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Distribution of zeros of the Hermite-Padé polynomials for a system of three functions, and the Nuttall condenser

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Abstract

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface \(\Re _3\) that has a canonical decomposition. We consider a system of three functions \(\mathfrak{f}_1 ,\mathfrak{f}_2 ,\mathfrak{f}_3\) that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ℂ̄ \ B of the open (possibly, disconnected) set B ⊂ ℂ̄ introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.

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Correspondence to R. K. Kovacheva.

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Original Russian Text © R.K. Kovacheva, S.P. Suetin, 2014, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 284, pp. 176–199.

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Kovacheva, R.K., Suetin, S.P. Distribution of zeros of the Hermite-Padé polynomials for a system of three functions, and the Nuttall condenser. Proc. Steklov Inst. Math. 284, 168–191 (2014). https://doi.org/10.1134/S008154381401012X

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  • DOI: https://doi.org/10.1134/S008154381401012X

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