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Efficient representation and accurate evaluation of oscillatory integrals and functions
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 |
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), 198-211.
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), 173-197.
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, Wiley-Interscience, 1999. |
[9] |
M. Goldstein and R. M. Thaler, Bessel functions for large arguments, Mathematical Tables and Other Aids to Computation, 12 (1958), 18-26.
doi: 10.2307/2002123. |
[10] |
L. Grafakos, Classical Fourier Analysis, Springer, New York, 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), 1-27.
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), 347-356.
doi: 10.1016/j.acha.2014.12.002. |
[14] |
L. Hörmader, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer, 1990.
doi: 10.1007/978-3-642-61497-2. |
[15] |
L. Hörmader, The Analysis of Linear Partial Differential Operators II, 2nd edition, Springer, 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), 516-528.
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), 565-586.
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. |
show all references
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), 198-211.
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), 173-197.
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, Wiley-Interscience, 1999. |
[9] |
M. Goldstein and R. M. Thaler, Bessel functions for large arguments, Mathematical Tables and Other Aids to Computation, 12 (1958), 18-26.
doi: 10.2307/2002123. |
[10] |
L. Grafakos, Classical Fourier Analysis, Springer, New York, 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), 1-27.
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), 347-356.
doi: 10.1016/j.acha.2014.12.002. |
[14] |
L. Hörmader, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer, 1990.
doi: 10.1007/978-3-642-61497-2. |
[15] |
L. Hörmader, The Analysis of Linear Partial Differential Operators II, 2nd edition, Springer, 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), 516-528.
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), 565-586.
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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