Nikishin Systems Are Perfect

K. Mahler introduced the concept of perfect systems in the general theory he developed for the simultaneous Hermite–Padé 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–Padé 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.