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

### **Elías Berriochoa 1,\*,†, Alicia Cachafeiro 1,\*,†, Alberto Castejón 1,† and José Manuel García-Amor 2,†**


Received: 28 February 2020; Accepted: 29 March 2020; Published: 2 April 2020

**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.

**Keywords:** lagrange interpolation; unit circle; nodal systems; separation properties; perturbed roots of the unity; convergence
