Average case complexity of weighted approximation and integration over ℝ₊

We study weighted approximation and integration of Gaussian stochastic processes X defined over ℝ₊ whose rth derivatives satisfy a Hölder condition with exponent β in the quadratic mean. We assume that the algorithms use samples of X at a finite number of points. We study the average case (information) complexity, i.e., the minimal number of samples that are sufficient to approximate/integrate X with the expected error not exceeding E. We provide sufficient conditions in terms of the weight and the parameters r and β for the weighted approximation and weighted integration problems to have finite complexity. For approximation, these conditions are necessary as well. We also provide sufficient conditions for these complexities to be proportional to the complexities of the corresponding problems defined over [0, 1], i.e., proportional to E^-1/α where α = r+β for the approximation and α=r+β+1/2 for the integration.