Improved estimates for nonoscillatory phase functions

TL;DR

This paper improves the theoretical understanding of nonoscillatory phase functions, enabling more accurate and efficient numerical solutions for highly oscillatory second order differential equations.

Contribution

It establishes an enhanced existence theorem showing solutions can be represented with exponentially decaying Fourier transforms in Schwartz space, improving approximation accuracy.

Findings

Solutions can be approximated with accuracy of order λ^{-1} exp(-μλ)

Solutions are representable using Schwartz functions with compactly supported Fourier transforms

The results facilitate a λ-independent numerical solution method for second order ODEs

Abstract

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $\lambda$ of the equation, there exists a function whose Fourier transform decays as $\exp(-\mu |\xi|)$ and which represents solutions of the differential equation with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solutions of second order linear ordinary differential equations can be represented with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$ using functions in the space of rapidly decaying Schwartz functions whose Fourier transforms are both exponentially decaying and compactly supported. These new observations play an important role in the analysis of a method for the numerical solution of second order ordinary differential equations whose running time is independent of the parameter $\lambda$. This algorithm will be reported at a later date.