Complexity of Oscillatory Integrals on the Real Line

TL;DR

This paper derives tight bounds for the worst-case error of algorithms approximating univariate oscillatory integrals on the real line for functions in Sobolev and $C^s$ spaces, considering the integral's frequency and density.

Contribution

It provides the first precise error bounds for optimal algorithms approximating oscillatory integrals with arbitrary frequency and smooth density functions.

Findings

Optimal error bounds are $ heta((n+ ext{max}(1,|k|))^{-s})$

Bounds depend only on the smoothness $s$ and the density $ ho$

Results apply to functions in Sobolev space $H^s$ and $C^s$ spaces

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 \[ I_k^\rho (f) = \int_{ {\mathbb{R}}} f(x) \,e^{-i\,kx} \rho(x) \, {\rm d} x\ \ \ \mbox{for}\ \ f\in H^s({\mathbb{R}})\ \ \mbox{or}\ \ f\in C^s({\mathbb{R}}) \] with $k\in {\mathbb{R}}$ and a smooth density function $\rho$ such as $ \rho(x) = \frac{1}{\sqrt{2 \pi}} \exp( -x^2/2) $. The optimal error bounds are $\Theta((n+\max(1,|k|))^{-s})$ with the factors in the $\Theta$ notation dependent only on $s$ and $\rho$.