Approximation by exponential sums revisited

Applied and Computational Harmonic Analysis

Volumn 28, Issue 2, 2010, Pages 131-149

  Beylkin, Gregory ^a   Monzón, Lucas ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Approximation by exponentials; Approximation by Gaussians; Models of gravitational potentials; Poisson kernel for half space; Poisson kernel for the sphere; Radial kernels; Separated representations

Indexed keywords

APPROXIMATION BY EXPONENTIALS; GAUSSIANS; GRAVITATIONAL POTENTIAL; HALF-SPACE; POISSON KERNEL; SEPARATED REPRESENTATION; SEPARATED REPRESENTATIONS;

EXPONENTIAL FUNCTIONS; FUNCTION EVALUATION; GRAVITATION; LAPLACE EQUATION; LAPLACE TRANSFORMS; MATHEMATICAL OPERATORS; POISSON DISTRIBUTION; POISSON EQUATION; SEPARATION; SPHERES;

APPROXIMATION ALGORITHMS;

EID: 75949087785     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2009.08.011     Document Type: Article

References (35)

1. Abramowitz M., and Stegun I.A. Handbook of Mathematical Functions. ninth ed. (1970), Dover Publ.
2. Ahrens C., and Beylkin G. Rotationally invariant quadratures for the sphere. Proc. R. Soc. A 465 2110 (2009) 3103-3125
3. Alzer H. Inequalities for the Gamma function. Proc. Amer. Math. Soc. 128 1 (2000) 141-147
4. Alzer H., and Berg C. Some classes of completely monotonic functions. II. Ramanujan J. 11 2 (2006) 225-248
5. Andrews G.E., Askey R., and Roy R. Special Functions. Encyclopedia Math. Appl. vol. 71 (1999), Cambridge Univ. Press, Cambridge
6. Beylkin G. On the fast Fourier transform of functions with singularities. Appl. Comput. Harmon. Anal. 2 4 (1995) 363-381
7. Beylkin G., Cheruvu V., and Pérez F. Fast adaptive algorithms in the non-standard form for multidimensional problems. Appl. Comput. Harmon. Anal. 24 3 (2008) 354-377
8. Beylkin G., and Cramer R. Toward multiresolution estimation efficient representation of gravitational fields. Celest. Mech. Dyn. Astronom. 84 1 (2002) 87-104
9. Beylkin G., Cramer R., Fann G.I., and Harrison R.J. Multiresolution separated representations of singular and weakly singular operators. Appl. Comput. Harmon. Anal. 23 2 (2007) 235-253
10. Beylkin G., Kurcz C., and Monzón L. Grids transforms for band-limited functions in a disk. Inverse Problems 23 5 (2007) 2059-2088
11. Beylkin G., Kurcz C., and Monzón L. Fast algorithms for Helmholtz Green's functions. Proc. R. Soc. A 464 2100 (2008) 3301-3326 10.1098/rspa.2008.0161
12. Beylkin G., Kurcz C., and Monzón L. Fast convolution with the free space Helmholtz Green's function. J. Comput. Phys. 228 8 (2009) 2770-2791
13. Beylkin G., Mohlenkamp M.J., and Pérez F. Preliminary results on approximating a wavefunction as an unconstrained sum of Slater determinants. Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting, Zürich, 2007. Proc. Appl. Math. Mech. 7 1 (2007) 1010301-1010302
14. Beylkin G., Mohlenkamp M.J., and Pérez F. Approximating a wavefunction as an unconstrained sum of Slater determinants. J. Math. Phys. 49 3 (2008) 032107
15. Beylkin G., and Monzón L. On generalized Gaussian quadratures for exponentials their applications. Appl. Comput. Harmon. Anal. 12 3 (2002) 332-373
16. Beylkin G., and Monzón L. On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19 1 (2005) 17-48
17. Beylkin G., and Monzón L. Nonlinear inversion of a band-limited Fourier transform. Appl. Comput. Harmon. Anal. 27 3 (2009) 351-366
18. Boys S.F. The integral formulae for the variational solution of the molecular many-electron wave equations in terms of Gaussian functions with direct electronic correlation. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 258 1294 (1960) 402
19. Braess D. Nonlinear Approximation Theory. Springer Ser. Comput. Math. vol. 7 (1986), Springer-Verlag, Berlin
20. Braess D. Asymptotics for the approximation of wave functions by exponential sums. J. Approx. Theory 83 1 (1995) 93-103
21. Braess D., and Hackbusch W. Approximation of 1 / x by exponential sums in [1, ∞). IMA J. Numer. Anal. 25 4 (2005) 685-697
22. Chen C.-P., and Qi F. Monotonicity results for the gamma function. JIPAM. J. Inequal. Pure Appl. Math. 4 2 (2003) article 44, 6 pp. (electronic)
23. Dubuc S. An approximation of the Gamma function. J. Math. Anal. Appl. 146 2 (1990) 461-468
24. F.G. Lemoine, et al., EGM96: The NASA GSFC and NIMA joint geopotential model, technical report, NASA/TP-1998-206861, 1998
25. Gradshteyn I.S., and Ryzhik I.M. Table of Integrals, Series, and Products. 5th ed. (1994), Academic Press
26. Harrison R.J., Fann G.I., Yanai T., and Beylkin G. Multiresolution quantum chemistry in multiwavelet bases. In: Sloot P.M.A., et al. (Ed). Computational Science - ICCS 2003. Lecture Notes in Comput. Sci. vol. 2660 (2003), Springer-Verlag 103-110
27. Harrison R.J., Fann G.I., Yanai T., Gan Z., and Beylkin G. Multiresolution quantum chemistry: Basic theory and initial applications. J. Chem. Phys. 121 23 (2004) 11587-11598
28. Kutzelnigg W. Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem. 51 (1994) 447-463
29. Li X., and Chen C.-P. Inequalities for the gamma function. JIPAM. J. Inequal. Pure Appl. Math. 8 1 (2007) article 28, 3 pp. (electronic)
30. Longstaff J.V.L., and Singer K. The use of Gaussian (exponential quadratic) wave functions in molecular problems. II. Wave functions for the ground state of the hydrogen atom and of hydrogen molecule. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 258 1294 (1960) 421
31. Qi F., and Mei J.-Q. Some inequalities of the incomplete gamma and related functions. J. Anal. Appl. 18 3 (1999) 793-799
32. Singer K. The use of Gaussian (exponential quadratic) wave functions in molecular problems. I. General formulae for the evaluation of integrals. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 258 1294 (1960) 412
33. Yanai T., Fann G.I., Gan Z., Harrison R.J., and Beylkin G. Multiresolution quantum chemistry: Analytic derivatives for Hartree-Fock and density functional theory. J. Chem. Phys. 121 7 (2004) 2866-2876
34. Yanai T., Fann G.I., Gan Z., Harrison R.J., and Beylkin G. Multiresolution quantum chemistry: Hartree-Fock exchange. J. Chem. Phys. 121 14 (2004) 6680-6688
35. Yarvin N., and Rokhlin V. Generalized Gaussian quadratures singular value decompositions of integral operators. SIAM J. Sci. Comput. 20 2 (1999) 699-718 (electronic)