Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity

Abstract

The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.

Mechanical Models for Hermite Interpolation on the Unit Circle

“…Secondly, we recall a separation property, given in [9], between both nodal systems W n (z) and W n (z).…”

“…This closeness can be established in terms of suitable separation properties. The advantages of using these types of nodal systems is that they can be obtained through a random uniform distribution (see the examples given in [9]).…”

“…In [9], we changed the focus of the interpolation problem, and our starting point was to use nodal systems that were not related to measures and that were only characterized to fulfil certain separation properties between the nodes. Thus, we studied the Lagrange interpolation problem on the unit circle by using nodal systems that are not connected with any measure, and they are only characterized by satisfying a separation property of the type:…”

“…In the present paper, we studied nodal systems like those used in [9] that meet the separation properties mentioned above, and we present mechanical models that fit these distributions. The aim of this work was to study the Hermite interpolation process on the unit circle by using these nodal systems characterized by a separation property between the nodes.…”

“…The target was to develop the corresponding interpolation theory that allows us to make practical use of these nodal systems linked to certain mechanical models. Thus, in the first part of the paper (see Sections 2 and 3), we recall the properties given in [9] satisfied by the nodal system, and we obtain some new ones that play an important role in the Hermite process. Our main result was to prove a new version of Fejér's theorem for continuous functions on the unit circle by using these nodal systems.…”