Complexity of oscillatory integration for univariate Sobolev spaces

We analyze univariate oscillatory integrals for the standard Sobolev spaces HsHs of periodic and non-periodic functions with an arbitrary integer s≥1s≥1. We find matching lower and upper bounds on the minimal worst case error of algorithms that use  nn function or derivative values. We also find sharp bounds on the information complexity which is the minimal nn for which the absolute or normalized error is at most  εε. We show surprising relations between the information complexity and the oscillatory weight. We also briefly consider the case of s=∞s=∞.