ is the k-th Fourier coefficient of f and $\mathrm{supp}\left(k\right):=\{j\in \{1,\dots ,d\left\}\right|k_j\ne 0\}$. Goda raised an open question as to whether the upper bound of the information complexity for integration in $F_d$ 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 $\Theta (d/{ϵ}^3)$, where $ϵ\in (0,1/2)$ 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 $\Theta (\max (\frac{d^2}{{ϵ}^2},\frac{d^{1/q}}{{ϵ}^{1/\alpha }}\left)\right)$, we present a new upper bound $\Theta \left(\right(\frac{d\log d}{q}+\frac{d\log (1/ϵ)}{\alpha })/{ϵ}^2)$, where $q\in [1,\infty ),\alpha \in (0,1]$ 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.