**5. Discussion**

Usually, the nodal systems used for interpolation problems are strongly connected with measures on the bounded interval and on the unit circle and their corresponding orthogonal or paraorthogonal polynomials. We must point out that these choices are very suitable to construct the whole theory but in some numerical applicatons it is possible that the nodal systems do not satisfy this requisite. So, the starting point of the paper is a distribution for the nodes that can be obtained through a perturbation of the uniform distribution and, in particular of the roots of the unity, and which is more general than that related to measures and orthogonality.

The results of this article contribute to elaborate a theory over these type of nodal systems, as well as to the Lagrange interpolation theory based on these interpolatory arrays. Moreover, a theory about the rate of convergence for some types of smooth functions is given. Finally, we translate the results to perturbed Chevyshev nodal systems and to Lagrange interpolation on the bounded interval.

We think that this research could be of interest for some mechanical models that generate these types of nodal systems. As an example we consider the next problem.

Let us suppose that we are studying a equatorial characteristic *<sup>F</sup>*(*ei<sup>θ</sup>* ) of a planet which depends on the angle *θ* and we have a theory which establishes that *<sup>F</sup>*(*ei<sup>θ</sup>* ) is an analytic function. We observe the phenomenon using an observatory in the boundary of a spatial station in an elliptic orbit of period *T* which rotates over itself with period *T*1 (with *T* = *n T*1 and *n* large enough). Moreover, we take our observations when the center of the planet, the observatory and the center of the station are aligned. We can translate the problem thinking that the planet is our Sun, the spatial station is our Earth and the observatory is our city. So the time between observations is the equivalent of a solar day. It is well known that the duration of a solar day is not constant (see https://en.wikipedia. org/wiki/Equation\_of\_time for a brief introduction about the so called **Equation of time**), in our case have a little oscillation on *T*1 and our observations are taken on a nodal system which satisfies (1). Notice that in this case we do not have a equispaced distribution nor the support of the theory of Orthogonal Polynomials. Therefore, before this paper we did not know how to use our data to reconstruct *<sup>F</sup>*(*ei<sup>θ</sup>* )and after this paper we can be confident about the use of Lagrange interpolation.

Some future research directions could be the study of other types of interpolation on the unit circle and on the bounded interval by using these general interpolatory arrays; as well as to study the correspondig Gibbs–Wilbraham phenomena.

### **6. Materials and Methods**

The experiments given in the section Numerical examples were obtained by using personal codes elaborated with Mathematica-R 12 (Wolfram Research Europe Ltd, Long Hanborough Oxfordshire, United Kingdom). These programs to obtain the nodal points and to compute the interpolation polynomials as well as the plots of the test functions and their interpolators are available at the public

repository https://www.dropbox.com/sh/0cx9chq3jfzov2w/AAA\_SvL2i7HlC7ChMGpuG-Ata?dl=0

There one can find the program related to Example 2. To obtain the other examples some minor changes must be done.

**Author Contributions:** Conceptualization, E.B., A.C. (Alicia Cachafeiro), A.C. (Alberto Castejón) and J.M.G.-A.; Investigation, E.B., A.C. (Alicia Cachafeiro), A.C. (Alberto Castejón) and J.M.G.-A.; Software, E.B., A.C. (Alicia Cachafeiro), A.C. (Alberto Castejón) and J.M.G.-A.; Writing—original draft, E.B., A.C. (Alicia Cachafeiro), A.C. (Alberto Castejón) and J.M.G.-A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
