# American Institute of Mathematical Sciences

August  2016, 36(8): 4101-4131. doi: 10.3934/dcds.2016.36.4101

## Improved estimates for nonoscillatory phase functions

 1 Department of Mathematics, University of California, Davis, Davis, CA 95616, United States 2 Department of Computer Science, Yale University, New Haven, CT 06511, United States

Received  May 2015 Published  March 2016

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 are used 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.
Citation: James Bremer, Vladimir Rokhlin. Improved estimates for nonoscillatory phase functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4101-4131. doi: 10.3934/dcds.2016.36.4101
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