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An extension of the Floater-Hormann family of barycentric rational interpolants. (English) Zbl 1275.65009

In 2007, M. S. Floater and K. Hormann [Numer. Math. 107, No. 2, 315–331 (2007; Zbl 1221.41002)] introduced a family of barycentric rational interpolants by considering blends of polynomial interpolants of fixed degree \(d\). In some cases, these rational functions achieve approximation of much higher quality than the classical polynomial interpolants, which, e.g., are ill-conditioned and lead to Runge’s phenomenon if the interpolation nodes are equispaced. However, for such nodes, it has been shown [L. Bos et al., Numer. Math. 121, No. 3, 461–471 (2012; Zbl 1252.41003)] that the Lebesgue constant of Floater-Hormann interpolation grows exponentially with increasing \(d\). One immediately sees the reason by plotting the Lebesgue function, which has at most \(d\) high oscillations at the ends of the interval and is much smaller in the remaining part.
In this paper, a modification of the Floater-Hormann interpolant is presented and investigated, for which the Lebesgue constant grows only logarithmically with increasing \(d\) for equispaced nodes. Similar to adding “fictitious points” in finite element approximation [B. Fornberg, A practical guide to pseudospectral methods. Cambridge Monographs on Applied and Computational Mathematics. Cambridge: Cambridge Univ. Press (1996; Zbl 0844.65084)], the idea is to add \(d\) new data values on each side of the interval by evaluating a smooth extension of the original data outside of the interval. The Lesbesgue function of the Floater-Hormann interpolant for this extended data will then have its high oscillations outside of the interval of interest.
The efficiency of its applications such as the approximation of derivatives, integrals and antiderivatives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants. In particular, “the rates of convergence stay roughly the same as with the original family of interpolants, but the constants involved in the error bounds are smaller in many cases”.

MSC:

65D05 Numerical interpolation
65D32 Numerical quadrature and cubature formulas
41A05 Interpolation in approximation theory
41A20 Approximation by rational functions
41A25 Rate of convergence, degree of approximation
65D25 Numerical differentiation
26A36 Antidifferentiation

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