Abel–Gontscharoff interpolation: continuous, discrete and time scale

In this paper we shall discuss the Abel–Gontscharoff interpolation problem in the continuous, discrete and time scale cases. Let Pn−1 denote the (n−1)th degree Abel–Gontscharoff interpolating polynomial of a given function x. The interpolating conditions in the continuous, discrete and time scale cases are, respectively, given by Pn−1(i)(ai+1)=x(i)(ai+1), ΔiPn−1(ai+1)=Δix(ai+1), and Pn−1Δi(ai+1)=xΔi(ai+1), 0⩽i⩽n−1 where ai, 1⩽i⩽n are the interpolating nodes. In each of the three cases we shall present the best possible error bounds under different settings of aiʹs. Furthermore, as an application of the error estimates obtained, criteria are developed for the right disfocality as well as disconjugacy for higher order equations in each of the three cases.