×

Lidstone polynomials and boundary value problems. (English) Zbl 0682.65049

The interpolation polynomial of degree 2m-1 of a given function \(y=y(x)\) given by G. J. Lidstone [Proc. Edinb. Math. Soc. 2(2), 16-19 (1930; JFM 56.1053.03)] matches \(y,y^{(2)},y^{(4)},...,y^{(m-1)}\) at \(x=0\) and \(x=1\). The authors provide a number of results on Lidstone interpolation and on the related two-point boundary value problem \(y^{(2m)}=F(x,y,y^{(1)},...,y^{(2m-1)})\) with \(y,y^{(2)},y^{(4)},...,y^{(m-1)}\) given at \(x=0\) and \(x=1\). (Note that when \(F\equiv 0\) this problem is solved by a Lidstone polynomial.)
Reviewer: J.M.Sanz-Serna

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
41A05 Interpolation in approximation theory
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
34B05 Linear boundary value problems for ordinary differential equations

Citations:

JFM 56.1053.03
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lidstone, G. J., Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types, (Proc. Edinburgh math. Soc., 2 (1929)), 16-19 · JFM 56.1053.03
[2] Boas, R. P., A note on functions of exponential type, Bull. Am. math. Soc., 47, 750-754 (1941) · Zbl 0063.00481
[3] Boas, R. P., Representation of functions by Lidstone series, Duke math. J., 10, 239-245 (1943) · Zbl 0061.11512
[4] Poritsky, H., On certain polynomial and other approximations to analytic functions, Trans. Am. math. Soc., 34, 274-331 (1932) · JFM 58.0314.01
[5] Schoenberg, I. J., On certain two-point expansions of integral of exponential type, Bull. Am. math. Soc., 42, 284-288 (1936) · Zbl 0014.02501
[6] Whittaker, J. M., On Lidstone’s series and two-point expansions of analytic functions, (Proc. Lond. math. Soc., 36 (1933-1934)), 451-469 · Zbl 0008.16901
[7] Widder, D. V., Functions whose even derivatives have a prescribed sign, (Proc. nat. Acad. Sci., 26 (1940)), 657-659 · Zbl 0061.11508
[8] Widder, D. V., Completely convex functions and Lidstone series, Trans. Am. math. Soc., 51, 387-398 (1942) · Zbl 0027.39202
[9] Davis, P. J., Interpolation and Approximation (1961), Blaisdell: Blaisdell Boston · Zbl 0111.06003
[10] Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations (1986), World Scientific: World Scientific Singapore · Zbl 0598.65062
[11] Noor, M. A.; Tirmizi, S. I., Numerical methods for unilateral problems, J. comp. appl. Math., 16, 387-395 (1986) · Zbl 0623.73120
[12] Timoshenko, S.; Woinowsky-Kreiger, S., Theory of Plates and Shells (1959), McGraw-Hill: McGraw-Hill New York
[13] Usmani, R. A., Solving boundary value problems in plate deflection theory, Simulation, 195-206 (1981), December · Zbl 0485.73070
[14] Agarwal, R. P.; Akrivis, G., Boundary value problems occuring in plate deflection theory, J. comp. appl. Math., 8, 145-154 (1982) · Zbl 0503.73061
[15] Chawla, M. M.; Katti, C. P., Finite difference methods for two-point boundary value problems involving higher order differential equations, BIT, 19, 27-33 (1979) · Zbl 0401.65053
[16] Jordan, C., Calculus of Finite Differences (1960), Chelsea: Chelsea New York
[17] Luke, Y. L., The Special Functions and their Approximations (1969), Academic Press: Academic Press New York · Zbl 0193.01701
[18] Fort, T., Finite Differences and Difference Equations in the Real Domain (1948), Oxford University Press: Oxford University Press London · Zbl 0030.11902
[19] Milne-Thomson, L. M., The Calculus of Finite Differences (1960), Macmillan: Macmillan London · Zbl 0008.01801
[20] Varma, A. K.; Howell, G., Best error bounds for derivatives in two point Birkhoff interpolation problem, J. Approximation Theory, 38, 258-268 (1983) · Zbl 0563.41004
[21] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman Boston · Zbl 0658.35003
[22] Agarwal, R. P., Monotone convergence of iterative methods for (n,p) and (p,n) boundary value problems, J. comp. appl. Math., 21, 223-230 (1988) · Zbl 0633.65077
[23] Agarwal, R. P.; Usmani, R. A., Monotone convergence of iterative methods for right focal point boundary value problems, J. math. Analysis Applic., 130, 451-459 (1988) · Zbl 0658.65069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.