Abstract
Although polynomials have several attractive features, polynomial interpolation of a given function often has the drawback of producing approximations that may be wildly oscillatory. To overcome this difficulty we divide the interval of interest into small subintervals and in each subinterval consider polynomials of relatively low degree and finally ‘piece together’ these polynomials. This subject has steadily developed over the past fifty years, and at present there are thousands of research papers on piecewise — polynomial interpolation and their applications. The plan of this chapter is as follows: In Section5.2 we collect some results from analysis which will be used repeatedly throughout the remaining monograph. In Section5.3 we shall follow the general treatment of Birkhoff, Ciarlet, Schultz and Varga [8,10,11] to provide an explicit representation of piecewise - Hermite interpolates. This representation is then used to obtain error bounds for the derivatives of piecewise — Hermite interpolates in L ∞ and L 2 — norms. These bounds improve on those given by Schultz [21], and extend to cases not considered by him. Section5.4 contains an explicit representation of piecewise — Lidstone interpolates and the error bounds for its derivatives in L ∞, and L 2 — norms. Results of Sections5.3 and 5.4 are extended to two variables in Sections5.5 and 5.6, respectively.
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References
R.P. Agarwal and R.C. Gupta, Essentials of Ordinary Differential Equations, McGraw-Hill, Singapore, 1991.
R.P. Agarwal and P.J.Y. Wong, Explicit error bounds for the derivatives of piecewise - Hermite interpolation in L 2 - norm, to appear.
R.P. Agarwal and P.J.Y. Wong, Explicit error bounds for the derivatives of piecewise — Lidstone interpolation, to appear.
J.H. Ahlberg, E.N. Nilson and J.L. Walsh, Convergence properties of generalized splines, Proc. Nat. Acad. Sci., U.S.A. 54(1965), 344–350.
J.H. Ahlberg, E.N. Nilson and J.L. Walsh, The Theory of Splines and their Applications, Academic Press, New York, 1967.
E. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin, 1965.
R. Bellman, A note on an inequality of E. Schmidt, Bull. Amer. Math. Soc. 50(1944), 734–736.
G. Birkhoff, M.H. Schultz and R.S. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations, Numerische Mathematik 11(1968), 232–256.
J. Brink, Inequalities involving ‖f ‖ p and ‖f (n) ‖q for f with n zeros, Pacific J. Math. 42(1972), 289–311.
P.G. Ciarlet, M.H. Schultz and R.S. Varga, Numerical methods of high order accuracy for nonlinear boundary value problems, I. one dimensional problem, Numerische Mathematik 9(1967), 394–430.
P.G. Ciarlet, M.H. Schultz and R.S. Varga, Numerical methods of high order accuracy for nonlinear boundary value problems, II. nonlinear boundary conditions, Numerische Mathematik 11(1968), 331–345.
G. Cimmino, Su una questione di minimo, Boll. Un. Mat. Ital 9(1930), 1–6.
K.M. Das and A.S. Vatsala, Green’s function for n - n boundary value problem and an analogue of Hartman’s result, J. Math. Anal. Appl. 51(1975), 670–677.
P. Dörfler, New inequalities of Markov type, SIAM J. Math. Anal. 18(1987), 490–494.
P. Forster, Existenzaussagen und Fehlerabschätzungen bei gewissen nichtlinearen Randwertaufgaben mit gewöhnlichen Differentialgleichungen, Numerishe Mathematik 10(1967), 410–422.
P. Goetgheluck, On the Markov inequality in L P - spaces, J. Approximation Theory 62(1990), 197–205.
G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Cambridge, 1934.
E. Hille, G. Szegö and J.D. Tamarkin, On some generalization of a theorem of A. Markoff, Duke Math. J. 3(1937), 729–739.
D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.
E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomensysteme und das zugehörige Extremum, Math. Ann. 119(1944), 165–204.
M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1973.
A.K. Varma and K.L. Katsifarakis, Optimal error bounds for Hermite interpolation, J. Approximation Theory 51(1987), 350–359.
P.J.Y. Wong and R.P. Agarwal, Explicit error estimates for quintic and biquintic spline interpolation, Computers Math. Applic. 18(1989), 701–722.
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© 1993 Springer Science+Business Media Dordrecht
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Agarwal, R.P., Wong, P.J.Y. (1993). Piecewise — Polynomial 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_5
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DOI: https://doi.org/10.1007/978-94-011-2026-5_5
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