In: Journal of Computational and Applied Mathematics, 2014, vol. 259, Part A, p. 95–107
Wellconditioned, stable and infinitely smooth interpolation in arbitrary nodes is by no means a trivial task, even in the univariate setting considered here; already the most important case, equispaced points, is not obvious. Certain approaches have nevertheless experienced significant developments in the last decades. In this paper we review one of them, linear barycentric rational...

In: SIAM Journal on Scientific Computing, 2014, vol. 36, no. 1, p. A105–A123
We introduce two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate as the rational quadrature rule but is costly on long integration intervals. The second, based on a...

Thèse de doctorat : Université de Fribourg, 2012 ; no. 1762.
This thesis is a collection of properties and applications of linear barycentric rational interpolation, mainly with the weights presented by Floater and Hormann in 2007. We are motivated by the counterintuitive and provable impossibility of constructing from equispaced data an approximation scheme that converges very rapidly to the approximated function and is simultaneously computationally...

In: SIAM Journal on Numerical Analysis, 2012, vol. 50, no. 2, p. 643–656
Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a...

In: Bit Numerical Mathematics, 2012, vol. 52, no. 2, p. 407424
Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational...

In: Applied Numerical Mathematics, 2011, vol. 61, no. 9, p. 9891000
In polynomial and spline interpolation the kth derivative of the interpolant, as a function of the mesh size h, typically converges at the rate of O(hd+1−k) as h→0, where d is the degree of the polynomial or spline. In this paper we establish, in the important cases k=1,2, the same convergence rate for a recently...

In: Numerical Algorithms, 2010, vol. 56, no. 1, p. 143157
Sinc interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. We give a formula for the error committed when the function neither decreases rapidly nor is periodic, so that the sinc series must be truncated for practical purposes. To do so, we first complete a previous result for an odd number of points, before deriving a formula...

In: Numerische Mathematik, 2009, vol. 112, no. 3, p. 341361
In former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval. In the present work we demonstrate the relevance of the formula through several applications: correction of the interpolant through the insertion of derivatives to increase its order of convergence, improvement of the...

In: Applied and Computational Harmonic Analysis, 2007, vol. 23, no. 3, p. 307320
We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the...

In: Numerical Algorithms, 2007, vol. 45, no. 14, p. 369374
Sincinterpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. It, however, requires that the interpolated function decreases rapidly or is periodic. We give an error formula for the case where neither of these conditions is satisfied.
