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Asymptotics of Chebyshev Rational Functions with Respect to Subsets of the Real Line

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Abstract

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an \(L^\infty \) norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of \(\overline{{{\mathbb {R}}}}\). We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szegő–Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau–Widom and DCT conditions.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, TU Wien, 1040, Wien, Austria

    Benjamin Eichinger

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48104, USA

    Milivoje Lukić & Giorgio Young

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  1. Benjamin Eichinger
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  2. Milivoje Lukić
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  3. Giorgio Young
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Correspondence to Giorgio Young.

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Communicated by Arno Kuijlaars.

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Benjamin Eichinger was supported by Austrian Science Fund FWF, projects: J 4138-N32 and P 33885.

Milivoje Lukić was supported in part by NSF grant. DMS–1700179.

Giorgio Young was supported in part by NSF grant. DMS–1745670

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Eichinger, B., Lukić, M. & Young, G. Asymptotics of Chebyshev Rational Functions with Respect to Subsets of the Real Line. Constr Approx (2024). https://doi.org/10.1007/s00365-023-09670-0

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  • DOI: https://doi.org/10.1007/s00365-023-09670-0

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