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Numerische Mathematik

, Volume 124, Issue 2, pp 361–394 | Cite as

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

  • Tomoaki Okayama
  • Takayasu Matsuo
  • Masaaki Sugihara
Article

Abstract

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.

Mathematics Subject Classification (1991)

41A05 41A55 65D05 65D30 

Notes

Acknowledgments

The authors are greatly indebted to Dr. Ken’ichiro Tanaka of Future University Hakodate for several helpful comments concerning Lemma 4.22. This work was supported by Global COE Program “The research and training center for new development in mathematics,” MEXT, Japan.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tomoaki Okayama
    • 1
    • 2
  • Takayasu Matsuo
    • 1
  • Masaaki Sugihara
    • 1
  1. 1.Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan
  2. 2.Graduate School of EconomicsHitotsubashi UniversityTokyoJapan

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