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To compute the optimal interpolation formula


Author: P. W. Gaffney
Journal: Math. Comp. 32 (1978), 763-777
MSC: Primary 65D05
DOI: https://doi.org/10.1090/S0025-5718-1978-0474698-2
MathSciNet review: 0474698
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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to explain how to compute the function $ \Omega $ which interpolates values of a function of one variable $ f(x)$ at n distinct points $ {x_1} < {x_2} < \cdots < {x_{n - 1}} < {x_n}$ and which, whenever $ \left\Vert{f^{(k)}}\right\Vert _\infty $ is bounded and the value of the bound is unknown, provides the smallest possible value of $ B(x)$ in the error bound

$\displaystyle \vert f(x) - \Omega (x)\vert \leqslant B(x)\left\Vert{f^{(k)}}\right\Vert _{\infty .}$


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  • [M] G. COX (1972), "The numerical evaluation of B-splines," J. Inst. Math. Appl., v. 10, pp. 134-149. MR 0334456 (48:12775)
  • [M] G. COX (1975), "An algorithm for spline interpolation," J. Inst. Math. Appl., v. 15, pp. 95-108. MR 0359265 (50:11720)
  • [C] de BOOR (1972), "On calculating with B-splines," J. Approximation Theory, v. 6, pp. 50-62. MR 0338617 (49:3381)
  • [C] de BOOR & A. PINKUS (1977), "Backward error analysis for totally positive linear systems," Numer. Math., v. 27, pp. 485-490. MR 0436558 (55:9501)
  • [P] W. GAFFNEY (1976a), "The calculation of indefinite integrals of B-splines," J. Inst. Math. Appl., v. 17, pp. 37-41. MR 0415985 (54:4062)
  • [P] W. GAFFNEY (1976b), Optimal Interpolation, D. Phil. Thesis, Oxford University.
  • [P] W. GAFFNEY (1977a), The Range of Possible Values of $ f(x)$, A.E.R.E. Report C.S.S. 51.
  • [P] W. GAFFNEY (1977b), Fortran Subroutines for Computing the Optimal Interpolation Formula, A.E.R.E. Report No. R.8781.
  • [P] W. GAFFNEY & M. J. D. POWELL (1975), "Optimal interpolation," Numerical Analysis (Proc. Conf., Univ. of Dundee, 1975), G. A. Watson (Editor), Lecture Notes in Math., vol. 506, Springer-Verlag, Berlin and New York. MR 0458001 (56:16204)
  • [C] A. MICCHELLI, T. J. RIVLIN & S. WINOGRAD (1976), "The optimal recovery of smooth functions," Numer. Math., v. 26, pp. 191-200. MR 0458005 (56:16208)
  • [J] M. ORTEGA & W. C. RHEINBOLDT (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York. MR 0273810 (42:8686)
  • [I] J. SCHOENBERG (1967), "On spline functions," Inequalities (Proc. Sympos., Wright-Patterson Air Force Base, Ohio, 1965), Oved Shisha (Editor), Academic Press, New York, pp. 255-291. MR 0223801 (36:6848)
  • [I] J. SCHOENBERG & A. WHITNEY (1953), "On Pólya frequency functions. III, The positivity of translation determinants with an application to the interpolation problem by spline curves," Trans. Amer. Math. Soc., v. 74, pp. 246-259. MR 0053177 (14:732g)
  • [J] F. STEFFENSON (1927), Interpolation, Chelsea, New York.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0474698-2
Article copyright: © Copyright 1978 American Mathematical Society

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