Polynomial tractability for integration in an unweighted function space with absolutely convergent Fourier series

Abstract

In this note, we prove that the following function space with absolutely convergent Fourier series \[ F d ≔ { f ∈ L 2 ( [ 0 , 1 ) d ) | ‖ f ‖ ≔ ∑ k ∈ Z d | f ^ ( k ) | max ( 1 , min j ∈ supp ⁡ ( k ) log ⁡ | k j | ) > ∞ } F_d≔\left \{ f\in L^2([0,1)^d)\middle | \|f\|≔\sum _{\boldsymbol {k}\in {\mathbb {Z}}^d}|\hat {f}({\boldsymbol {k}}) |\max \left (1,\min _{j\in \operatorname {supp}({\boldsymbol {k}})}\log |k_j|\right )>\infty \right \} \] with f ^ ( k ) \hat {f}({\boldsymbol {k}}) being the k {\boldsymbol {k}} -th Fourier coefficient of f f and supp ⁡ ( k ) ≔ { j ∈ { 1 , … , d } ∣ k j ≠ 0 } \operatorname {supp}({\boldsymbol {k}}) ≔\{j\!\in \!\{1,\ldots ,d\}\mid k_j\neq 0\} is polynomially tractable for multivariate integration in the worst-case setting. Here polynomial tractability means that the minimum number of function evaluations required to make the worst-case error less than or equal to a tolerance ε \varepsilon grows only polynomially with respect to ε − 1 \varepsilon ^{-1} and d d . It is important to remark that the function space F d F_d is unweighted, that is, all variables contribute equally to the norm of functions. Our tractability result is in contrast to those for most of the unweighted integration problems studied in the literature, in which polynomial tractability does not hold and the problem suffers from the curse of dimensionality. Our proof is constructive in the sense that we provide an explicit quasi-Monte Carlo rule that attains a desired worst-case error bound.