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Error Inequalities in Polynomial Interpolation and Their Applications pp 1–61Cite as

Lidstone Interpolation

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Part of the book series: Mathematics and Its Applications ((MAIA,volume 262))

Abstract

In the year 1929 Lidstone [15] introduced a generalization of Taylor’s series, it approximates a given function in the neighborhood of two points instead of one. From the practical point of view such a development is very useful; and in terms of completely continuous functions it has been characterized in the work of Boas [9], Poritsky [19], Schoenberg [20], Whittaker [28, 29], Widder [30, 31], and others. In the field of approximation theory [12,27] the Lidstone interpolating polynomial P (1.1.1)(t) of degree (2m — 1) satisfies the Lidstone conditions

$$ P_{(1.1.1)}^{(2i)} (0) = \alpha _i , P_{(1.1.1)}^{(2i)} (1) = \beta _i ,0 \leqslant i \leqslant m - 1. $$
(1.1.1)

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References

  1. R.P.Agarwal and G.Akrivis Boundary value problems occurring in plate deflection theory, J. Comp. Appl. Math. 8(1982), 145–154.

    Article  MathSciNet  Google Scholar 

  2. R.P.Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore • Philadelphia, 1986.

    MATH  Google Scholar 

  3. R.P.Agarwal and P.J.Y.Wong Lidstone polynomials and boundary value problems, Computers Math. Applic. 17(1989), 1397–1421.

    Article  MathSciNet  MATH  Google Scholar 

  4. R.P.Agarwal Sharp inequalities in polynomial interpolation, in General Inequalities 6, Ed. W.Walter, International Series of Numerical Mathematics, Birkhäuser Verlag 103(1992), 73–92.

    Google Scholar 

  5. R.P.Agarwal and P.J.Y.Wong Quasilinearization and approximate quasilinearization for Lidstone boundary value problems, Intern. J. Computer Math. 42(1992), 99–116.

    Article  MATH  Google Scholar 

  6. P.Baldwin Asymptotic estimates of the eigenvalues of a sixth - order boundary - value problem obtained by using global phase - integral method, Phil. Trans. R. Soc. London A322(1987), 281–305.

    MathSciNet  Google Scholar 

  7. P.Baldwin Localized instability in a Bénard layer, Applicable Analysis 24(1987), 117–156.

    Article  MathSciNet  MATH  Google Scholar 

  8. R.P.Boas A note on functions of exponential type, Bull. Amer. Math. Soc. 47(1941), 750–754.

    Article  MathSciNet  Google Scholar 

  9. R.P.Boas Representation of functions by Lidstone series, Duke Math. J. 10(1943), 239–245.

    Article  MathSciNet  MATH  Google Scholar 

  10. A.Boutayeb and E.H.Twizell Finite - difference methods for twelfth - order boundary value problems, J. Comp. Appl. Math. 35(1991), 133–138.

    Article  MathSciNet  MATH  Google Scholar 

  11. M.M.Chawla and C.P.Katti Finite difference methods for two - point boundary value problems involving higher order differential equations, BIT 19(1979), 27–33.

    Article  MathSciNet  MATH  Google Scholar 

  12. P.J.Davis Interpolation and Approximation, Blaisdell Publishing Co., Boston, 1961.

    Google Scholar 

  13. T.Fort Finite Differences and Difference Equations in the Real Domain, Oxford University Press, London, 1948.

    MATH  Google Scholar 

  14. C.Jordan Calculus of Finite Differences, Chelsea Pub. Co., New York, 1960.

    Google Scholar 

  15. G.J.Lidstone Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types, Proc. Edinburgh Math. Soc. (2), 2(1929), 16–19.

    Article  Google Scholar 

  16. Y.L.Luke The Special Functions and their Approximations, Academic Press, New York, 1969.

    MATH  Google Scholar 

  17. L.M.Milne - Thomson The Calculus of Finite Differences, MacMillan, London, 1960.

    Google Scholar 

  18. Muhammad Aslam Noor and S.I.Tirmizi Numerical methods for unilateral problems, J. Comp. Appl. Math. 16(1986), 387–395.

    Article  MathSciNet  MATH  Google Scholar 

  19. H.Poritsky On certain polynomial and other approximations to analytic functions, Trans. Amer. Math. Soc. 34(1932), 274–331.

    Article  MathSciNet  Google Scholar 

  20. I.J.Schoenberg On certain two - point expansions of integral functions of exponential type, Bull. Amer. Math. Soc. 42(1936), 284–288.

    Article  MathSciNet  Google Scholar 

  21. J.Toomre J.R.Jahn, J.Latour and E.A.Spiegel Stellar convection theory II: single - mode study of the second convection zone in an A - type star, Astrophys. J. 207(1976), 545–563.

    Google Scholar 

  22. E.H.Twizell and S.I.A.Tirmizi A sixth order multiderivative method for two beam problems, Int. J. Numer. Methods Engg. 23(1986), 2089–2102.

    Article  MathSciNet  MATH  Google Scholar 

  23. E.H.Twizell Numerical methods for sixth - order boundary value problems, International Series of Numerical Mathematics 86(1988), 495–506.

    MathSciNet  Google Scholar 

  24. E.H.Twizell and S.I.A.Tirmizi Multiderivative methods for nonlinear beam problems, Comm. Appl. Numer. Methods 4(1988), 43–50.

    Article  MathSciNet  MATH  Google Scholar 

  25. E.H.Twizell and A.Boutayeb Numerical methods for the solution of special and general sixth - order boundary value problems, with applications to Bénard layer eigenvalue problems, Proc. R. Soc. London A431(1990), 433–450.

    MathSciNet  Google Scholar 

  26. R.A.Usmani Solving boundary value problems in plate deflection theory, Simulation, December(1981), 195–206.

    MathSciNet  Google Scholar 

  27. A.K.Varma and G.Howell Best error bounds for derivatives in two point Birkhoff interpolation problem, J. Approximation Theory 38(1983), 258–268.

    Article  MathSciNet  MATH  Google Scholar 

  28. J.M.Whittaker On Lidstone’s series and two-point expansions of analytic functions, Proc. London Math. Soc. (2), 36(1933–34), 451–459.

    Article  MathSciNet  Google Scholar 

  29. J.M.Whittaker, Interpolatory Function Theory, Cambridge, 1935.

    Google Scholar 

  30. D.V.Widder Functions whose even derivatives have a prescribed sign, Proc. National Acad. Sciences 26(1940), 657–659.

    Article  MathSciNet  Google Scholar 

  31. D.V.Widder Completely convex functions and Lidstone series, Trans. Amer. Math. Soc. 51(1942), 387–398.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Kent Ridge, Singapore

    Ravi P. Agarwal

  2. Division of Mathematics, Nanyang Technological University, Singapore

    Patricia J. Y. Wong

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  1. Ravi P. Agarwal
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  2. Patricia J. Y. Wong
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© 1993 Springer Science+Business Media Dordrecht

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Agarwal, R.P., Wong, P.J.Y. (1993). Lidstone Interpolation. In: Error Inequalities in Polynomial Interpolation and Their Applications. Mathematics and Its Applications, vol 262. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2026-5_1

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  • DOI: https://doi.org/10.1007/978-94-011-2026-5_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4896-5

  • Online ISBN: 978-94-011-2026-5

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