Sharp Bounds for Lebesgue Constants of Barycentric Rational Interpolation at Equidistant Points

Abstract

A rough analysis of the growth of the Lebesgue constant in the case of barycentric rational interpolation at equidistant interpolation points was made in [Bos et al. 11] and [Bos et al. 12], leading to the conclusion that it only grows logarithmically. Herewe give a fine analysis, obtaining the precise growth formula 2/pi (In(n+1) + In 2 + gamma) + o(1) for the Lebesgue constant under consideration, with gamma being the Euler constant. The similarity between barycentric rational interpolation at equidistant points and polynomial interpolation at Chebyshev nodes (or the like) is remarkable. After revisiting the polynomial interpolation case in Section 1 and introducing the barycentric rational interpolation case in Section 2, tight lower and upper bound estimates are given in Section 3. These fine results could only be formulated after performing very high-order numerical experiments in exact arithmetic. In Section 4, we indicate that the result can be extended to the rational interpolants introduced by Floater and Hormann in [ Floater and Hormann 07]. Finally, the proof of the new tight bounds is detailed in Section 5.