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Mathematics of Computation

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Near optimality of the sinc approximation

Author: Masaaki Sugihara
Journal: Math. Comp. 72 (2003), 767-786
MSC (2000): Primary 41A30, 41A25, 65D15
Published electronically: June 4, 2002
MathSciNet review: 1954967
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Abstract | References | Similar Articles | Additional Information

Abstract: Near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of the infinity.

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  • 1. M. Abramowitz and I. Stegun, Handbook of mathematical functions, NBS Applied Math. Series 53, 1964. MR 29:4914
  • 2. J.-E. Andersson, Optimal quadrature of $H^p$ functions, Math. Z. 172 (1980), 55-62. MR 83m:41023
  • 3. S. N. Bernstein, Sur la meilleure approximation de $\vert x\vert^{p}$par des polynômes de degrées très élevés, Bull. Acad. Sci. USSR, Cl. Sci. Math. Nat. 2 (1938), 181-190.
  • 4. H. G. Burchard and K. Höllig, $n$-width and entropy of $H^p$-classes in $L_q(-1,1)$, SIAM J. Math. Anal. 16 (1985), 405-421. MR 86i:41021
  • 5. R. A. DeVore and K. Scherer, Variable knot, variable degree spline approximation to $x^{\mu}$, in: Quantitative Approximation (R. A. DeVore and K. Scherer, eds.), Academic Press, London, 1980, 121-132. MR 81m:41008
  • 6. T. H. Ganelius, Rational approximation in the complex plane and on the line, Ann. Acad. Sci. Fenn. Ser. A.I. 2 (1976), 129-145. MR 58:17106
  • 7. S.-Å. Gustafson and F. Stenger, Convergence acceleration applied to sinc approximation with application to approximation of $\vert x\vert^{\alpha}$, in: Computation and Control II(K. L. Bowers and J. Lund, eds.), Birkhäuser, Basel, 1991, 161-171. MR 92g:93006
  • 8. F. Keinert, Uniform approximation to $\vert x\vert^\beta$ by sinc functions, J. Approx. Th. 66 (1991), 44-52. MR 92d:41047
  • 9. M. A. Kowalski, K. A. Sikorski, and F. Stenger, Selected topics in approximation and computation, Oxford Univ. Press, Oxford, 1995. MR 97k:41001
  • 10. J. Lund and K. L. Bowers, Sinc methods for quadrature and differential equations, SIAM, Philadelphia, PA, 1992. MR 93i:65004
  • 11. M. Mori and M. Sugihara, The double exponential transformation in numerical analysis, in: Numerical Analysis in the 20th Century Vol. V: Quadrature and Orthogonal Polynomials, J. Comput. Appl. Math. 127 (2001), 287-296. MR 2001k:65041
  • 12. D. J. Newman, Quadrature formulae for $H^p$ functions, Math. Z. 166 (1979), 111-115. MR 80g:41022
  • 13. J. R. Rice, On the degree of convergence of nonlinear spline approximation, in: Approximations with Special Emphasis on Spline Functions (I. J. Schoenberg, ed.), Academic Press, New York, 1969, 349-365. MR 42:2226
  • 14. F. Stenger, Optimal convergence of minimum norm approximations in $H_p$, Numer. Math. 29 (1978), 345-362. MR 58:3342
  • 15. F. Stenger, Numerical methods based on Whittaker cardinal or sinc functions, SIAM Rev. 23 (1981), 165-224. MR 83g:65027
  • 16. F. Stenger, Explicit nearly optimal linear rational approximation with preassigned poles, Math. Comput. 47 (1986), 225-252. MR 87g:41034
  • 17. F. Stenger, Numerical methods based on sinc and analytic functions, Springer-Verlag, New York, 1993. MR 94k:65003
  • 18. F. Stenger, Summary of sinc numerical methods, in: Numerical Analysis in the 20th Century Vol. I: Approximation Theory, J. Comput. Appl. Math. 121 (2000), 379-420. MR 2001d:65018
  • 19. M. Sugihara, Optimality of the double exponential formula -- functional analysis approach, Numer. Math. 75 (1997), 379-395. MR 97i:41041
  • 20. H. Takahasi and M. Mori, Double exponential formulas for numerical integration, Publ. RIMS Kyoto Univ. 9 (1974), 721-741. MR 49:11781
  • 21. K. Wilderotter, $n$-widths of $H^p$-spaces in $L_q(-1,1)$, J. Complexity 8 (1992), 324-335. MR 94e:46051

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Additional Information

Masaaki Sugihara
Affiliation: Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan

Keywords: Sinc approximation, near optimality, variable transformation, double exponential formula
Received by editor(s): July 10, 2000
Received by editor(s) in revised form: August 27, 2001
Published electronically: June 4, 2002
Additional Notes: The author was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Sports, Culture and Science, and by the Japan Society for Promotion of Science.
Article copyright: © Copyright 2002 American Mathematical Society

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