A periodic map for linear barycentric rational trigonometric interpolation

Consider the set of equidistant nodes in [0, 2 pi), theta(k) :=k . 2 pi/n, k = 0, ..., n-1. For an arbitrary 2 pi-periodic function f(theta), the barycentric formula for the corresponding trigonometric interpolant between the theta(k)'s is T[f](theta) = Sigma(n-1)(k=0)(-1)(k) cst(theta-theta(k)/2) f(theta(k))/Sigma(n-1)(k=0)(-1)(k) cst(theta-theta(k)/2), 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 right-hand side of the above equation for arbitrary nodes introduced in [9] converges exponentially toward f when the nodes are the images of the theta(k)'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. (C) 2019 Elsevier Inc. All rights reserved.