# 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
##### References:
 [1] G. Andrews, R. Askey and R. Roy, Special Functions,, Cambridge University Press, (1999). doi: 10.1017/CBO9781107325937. [2] R. Bellman, Stability Theory of Differential Equations,, Dover Publications, (1953). [3] O. Borůvka, Linear Differential Transformations of the Second Order,, The English University Press, (1971). [4] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Krieger Publishing Company, (1955). [5] A. O. Daalhuis, Hyperasymptotic solutions of second-order linear differential equations. II,, Methods and Applications of Analysis, 2 (1995), 198. doi: 10.4310/MAA.1995.v2.n2.a5. [6] A. O. Daalhuis and F. W. J. Olver, Hyperasymptotic solutions of second-order linear differential equations. I,, Methods and Applications of Analysis, 2 (1995), 173. doi: 10.4310/MAA.1995.v2.n2.a4. [7] M. V. Fedoryuk, Asymptotic Analysis,, Springer-Verlag, (1993). doi: 10.1007/978-3-642-58016-1. [8] G. B. Folland, Real Analysis: Modern Techniques and Their Application,, 2nd edition, (1999). [9] M. Goldstein and R. M. Thaler, Bessel functions for large arguments,, Mathematical Tables and Other Aids to Computation, 12 (1958), 18. doi: 10.2307/2002123. [10] L. Grafakos, Classical Fourier Analysis,, Springer, (2014). doi: 10.1007/978-1-4939-1194-3. [11] L. Grafakos, Modern Fourier Analysis,, Springer, (2009). doi: 10.1007/978-0-387-09434-2. [12] Z. Heitman, J. Bremer and V. Rokhlin, On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime,, Journal of Computational Physics, 290 (2015), 1. doi: 10.1016/j.jcp.2015.02.028. [13] Z. Heitman, J. Bremer, V. Rokhlin and B. Vioreanu, On the asymptotics of Bessel functions in the Fresnel regime,, Applied and Computational Harmonic Analysis, 39 (2015), 347. doi: 10.1016/j.acha.2014.12.002. [14] L. Hörmader, The Analysis of Linear Partial Differential Operators I,, 2nd edition, (1990). doi: 10.1007/978-3-642-61497-2. [15] L. Hörmader, The Analysis of Linear Partial Differential Operators II,, 2nd edition, (1990). [16] E. Kummer, De generali quadam aequatione differentiali tertti ordinis,, Progr. Evang. Köngil. Stadtgymnasium Liegnitz., (). [17] F. Neuman, Global Properties of Linear Ordinary Differential Equations,, Kluwer Academic Publishers, (1991). [18] F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions,, Cambridge University Press, (2010). [19] W. Rudin, Principles of Mathematical Analysis,, McGraw-Hill, (1976). [20] J. Segura, Bounds for the ratios of modified Bessel functions and associated Turán-type inequalities,, Journal of Mathematics Analysis and Applications, 374 (2011), 516. doi: 10.1016/j.jmaa.2010.09.030. [21] R. Spigler and M. Vianello, The phase function method to solve second-order asymptotically polynomial differential equations,, Numerische Mathematik, 121 (2012), 565. doi: 10.1007/s00211-011-0441-9. [22] N. Trefethen, Approximation Theory and Approximation Practice,, Society for Industrial and Applied Mathematics, (2013). [23] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Volume I: Fixed-point Theorems,, Springer-Verlag, (1986). doi: 10.1007/978-1-4612-4838-5.

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##### References:
 [1] G. Andrews, R. Askey and R. Roy, Special Functions,, Cambridge University Press, (1999). doi: 10.1017/CBO9781107325937. [2] R. Bellman, Stability Theory of Differential Equations,, Dover Publications, (1953). [3] O. Borůvka, Linear Differential Transformations of the Second Order,, The English University Press, (1971). [4] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Krieger Publishing Company, (1955). [5] A. O. Daalhuis, Hyperasymptotic solutions of second-order linear differential equations. II,, Methods and Applications of Analysis, 2 (1995), 198. doi: 10.4310/MAA.1995.v2.n2.a5. [6] A. O. Daalhuis and F. W. J. Olver, Hyperasymptotic solutions of second-order linear differential equations. I,, Methods and Applications of Analysis, 2 (1995), 173. doi: 10.4310/MAA.1995.v2.n2.a4. [7] M. V. Fedoryuk, Asymptotic Analysis,, Springer-Verlag, (1993). doi: 10.1007/978-3-642-58016-1. [8] G. B. Folland, Real Analysis: Modern Techniques and Their Application,, 2nd edition, (1999). [9] M. Goldstein and R. M. Thaler, Bessel functions for large arguments,, Mathematical Tables and Other Aids to Computation, 12 (1958), 18. doi: 10.2307/2002123. [10] L. Grafakos, Classical Fourier Analysis,, Springer, (2014). doi: 10.1007/978-1-4939-1194-3. [11] L. Grafakos, Modern Fourier Analysis,, Springer, (2009). doi: 10.1007/978-0-387-09434-2. [12] Z. Heitman, J. Bremer and V. Rokhlin, On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime,, Journal of Computational Physics, 290 (2015), 1. doi: 10.1016/j.jcp.2015.02.028. [13] Z. Heitman, J. Bremer, V. Rokhlin and B. Vioreanu, On the asymptotics of Bessel functions in the Fresnel regime,, Applied and Computational Harmonic Analysis, 39 (2015), 347. doi: 10.1016/j.acha.2014.12.002. [14] L. Hörmader, The Analysis of Linear Partial Differential Operators I,, 2nd edition, (1990). doi: 10.1007/978-3-642-61497-2. [15] L. Hörmader, The Analysis of Linear Partial Differential Operators II,, 2nd edition, (1990). [16] E. Kummer, De generali quadam aequatione differentiali tertti ordinis,, Progr. Evang. Köngil. Stadtgymnasium Liegnitz., (). [17] F. Neuman, Global Properties of Linear Ordinary Differential Equations,, Kluwer Academic Publishers, (1991). [18] F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions,, Cambridge University Press, (2010). [19] W. Rudin, Principles of Mathematical Analysis,, McGraw-Hill, (1976). [20] J. Segura, Bounds for the ratios of modified Bessel functions and associated Turán-type inequalities,, Journal of Mathematics Analysis and Applications, 374 (2011), 516. doi: 10.1016/j.jmaa.2010.09.030. [21] R. Spigler and M. Vianello, The phase function method to solve second-order asymptotically polynomial differential equations,, Numerische Mathematik, 121 (2012), 565. doi: 10.1007/s00211-011-0441-9. [22] N. Trefethen, Approximation Theory and Approximation Practice,, Society for Industrial and Applied Mathematics, (2013). [23] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Volume I: Fixed-point Theorems,, Springer-Verlag, (1986). doi: 10.1007/978-1-4612-4838-5.
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