Resumen de Convergence and asymptotic of multi-level hermite-padé polynomials

Luis Giraldo González Ricardo

• The dissertation focuses on the study of multilevel Hermite-Padé approximants. In the first chapter we introduce the theoretical background needed for a better comprehension of the thesis, as well as some historical notes. In Chapter 2 it is studied the convergence3 of the multilevel Hermite-Padé approximants to certain class of meromorphic functions. Moreover, it is studied the logarithmic asymptotic of the multilevel Hermite-Padé polynomials and its associated multi-orthogonal polynomials, in order to give better estimates of the rates of convergence. In Chapter 3, it analyzed the convergence of a generalization of multilevel Hermite-Padé approximation scheme to a Nikishin system, as well as the ratio asymptotic of its associated multi-orthogonal polynomials. Finally, Chapter 4 is devoted to the study of the strong asymptotic of Cauchy biorthogonal polynomials, where its generating measures satisfies Szego's conditions. The main tool is the relationship between Cauchy biorthogonal polynomials and multilevel Hermite-Padé approximation problems