Complexity of Oscillatory Integrals on the Real Line
Provides fundamental complexity results for numerical integration of oscillatory functions, relevant to computational mathematics and physics.
The paper establishes tight upper and lower bounds for the worst-case error of optimal algorithms approximating univariate oscillatory integrals over the real line for functions in Sobolev or Hölder spaces, showing the error scales as Θ((n + max(1,|k|))^{-s}).
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 $ρ$.