Una aproximación a los polinomios ortogonales en el círculo unitario.

Abstract

In this work is presented an approach to the orthogonal polynomials on the unit circle via the general properties of these. At first, the necessary contents is studied to understand the general theory of the orthogonal polynomials, as algebraic concepts, functional analysis and measure theory; the construction of Chebyshev's polynomials is shown, to present some orthogonal polynomial properties, like the recurrence relation that they satisfy (called recurrence relation of the minimum and maximum and the Christoffel- Darboux identity) with the maximum and minimum problem; all this to appreciate that the seen properties, have their analogous in terms of the orthogonal polynomials on the unit circle. Finally, a theorem is shown in which it can be seen the relation that the orthogonal polynomials on R have with the orthogonal polynomials on the unit circle. This last fact is quite interesting, since in order to construct the sequences that relate this two systems, the Chebyshev polynomials are used.