Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration
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
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.
- 9.Muhammad, M., Mori, M.: Double exponential formulas for numerical indefinite integration. J. Comput. Appl. Math. 161, 431–448 (2003)Google Scholar
- 21.Sugihara, M.: Double exponential transformation in the sinc-collocation method for two-point boundary value problems. J. Comput. Appl. Math. 149, 239–250 (2002)Google Scholar
- 25.Tanaka, K., Sugihara, M., Murota, K.: Function classes for successful DE-Sinc approximations. Math. Comput. 78, 1553–1571 (2009) Google Scholar
- 27.Yamanaka, N., Okayama, T., Oishi, S., Ogita, T.: A fast verified automatic integration algorithm using double exponential formula. In: Nonlinear Theory and Its Applications, vol. 1, pp. 119–132, IEICE (2010)Google Scholar