Un acercamiento a los polinomios ortogonales de Chebyshev

Abstract

Orthogonal polynomials appeared to provide solutions to multiple application and theoretical problems, thus leading to a number of applications in mathematics and physics.In 1858, Pafnuti Chebyshev provided Chebyshev's orthogonal polynomials, coming to be considered one of the parents of the general theory of orthogonal polynomials exposed in the early 19th century. Hence the importance of the study of these specific polynomials, which are of four classes that are represented by a respective recurrence relationship and initial conditions, these classes are related to each other and defined in terms of cosine and sine, hence the facility to work on them because there are multiple trigonometric equalities, obtaining that the first class is the most important since it is related to the other three classes and is defined in terms simply of cosine of theta, where the range of the variable theta can vary on any closed interval a, b by means of a linear transformation that maps it on the closed interval -1, 1 where equalities are provided to find the zeros, extremes, express the powers of x in terms of first class polynomials and vice versa, like evaluating sums, products, integrals, and derivatives of Chebyshev polynomials.