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Preface
Efficient representation and accurate evaluation of oscillatory integrals and functions
1. | Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO 80309-0526, United States, United States |
References:
[1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th edition, Dover Publications, 1970. |
[2] |
G. E. Andrews, R. Askey and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9781107325937. |
[3] |
G. Beylkin and T. S. Haut, Nonlinear approximations for electronic structure calculations, Proc. R. Soc. A, 469 (2013), 20130408.
doi: 10.1098/rspa.2013.0231. |
[4] |
G. Beylkin and L. Monzón, On generalized Gaussian quadratures for exponentials and their applications, Appl. Comput. Harmon. Anal., 12 (2002), 332-373.
doi: 10.1006/acha.2002.0380. |
[5] |
G. Beylkin and L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal., 19 (2005), 17-48.
doi: 10.1016/j.acha.2005.01.003. |
[6] |
G. Beylkin and L. Monzón, Nonlinear inversion of a band-limited Fourier transform, Appl. Comput. Harmon. Anal., 27 (2009), 351-366.
doi: 10.1016/j.acha.2009.04.003. |
[7] |
G. Beylkin and L. Monzón, Approximation of functions by exponential sums revisited, Appl. Comput. Harmon. Anal., 28 (2010), 131-149.
doi: 10.1016/j.acha.2009.08.011. |
[8] |
G. Beylkin and K. Sandberg, Wave propagation using bases for bandlimited functions, Wave Motion, 41 (2005), 263-291.
doi: 10.1016/j.wavemoti.2004.05.008. |
[9] |
G. Beylkin and K. Sandberg, O{DE} solvers using bandlimited approximations, J. Comp. Phys., 265 (2014), 156-171.
doi: 10.1016/j.jcp.2014.02.001. |
[10] |
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 2nd edition, Dover Publications Inc., New York, 1986. |
[11] |
A. Bonami and A. Karoui, Uniform bounds of prolate spheroidal wave functions and eigenvalues decay, Comptes Rendus Mathematique, 352 (2014), 229-234.
doi: 10.1016/j.crma.2014.01.004. |
[12] |
E. Candès and L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris, 336 (2003), 395-398,
doi: 10.1016/S1631-073X(03)00095-5. |
[13] |
E. Candès, L. Demanet and L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput., 29 (2007), 2464-2493.
doi: 10.1137/060671139. |
[14] |
E. J. Candès and L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58 (2005), 1472-1528.
doi: 10.1002/cpa.20078. |
[15] |
M. Condon, A. Deaño and A. Iserles, On highly oscillatory problems arising in electronic engineering, ESAIM Mathematical Modelling and Numerical Analysis, 43 (2009), 785-804.
doi: 10.1051/m2an/2009024. |
[16] |
M. Condon, A. Deaño and A. Iserles, On second-order differential equations with highly oscillatory forcing terms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1809-1828.
doi: 10.1098/rspa.2009.0481. |
[17] |
M. Condon, A. Deaño and A. Iserles, Asymptotic solvers for oscillatory systems of differential equations, SeMA J., 53 (2011), 79-101. |
[18] |
L. Demanet and L. Ying, Fast wave computation via fourier integral operators, Math. Comp., 81 (2012), 1455-1486.
doi: 10.1090/S0025-5718-2012-02557-9. |
[19] |
B. Engquist, A. Fokas, E. Hairer and A. Iserles, Highly Oscillatory Problems, vol. 366 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9781139107136. |
[20] |
G. B. Folland, Real Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1984, Modern techniques and their applications, A Wiley-Interscience Publication. |
[21] |
F. G. Friedlander and J. B. Keller, Asymptotic expansions of solutions of $(\nabla^2+k^2)u=0$, Comm. Pure Appl. Math., 8 (1955), 387-394.
doi: 10.1002/cpa.3160080306. |
[22] |
A. Gil, J. Segura and N. M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
doi: 10.1137/1.9780898717822. |
[23] |
T. S. Haut and G. Beylkin, Fast and accurate con-eigenvalue algorithm for optimal rational approximations, SIAM J. Matrix Anal. Appl., 33 (2012), 1101-1125.
doi: 10.1137/110821901. |
[24] |
T. S. Haut, G. Beylkin and L. Monzón, Solving Burgers' equation using optimal rational approximations, Appl. Comput. Harmon. Anal., 34 (2013), 83-95.
doi: 10.1016/j.acha.2012.03.004. |
[25] |
A. Iserles and D. Levin, Asymptotic expansion and quadrature of composite highly oscillatory integrals, Mathematics of Computation, 80 (2011), 279-296.
doi: 10.1090/S0025-5718-2010-02386-5. |
[26] |
A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772.
doi: 10.1007/s10543-004-5243-3. |
[27] |
A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1383-1399.
doi: 10.1098/rspa.2004.1401. |
[28] |
A. Iserles and S. P. Nørsett, From high oscillation to rapid approximation. III. Multivariate expansions, IMA J. Numer. Anal., 29 (2009), 882-916.
doi: 10.1093/imanum/drn020. |
[29] |
A. Iserles, S. P. Nørsett and S. Olver, Highly oscillatory quadrature: The story so far, in Numerical mathematics and advanced applications, Springer, Berlin, 2006, 97-118.
doi: 10.1007/978-3-540-34288-5_6. |
[30] |
H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell System Tech. J., 40 (1961), 65-84.
doi: 10.1002/j.1538-7305.1961.tb03977.x. |
[31] |
H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty III, Bell System Tech. J., 41 (1962), 1295-1336.
doi: 10.1002/j.1538-7305.1962.tb03279.x. |
[32] |
P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646.
doi: 10.1215/S0012-7094-57-02471-7. |
[33] |
R. D. Lewis, G. Beylkin and L. Monzón, Fast and accurate propagation of coherent light, Proc. R. Soc. A, 469 (2013), 20130458.
doi: 10.1098/rspa.2013.0323. |
[34] |
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974, Computer Science and Applied Mathematics. |
[35] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010. |
[36] |
A. Osipov, Certain inequalities involving prolate spheroidal wave functions and associated quantities, Applied and Computational Harmonic Analysis, 35 (2013), 359-393.
doi: 10.1016/j.acha.2012.10.002. |
[37] |
A. Osipov, V. Rokhlin and H. Xiao, Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation, vol. 187, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4614-8259-8. |
[38] |
M. Reynolds, G. Beylkin and L. Monzón, On generalized Gaussian quadratures for bandlimited exponentials, Appl. Comput. Harmon. Anal., 34 (2013), 352-365.
doi: 10.1016/j.acha.2012.07.002. |
[39] |
D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J., 43 (1964), 3009-3057.
doi: 10.1002/j.1538-7305.1964.tb01037.x. |
[40] |
D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty V. The discrete case, Bell System Tech. J., 57 (1978), 1371-1430.
doi: 10.1002/j.1538-7305.1978.tb02104.x. |
[41] |
D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Review, 25 (1983), 379-393.
doi: 10.1137/1025078. |
[42] |
D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell System Tech. J., 40 (1961), 43-63.
doi: 10.1002/j.1538-7305.1961.tb03976.x. |
[43] |
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. |
[44] |
H. Xiao, V. Rokhlin and N. Yarvin, Prolate spheroidal wavefunctions, quadrature and interpolation, Inverse Problems, 17 (2001), 805-838.
doi: 10.1088/0266-5611/17/4/315. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th edition, Dover Publications, 1970. |
[2] |
G. E. Andrews, R. Askey and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9781107325937. |
[3] |
G. Beylkin and T. S. Haut, Nonlinear approximations for electronic structure calculations, Proc. R. Soc. A, 469 (2013), 20130408.
doi: 10.1098/rspa.2013.0231. |
[4] |
G. Beylkin and L. Monzón, On generalized Gaussian quadratures for exponentials and their applications, Appl. Comput. Harmon. Anal., 12 (2002), 332-373.
doi: 10.1006/acha.2002.0380. |
[5] |
G. Beylkin and L. Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal., 19 (2005), 17-48.
doi: 10.1016/j.acha.2005.01.003. |
[6] |
G. Beylkin and L. Monzón, Nonlinear inversion of a band-limited Fourier transform, Appl. Comput. Harmon. Anal., 27 (2009), 351-366.
doi: 10.1016/j.acha.2009.04.003. |
[7] |
G. Beylkin and L. Monzón, Approximation of functions by exponential sums revisited, Appl. Comput. Harmon. Anal., 28 (2010), 131-149.
doi: 10.1016/j.acha.2009.08.011. |
[8] |
G. Beylkin and K. Sandberg, Wave propagation using bases for bandlimited functions, Wave Motion, 41 (2005), 263-291.
doi: 10.1016/j.wavemoti.2004.05.008. |
[9] |
G. Beylkin and K. Sandberg, O{DE} solvers using bandlimited approximations, J. Comp. Phys., 265 (2014), 156-171.
doi: 10.1016/j.jcp.2014.02.001. |
[10] |
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, 2nd edition, Dover Publications Inc., New York, 1986. |
[11] |
A. Bonami and A. Karoui, Uniform bounds of prolate spheroidal wave functions and eigenvalues decay, Comptes Rendus Mathematique, 352 (2014), 229-234.
doi: 10.1016/j.crma.2014.01.004. |
[12] |
E. Candès and L. Demanet, Curvelets and Fourier integral operators, C. R. Math. Acad. Sci. Paris, 336 (2003), 395-398,
doi: 10.1016/S1631-073X(03)00095-5. |
[13] |
E. Candès, L. Demanet and L. Ying, Fast computation of Fourier integral operators, SIAM J. Sci. Comput., 29 (2007), 2464-2493.
doi: 10.1137/060671139. |
[14] |
E. J. Candès and L. Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58 (2005), 1472-1528.
doi: 10.1002/cpa.20078. |
[15] |
M. Condon, A. Deaño and A. Iserles, On highly oscillatory problems arising in electronic engineering, ESAIM Mathematical Modelling and Numerical Analysis, 43 (2009), 785-804.
doi: 10.1051/m2an/2009024. |
[16] |
M. Condon, A. Deaño and A. Iserles, On second-order differential equations with highly oscillatory forcing terms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 1809-1828.
doi: 10.1098/rspa.2009.0481. |
[17] |
M. Condon, A. Deaño and A. Iserles, Asymptotic solvers for oscillatory systems of differential equations, SeMA J., 53 (2011), 79-101. |
[18] |
L. Demanet and L. Ying, Fast wave computation via fourier integral operators, Math. Comp., 81 (2012), 1455-1486.
doi: 10.1090/S0025-5718-2012-02557-9. |
[19] |
B. Engquist, A. Fokas, E. Hairer and A. Iserles, Highly Oscillatory Problems, vol. 366 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9781139107136. |
[20] |
G. B. Folland, Real Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1984, Modern techniques and their applications, A Wiley-Interscience Publication. |
[21] |
F. G. Friedlander and J. B. Keller, Asymptotic expansions of solutions of $(\nabla^2+k^2)u=0$, Comm. Pure Appl. Math., 8 (1955), 387-394.
doi: 10.1002/cpa.3160080306. |
[22] |
A. Gil, J. Segura and N. M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
doi: 10.1137/1.9780898717822. |
[23] |
T. S. Haut and G. Beylkin, Fast and accurate con-eigenvalue algorithm for optimal rational approximations, SIAM J. Matrix Anal. Appl., 33 (2012), 1101-1125.
doi: 10.1137/110821901. |
[24] |
T. S. Haut, G. Beylkin and L. Monzón, Solving Burgers' equation using optimal rational approximations, Appl. Comput. Harmon. Anal., 34 (2013), 83-95.
doi: 10.1016/j.acha.2012.03.004. |
[25] |
A. Iserles and D. Levin, Asymptotic expansion and quadrature of composite highly oscillatory integrals, Mathematics of Computation, 80 (2011), 279-296.
doi: 10.1090/S0025-5718-2010-02386-5. |
[26] |
A. Iserles and S. P. Nørsett, On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44 (2004), 755-772.
doi: 10.1007/s10543-004-5243-3. |
[27] |
A. Iserles and S. P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 1383-1399.
doi: 10.1098/rspa.2004.1401. |
[28] |
A. Iserles and S. P. Nørsett, From high oscillation to rapid approximation. III. Multivariate expansions, IMA J. Numer. Anal., 29 (2009), 882-916.
doi: 10.1093/imanum/drn020. |
[29] |
A. Iserles, S. P. Nørsett and S. Olver, Highly oscillatory quadrature: The story so far, in Numerical mathematics and advanced applications, Springer, Berlin, 2006, 97-118.
doi: 10.1007/978-3-540-34288-5_6. |
[30] |
H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell System Tech. J., 40 (1961), 65-84.
doi: 10.1002/j.1538-7305.1961.tb03977.x. |
[31] |
H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty III, Bell System Tech. J., 41 (1962), 1295-1336.
doi: 10.1002/j.1538-7305.1962.tb03279.x. |
[32] |
P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646.
doi: 10.1215/S0012-7094-57-02471-7. |
[33] |
R. D. Lewis, G. Beylkin and L. Monzón, Fast and accurate propagation of coherent light, Proc. R. Soc. A, 469 (2013), 20130458.
doi: 10.1098/rspa.2013.0323. |
[34] |
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York-London, 1974, Computer Science and Applied Mathematics. |
[35] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010. |
[36] |
A. Osipov, Certain inequalities involving prolate spheroidal wave functions and associated quantities, Applied and Computational Harmonic Analysis, 35 (2013), 359-393.
doi: 10.1016/j.acha.2012.10.002. |
[37] |
A. Osipov, V. Rokhlin and H. Xiao, Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation, vol. 187, Springer Science & Business Media, 2013.
doi: 10.1007/978-1-4614-8259-8. |
[38] |
M. Reynolds, G. Beylkin and L. Monzón, On generalized Gaussian quadratures for bandlimited exponentials, Appl. Comput. Harmon. Anal., 34 (2013), 352-365.
doi: 10.1016/j.acha.2012.07.002. |
[39] |
D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J., 43 (1964), 3009-3057.
doi: 10.1002/j.1538-7305.1964.tb01037.x. |
[40] |
D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty V. The discrete case, Bell System Tech. J., 57 (1978), 1371-1430.
doi: 10.1002/j.1538-7305.1978.tb02104.x. |
[41] |
D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Review, 25 (1983), 379-393.
doi: 10.1137/1025078. |
[42] |
D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell System Tech. J., 40 (1961), 43-63.
doi: 10.1002/j.1538-7305.1961.tb03976.x. |
[43] |
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. |
[44] |
H. Xiao, V. Rokhlin and N. Yarvin, Prolate spheroidal wavefunctions, quadrature and interpolation, Inverse Problems, 17 (2001), 805-838.
doi: 10.1088/0266-5611/17/4/315. |
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