Exponential convergence of a linear rational interpolant between transformed Chebyshev points

In 1988 the second author presented experimentally well-conditioned linear rational functions for global interpolation. We give here arrays of nodes for which one of these interpolants converges exponentially for analytic functions Introduction Let f be a complex function defined on an interval I of the real axis and let x0, x1, . . . , xn be n + 1 distinct points of I, which we do not assume equidistant or ordered. Let fk := f(xk), k = 0(1)n. Then Pn[f ](x) := n ∑ k=0 fkLk(x), Lk(x) := n ∏ j=0,j 6=k x− xj xk − xj , (1) is the Lagrangian representation of the unique polynomial of degree at most n interpolating f between the points xk, k = 0(1)n. Introducing the notations [Sch] λk := 1 ∏ j 6=k (xk − xj) , k = 0(1)n, (2) and L(x) := (x− x0)(x − x1) · · · (x− xn), (3) we can rewrite (1) as Pn[f ](x) = L(x) n ∑ k=0 λk x− xk fk. (4) Pn[f ] can also be written in its barycentric form by making use of the relation