A Multidimensional Generalization of the Padé table. VI

“…These lectures were published years later in [32]. Important contributions in this respect are also due to his students J. Coates and H. Jager, see [12] and [29]. K. Mahler's approach to the simultaneous approximation of finite systems of analytic functions may be reformulated in the following terms.…”

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confidence: 99%

Nikishin Systems Are Perfect

Prieto

Lagomasino

2011

Constr Approx

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Abstract. 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.

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“…The concept of perfect systems was introduced and developed by Mahler in lectures delivered at the University of Groningen in 1934-35 which were published much later in [22] (see also [6,20]). This notion plays a central role in the general theory of simultaneous approximation of systems of analytic functions.…”

Section: Perfect Systemsmentioning

confidence: 99%

Nikishin systems are perfect. The case of unbounded and touching supports

Prieto

Lagomasino

2011

Journal of Approximation Theory

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K. Mahler introduced the concept of perfect systems in the theory of simultaneous Hermite-Padé approximation of analytic functions. Recently, we proved that Nikishin systems, generated by measures with bounded support and non-intersecting consecutive supports contained on the real line, are perfect. Here, we prove that they are also perfect when the supports of the generating measures are unbounded or touch at one point. As an application, we give a version of the Stieltjes theorem in the context of simultaneous Hermite-Padé approximation.

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A Multidimensional Generalization of the Padé table. VI

Cited by 13 publications

References 1 publication

Auxiliary functions in transcendental number theory

Auxiliary functions in transcendental number theory

Nikishin Systems Are Perfect

Nikishin systems are perfect. The case of unbounded and touching supports

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