Quadrature Error Bounds with Applications to Lattice Rules

Reproducing kernel Hilbert spaces are used to derive error bounds and worst-case integrands for a large family of quadrature rules. In the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature. However, the theory developed here does not require periodicity and is not restricted to lattice rules. An analysis of variance (ANOVA) decomposition is employed in defining the inner product. It is shown that imbedded rules are superior when integrating functions with large high-order ANOVA effects.