Complexity of Oscillatory Integrals on the Real Line

Abstract

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. 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 \[ I_k^ρ(f) = \int_{ {\mathbb{R}}} f(x) \,e^{-i\,kx} ρ(x) \, {\rm d} x\ \ \ \mbox{for}\ \ f\in H^s({\mathbb{R}})\ \ \mbox{or}\ \ f\in C^s({\mathbb{R}}) \] with $k\in {\mathbb{R}}$ and a smooth density function $ρ$ such as $ ρ(x) = \frac{1}{\sqrt{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 $ρ$.