Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces

It is known from the analysis by Sloan and Woz´niakowski that under appropriate conditions on the weights, the optimal rate of convergence for multivariate integration in weighted Korobov spaces is <f>O(n−α/2+δ)</f> where <f>α>1</f> is some parameter of the spaces, <f>δ</f> is an arbitrary positive number, and the implied constant in the big O notation depends only on <f>δ</f>, and is independent on the number of variables. Similarly, the optimal rate for weighted Sobolev spaces is <f>O(n−1+δ)</f>. However, their work did not show how to construct rules achieving these rates of convergence. The existing theory of the component-by-component constructions developed by Sloan, Kuo and Joe for the Sobolev case yields the rules achieving <f>O(n−1/2)</f> error bounds. Here we present theorems which show that those lattice rules constructed by the component-by-component algorithms in fact achieve the optimal rate of convergence under appropriate conditions on the weights. [Copyright &y& Elsevier]