Complexity of oscillatory integration for univariate Sobolev spaces

We analyze univariate oscillatory integrals for the standard Sobolev spaces Hs of periodic and non-periodic functions with an arbitrary integer s ≥ 1. We find matching lower and upper bounds on the minimal worst case error of algorithms that use n function or derivative values. We also find sharp bounds on the information complexity which is the minimal n for which the absolute or normalized error is at most ε. We show surprising relations between the information complexity and the oscillatory weight. We also briefly consider the case of s =∞. ∗This author was partially supported by the DFG-Priority Program 1324. †This author was partially supported by the DFG GRK 1523. ‡This author was partially supported by the National Science Foundation.