Function classes for successful DE-Sinc approximations
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- by Ken’ichiro Tanaka, Masaaki Sugihara and Kazuo Murota PDF
- Math. Comp. 78 (2009), 1553-1571 Request permission
Abstract:
The DE-Sinc formulas, resulting from a combination of the Sinc approximation formula with the double exponential (DE) transformation, provide a highly efficient method for function approximation. In many cases they are more efficient than the SE-Sinc formulas, which are the Sinc approximation formulas combined with the single exponential (SE) transformations. Function classes suited to the SE-Sinc formulas have already been investigated in the literature through rigorous mathematical analysis, whereas this is not the case with the DE-Sinc formulas. This paper identifies function classes suited to the DE-Sinc formulas in a way compatible with the existing theoretical results for the SE-Sinc formulas. Furthermore, we identify alternative function classes for the DE-Sinc formulas, as well as for the SE-Sinc formulas, which are more useful in applications in the sense that the conditions imposed on the functions are easier to verify.References
- Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
- J. McNamee, F. Stenger, and E. L. Whitney, Whittaker’s cardinal function in retrospect, Math. Comp. 25 (1971), 141–154. MR 301428, DOI 10.1090/S0025-5718-1971-0301428-0
- Masatake Mori and Masaaki Sugihara, The double-exponential transformation in numerical analysis, J. Comput. Appl. Math. 127 (2001), no. 1-2, 287–296. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808579, DOI 10.1016/S0377-0427(00)00501-X
- Mayinur Muhammad and Masatake Mori, Double exponential formulas for numerical indefinite integration, J. Comput. Appl. Math. 161 (2003), no. 2, 431–448. MR 2017024, DOI 10.1016/j.cam.2003.05.002
- Frank Stenger, Approximations via Whittaker’s cardinal function, J. Approximation Theory 17 (1976), no. 3, 222–240. MR 481786, DOI 10.1016/0021-9045(76)90086-1
- Frank Stenger, Numerical methods based on sinc and analytic functions, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR 1226236, DOI 10.1007/978-1-4612-2706-9
- Frank Stenger, Summary of Sinc numerical methods, J. Comput. Appl. Math. 121 (2000), no. 1-2, 379–420. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1780056, DOI 10.1016/S0377-0427(00)00348-4
- Masaaki Sugihara, An approximation to the Sinc function that uses a transformation of variables of double exponential function type, Sūrikaisekikenkyūsho K\B{o}kyūroku 990 (1997), 125–134 (Japanese). Theory and application of numerical calculation in science and technology, II (Japanese) (Kyoto, 1996). MR 1608743
- Masaaki Sugihara, Optimality of the double exponential formula—functional analysis approach, Numer. Math. 75 (1997), no. 3, 379–395. MR 1427714, DOI 10.1007/s002110050244
- Masaaki Sugihara, Near optimality of the sinc approximation, Math. Comp. 72 (2003), no. 242, 767–786. MR 1954967, DOI 10.1090/S0025-5718-02-01451-5
- Masaaki Sugihara and Takayasu Matsuo, Recent developments of the Sinc numerical methods, Proceedings of the 10th International Congress on Computational and Applied Mathematics (ICCAM-2002), 2004, pp. 673–689. MR 2056907, DOI 10.1016/j.cam.2003.09.016
- Hidetosi Takahasi and Masatake Mori, Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci. 9 (1973/74), 721–741. MR 0347061, DOI 10.2977/prims/1195192451
- Ken’ichiro Tanaka, Masaaki Sugihara, and Kazuo Murota, Numerical indefinite integration by double exponential sinc method, Math. Comp. 74 (2005), no. 250, 655–679. MR 2114642, DOI 10.1090/S0025-5718-04-01724-7
- K. Tanaka, M. Sugihara, and K. Murota, Function classes for successful DE-Sinc approximations, Technical report METR 2007-08, University of Tokyo, February 2007.
Additional Information
- Ken’ichiro Tanaka
- Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
- Masaaki Sugihara
- Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
- Email: m_sugihara@mist.i.u-tokyo.ac.jp
- Kazuo Murota
- Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
- Email: murota@mist.i.u-tokyo.ac.jp
- Received by editor(s): February 8, 2007
- Received by editor(s) in revised form: June 19, 2008
- Published electronically: October 28, 2008
- Additional Notes: This work was supported by the 21st Century COE Program on Information Science and Technology Strategic Core and a Grant-in-Aid of the Ministry of Education, Culture, Sports, Science and Technology of Japan. The first author was supported by the Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. Technical details omitted in this paper can be found in [14]
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 78 (2009), 1553-1571
- MSC (2000): Primary 65D05; Secondary 41A25, 41A30
- DOI: https://doi.org/10.1090/S0025-5718-08-02196-0
- MathSciNet review: 2501063