A posteriori improvements for interpolating periodic splines

Author:
Thomas R. Lucas

Journal:
Math. Comp. **40** (1983), 243-251

MSC:
Primary 41A15; Secondary 65D07

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679443-5

MathSciNet review:
679443

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Abstract | References | Similar Articles | Additional Information

Abstract: A method of a posteriori improvements of interpolating periodic splines of order 2*r* and their derivatives over a uniform mesh is developed using polynomial-type correction terms. These improvements enhance the order of convergence by several powers of the step size *h* and are convenient and inexpensive to implement. The polynomials are specified in closed form using the Bernoulli numbers. That the first of these is related to the Bernoulli polynomial of degree 2*r* is due to Swartz [10], but no general development beyond the first has previously been made. These polynomials are multiplied by high order derivatives of the function evaluated at the mesh points. Some recent results by Lucas [8] are used to accurately estimate these values. Some numerical results are given which correspond closely with the predictions of the theory.

**[1]**M. Abramowitz & I. A. Stegun,*Handbook of Mathematical Functions*, Nat. Bur. Standards Appl. Math. Ser. No. 55, Washington, D. C., 1964.**[2]**J. H. Ahlberg, E. N. Nilson, and J. L. Walsh,*The theory of splines and their applications*, Academic Press, New York-London, 1967. MR**0239327****[3]**A. R. Curtis & M. J. D. Powell,*Using Cubic Splines to Approximate Functions of One Variable to Prescribed Accuracy*, Atomic Energy Research Establishment, R5602, Harwell, England, 1967.**[4]**D. J. Fyfe,*Linear dependence relations connecting equal interval 𝑁𝑡ℎ degree splines and their derivatives*, J. Inst. Math. Appl.**7**(1971), 398–406. MR**0284748****[5]**Michael Golomb,*Approximation by periodic spline interpolants on uniform meshes*, J. Approximation Theory**1**(1968), 26–65. MR**0233121****[6]**N. F. Innes,*High Order End Conditions and Convergence Results for Uniformly Spaced Quintic Splines*, Research Report 1, Dept. of Mathematics, University of North Carolina at Charlotte, 1979.**[7]**Thomas R. Lucas,*Error bounds for interpolating cubic splines under various end conditions*, SIAM J. Numer. Anal.**11**(1974), 569–584. MR**0351039**, https://doi.org/10.1137/0711049**[8]**Thomas R. Lucas,*Asymptotic expansions for interpolating periodic splines*, SIAM J. Numer. Anal.**19**(1982), no. 5, 1051–1066. MR**672577**, https://doi.org/10.1137/0719076**[9]**Murray Rosenblatt,*Asymptotics and representation of cubic splines*, J. Approximation Theory**17**(1976), no. 4, 332–343. MR**0417632****[10]**Blair Swartz,*𝑂(ℎ^{2𝑛+2-𝑙}) bounds on some spline interpolation errors*, Bull. Amer. Math. Soc.**74**(1968), 1072–1078. MR**0236574**, https://doi.org/10.1090/S0002-9904-1968-12052-X**[11]**Blair Swartz,*𝑂(ℎ^{2𝑛+2-𝑙}) bounds on some spline interpolation errors*, Bull. Amer. Math. Soc.**74**(1968), 1072–1078. MR**0236574**, https://doi.org/10.1090/S0002-9904-1968-12052-X

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679443-5

Keywords:
Splines,
periodic,
asymptotic expansion,
interpolation,
a posteriori corrections,
Bernoulli polynomials,
Bernoulli numbers

Article copyright:
© Copyright 1983
American Mathematical Society