On the interval Legendre polynomials. (Q1874202)

This paper deals with the extension of the classical Legendre polynomials to the interval theory by considering the family of interval polynomials \(\mathbb L_{n,k}(x) \) satisfying, for each natural number \(k\), the recursive formula \(\mathbb L_{0,k}(x)=[1-\frac 1k,1+\frac 1k]\), \(\mathbb L_{1,k}(x)=[1-\frac 1k,1+\frac 1k]x\), \(\mathbb L_{n+1,k}(x) =\frac {2n+1}{n+1}\mathbb L_{n,k}(x) -\frac {n}{n+1}\mathbb L_{n-1,k}(x)\) , for \(n\in \mathbb N\). These functions \(\mathbb L_{n,k}(x) \), for each \(k\in \mathbb N\) and \(n\in \mathbb N,\) are called interval Legendre polynomials and many of their properties are studied in this paper. A minimum square approximation for continuous and discrete data based on these interval polynomials is introduced. Some interesting examples showing the behaviour of the defined approximations, and also an application to the computation of an estimate in the Hertzsprung-Russel diagram in Astrophysics, are given.