Quadrature for Self-affine Distributions on Rd

This article presents a systematic treatment of quadrature problems for self-similar probability distributions. We introduce explicit deterministic and randomized algorithms and study their errors for integrands of fractional smoothness of Holder-Lipschitz type. Conversely, we derive lower bounds for worst-case errors of arbitrary integration schemes that prove optimality of our algorithms in many cases. In particular, we see that the effective dimension of the quadrature problem for functions of smoothness q > 0 is given by the quantization dimension of order q of the fractal measure.