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 ≥ 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

$$ I_{k}^{\varrho} (f) = {\int}_{\mathbb{R}} f(x) \,\mathrm{e}^{-i\,kx} \varrho(x) \, \mathrm{d} x\ \ \ \text{for}\ \ f\in H^{s}(\mathbb{R})\ \ \text{or}\ \ f\in C^{s}(\mathbb{R}) $$

(1)

with $k\in {\mathbb {R}}$ and a smooth density function ρ 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 Θ notation dependent only on s and ϱ.