Optimal quadrature for Haar wavelet spaces

This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, Hwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ ,t,m,s)-nets and $(t,s)$-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands Hwav.