An interpolation problem on the circle between Lagrange and Hermite problems

A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

“…Remark 3. In [15], when considering the roots of unimodular complex numbers as nodal systems, we obtained similar results to those given in Propositions 5 and 6 although the order of convergence given in [15] is better than those.…”

“…Proof. This result is a consequence of Jackson's theorem and the proof can be seen in Lemma 2 of [15].…”

“…). Now, by taking into account the expressions given in [15], we can state the following result: Proposition 1. (i) The Hermite-Fejér interpolation polynomial HF −p,q (z) satisfying (1) has the expression…”

“…Our first aim is to obtain the expressions of these interpolation polynomials when we use nodal systems more general than the equally spaced ones. Indeed, the case in which the nodal points are equally spaced was studied in [15]. The novelty of the present paper is that the considered ones are not related to para-orthogonal polynomials on the unit circle, being only characterized by satisfying some suitable separation properties.…”

“…In some recent papers, we considered another variation of the classical methods. Indeed, in [14,15], we studied a problem of interpolation between those of Lagrange and Hermite. There, we use all the nodes for the data values and half of them for the derivative values.…”