Nikishin systems are perfect

TL;DR

This paper proves that Nikishin systems are perfect, expanding the class of systems with this property and deriving several important approximation and asymptotic results in the context of Hermite-Pade approximation.

Contribution

It establishes that Nikishin systems are perfect, enabling new results in Hermite-Pade approximation, quadrature convergence, and asymptotic analysis of multiple orthogonal polynomials.

Findings

Nikishin systems are perfect.

Extension of Markov's theorem to Nikishin systems.

Asymptotic behavior of multiple orthogonal polynomials.

Abstract

K. Mahler introduced the concept of perfect systems in the general theory he developed for the simultaneous Hermite-Pade approximation of analytic functions. We prove that Nikishin systems are perfect providing, by far, the largest class of systems of functions for which this important property holds. As consequences, in the context of Nikishin systems, we obtain: an extension of Markov's theorem to simultaneous Hermite-Pade approximation, a general result on the convergence of simultaneous quadrature rules of Gauss-Jacobi type, the logarithmic asymptotics of general sequences of multiple orthogonal polynomials, and an extension of the Denisov-Rakhmanov theorem for the ratio asymptotics of mixed type multiple orthogonal polynomials.