Complexity of oscillatory integrals on the real line

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space H^s (ℝ) and from the space C^s(ℝ) with an arbitrary integer s ≥ 1. We find tight upper and lower bounds for the worst case error of optimal algorithms that use n function values. More specifically, we study integrals of the form 1 I_k^ϱ (f) = ∫_ℝ f(x) e^-i kxϱ(x) d x for f∈ H^s(ℝ) or f∈ C^s(ℝ) with k∈ℝ and a smooth density function ρ such as ρ (x) = 1/√(2 π)exp (-x^2/2) . The optimal error bounds are Θ((n+max (1,|k|))^-s) with the factors in the Θ notation dependent only on s and ϱ .