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Mathematics of Computation

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A posteriori improvements for interpolating periodic splines

Author: Thomas R. Lucas
Journal: Math. Comp. 40 (1983), 243-251
MSC: Primary 41A15; Secondary 65D07
MathSciNet review: 679443
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Abstract: A method of a posteriori improvements of interpolating periodic splines of order 2r 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 2r 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.

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Keywords: Splines, periodic, asymptotic expansion, interpolation, a posteriori corrections, Bernoulli polynomials, Bernoulli numbers
Article copyright: © Copyright 1983 American Mathematical Society