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We study the problem of constructing rank-1 lattice rules which have good bounds on the " weighted star discrepancy ". Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used… (More)

We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy , for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy-Krause variations over all lower dimensional projections of the… (More)

We study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions. We consider integrands that are square integrable but whose Fourier series are not necessarily absolutely convergent. For such integrands, a bound on the variance is expressed through a… (More)

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