A periodic map for linear barycentric rational trigonometric interpolation

Berrut, JeanPaul
Department of Mathematics, University of Fribourg, Pérolles, Fribourg 1700, Switzerland

Elefante, Giacomo
Department of Mathematics, University of Fribourg, Pérolles, Fribourg 1700, Switzerland
Published in:
 Applied Mathematics and Computation.  2020, vol. 371, p. 124924
English
Consider the set of equidistant nodes in [0, 2π),θk:=k·2πn,k=0,⋯,n−1.For an arbitrary 2π–periodic function f(θ), the barycentric formula for the corresponding trigonometric interpolant between the θk’s isT[f](θ)=∑k=0n−1(−1)kcst(θ−θk2)f(θk)∑k=0n−1(−1)kcst(θ−θk2),where cst(·):=ctg(·) if the number of nodes n is even, and cst(·):=csc(·) if n is odd. Baltensperger [3] has shown that the corresponding barycentric rational trigonometric interpolant given by the righthand side of the above equation for arbitrary nodes introduced in [9] converges exponentially toward f when the nodes are the images of the θk’s under a periodic conformal map. In the present work, we introduce a simple periodic conformal map which accumulates nodes in the neighborhood of an arbitrarily located front, as well as its extension to several fronts. Despite its simplicity, this map allows for a very accurate approximation of smooth periodic functions with steep gradients.

Faculty
 Faculté des sciences et de médecine

Department
 Département de Mathématiques

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Mathematics

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https://folia.unifr.ch/unifr/documents/308513
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