LAUSR

AbstractWe analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space Hs(ℝ)$H^{s} (\mathbb {R})$ and from the space Cs(ℝ)$C^{s}(\mathbb {R})$ 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 1Ikϱ(f)=∫ℝf(x)e−ikxϱ(x)dxforf∈Hs(ℝ)orf∈Cs(ℝ)$$ I_{k}^{\varrho} (f) = {\int}_{\mathbb{R}} f(x) \,\mathrm{e}^{-i\,kx} \varrho(x) \, \mathrm{d} x\ \ \ \text{for}\ \ f\in H^{s}(\mathbb{R})\ \ \text{or}\ \ f\in C^{s}(\mathbb{R}) $$ with k∈ℝ$k\in {\mathbb {R}}$ and a smooth density function ρ such as ρ(x)=12πexp(−x2/2)$ \rho (x) = \frac {1}{\sqrt {2 \pi }} \exp (-x^{2}/2)$. The optimal error bounds are Θ((n+max(1,|k|))−s)${\Theta }((n+\max (1,|k|))^{-s})$ with the factors in the Θ notation dependent only on s and ϱ.