Improved estimates for nonoscillatory phase functions (Q262081)

The authors consider the linear differential equation NEWLINE\[NEWLINEy''(t)+\lambda^2 q(t)y(t)=0,\eqno{(*)}NEWLINE\]NEWLINE \(a\leq t\leq b\), where \(\lambda\) is a real number and \(q\) is a smooth and positive function, along with a phase function. The results presented in this article improve those of [\textit{Z. Heitman} and the authors, ``On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime'', J. Comput. Phys. 290, 1--27 (2015; \url{doi:10.1016/j.jcp.2015.02.028})], which observed that solutions of \((*)\) can be represented with accuracy of the order of \(\exp(-\mu\lambda)\) using functions which are in \(L^2(\mathbb R)\cap C^\infty(\mathbb R)\) and whose Fourier transform decay exponentially. In this article, it is established an improved existence theorem for nonoscillatory phase functions. Among other things, it is shown that solutions of \((*)\) can be represented with accuracy of 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 are used in the analysis of a method for numerical solutions of second order ordinary differential equations whose running time is independent of the parameter \(\lambda\).